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BEGIN:VEVENT
SUMMARY:Michael Roop
DTSTART:20200427T120000Z
DTEND:20200427T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/1
DESCRIPTION:Title: Shock waves in Euler flows of gases\nby Michael Roop as part of Ge
ometry of differential equations seminar\n\n\nAbstract\nNon-stationary one
-dimensional Euler flows of gases are studied. The system of differential
equations describing such flows can be represented by means of 2-forms on
zero-jet space and we get some exact solutions by means of such a represen
tation. Solutions obtained are multivalued and we provide a method of find
ing caustics\, as well as wave front displacement. The method can be appli
ed to any model of thermodynamic state as well as to any thermodynamic pro
cess. We illustrate the method on adiabatic ideal gas flows.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20200504T120000Z
DTEND:20200504T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/2
DESCRIPTION:Title: On structure of linear differential operators of the first order\n
by Valentin Lychagin as part of Geometry of differential equations seminar
\n\n\nAbstract\nWe'll discuss the equivalence problem (local as well as gl
obal) for linear differential operators of the first order\, acting in vec
tor bundles.\n\nThe slides will be in English and if preferred by anyone i
n the audience the talk itself can be switched from Russian to English.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valery Yumaguzhin
DTSTART:20200511T120000Z
DTEND:20200511T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/3
DESCRIPTION:Title: Invariants of forth order linear differential operators\nby Valery
Yumaguzhin as part of Geometry of differential equations seminar\n\n\nAbs
tract\nThe report is devoted to linear scalar differential operators of th
e fourth order on 2-dimensional manifolds. The field of rational different
ial invariants of such operators will be described and their application t
o the equivalence problem with respect to the group of diffeomorphisms of
the manifold will be shown.\n\nAlthough the talk will be in Russian\, the
slides will be in English and the discussion will be in both languages.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Samokhin
DTSTART:20200518T120000Z
DTEND:20200518T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/4
DESCRIPTION:Title: Using the KdV conserved quantities in problems of splitting of initial
data and reflection / refraction of solitons in varying dissipation and/
or dispersion media\nby Alexey Samokhin as part of Geometry of differ
ential equations seminar\n\n\nAbstract\nAn arbitrary compact-support initi
al datum for the Korteweg-de Vries equation asymptotically splits into sol
itons and a radiation tail\, moving in opposite direction. We give a simpl
e method to predict the number and amplitudes of resulting solitons and so
me integral characteristics of the tail using only conservation laws.\n\nA
similar technique allows to predict details of the behavior of a soliton
which\, while moving in non-dissipative and dispersion-constant medium en
counters a finite-width barrier with varying dissipation and/or dispersi
on\; beyond the layer dispersion is constant (but not necessarily of the s
ame value) and dissipation is null. The process is described with a spec
ial type generalized KdV-Burgers equation $u_t=(u^2+f(x)u_{xx})_x$.\n\nThe
transmitted wave either retains the form of a soliton (though of differen
t parameters) or scatters a into a number of them. And a reflection wave m
ay be negligible or absent. This models a situation similar to a light pas
sing from a humid air to a dry one through the vapor saturation/condensati
on area. Some rough estimations for a prediction of an output are given us
ing the relative decay of the KdV conserved quantities\; in particular a f
ormula for a number of solitons in the transmitted signal is given.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Tychkov
DTSTART:20200525T120000Z
DTEND:20200525T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/5
DESCRIPTION:Title: Continuum mechanics of media with inner structures\nby Sergey Tych
kov as part of Geometry of differential equations seminar\n\n\nAbstract\nW
e propose a geometrical approach to the mechanics of continuous media equi
pped with inner structures and give the basic equations of their motion: t
he mass conservation law\, the Navier-Stokes equation and the energy conse
rvation law.\n\nThis is a joint work with Anna Duyunova and Valentin Lycha
gin.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aleks Kleyn
DTSTART:20200601T120000Z
DTEND:20200601T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/6
DESCRIPTION:Title: System of differential equations over quaternion algebra\nby Aleks
Kleyn as part of Geometry of differential equations seminar\n\n\nAbstract
\nThe talk is based on the file\nhttps://gdeq.org/files/Aleks_Kleyn-2020.0
6.01.English.pdf (Russian transl.: https://gdeq.org/files/Aleks_Kleyn-2020
.06.01.Russian.pdf)\n\nIn order to study homogeneous system of linear diff
erential equations\, I considered vector space over division D-algebra and
the theory of eigenvalues in non commutative division D-algebra. I starte
d from section 1 dedicated to product of matrices. Since product in algebr
a is non-commutative\, I considered two forms of product of matrices and t
wo forms of eigenvalues (section 4). In sections 5\, 6\, 7\, I considered
solving of homogeneous system of differential equations. In the section 8\
, I considered the system of differential equations which has infinitely m
any fundamental solutions. Following sections are dedicated to analysis of
solutions of system of differential equations. In particular\, if a syste
m of differential equations has infinitely many fundamental solutions\, th
en each solution is envelope of a family of solutions of considered system
of differential equations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irina Bobrova (HSE\, Moscow)
DTSTART:20200608T120000Z
DTEND:20200608T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/7
DESCRIPTION:Title: On the second Painlevé equation and its higher analogues\nby Irin
a Bobrova (HSE\, Moscow) as part of Geometry of differential equations sem
inar\n\n\nAbstract\nSix Painlevé equations were obtained by Paul Painlev
é and his school during the classification of ODE's of the form $w'' = P
(z\, w\, w')$\, where the function $P (z\, w\, w')$ is a polynomial in $w$
and $w'$ and is an analytic function of $z$. These equations are widely u
sed in physics and have beautiful mathematical structures. My talk is devo
ted to the second Painlevé equation.\n\nWe will discuss the integrability
of this equation and introduce its Hamiltonian representation in terms of
the Kazuo Okamoto variables. On the other hand\, the PII equation is inte
grable in the sense of the Lax pair and the isomonodromic representation\,
that I will present.\n\nThe Bäcklund transformation and the affine Weyl
group are another interesting question. Using these symmetries\, we are ab
le to construct various rational solutions for the integer parameter PII e
quation.\n\nThe second Painlevé equation has one more important represent
ation in terms of $\\sigma$-coordinates which are $log$-symplectic.\n\nThe
re are higher analogues of the PII equation\, which we will obtain by self
-similar reduction of the modified Korteveg-de Vries hierarchy.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hovhannes Khudaverdian
DTSTART:20200615T120000Z
DTEND:20200615T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/8
DESCRIPTION:Title: Non-linear homomorphisms and thick morphisms\nby Hovhannes Khudave
rdian as part of Geometry of differential equations seminar\n\n\nAbstract\
nIn 2014\, Voronov introduced the notion of thick morphisms of (super)mani
folds as a tool for constructing $L_{\\infty}$-morphisms of homotopy Poiss
on algebras. Thick morphisms generalise ordinary smooth maps\, but are not
maps themselves. Nevertheless\, they induce pull-backs on $C^{\\infty}$ f
unctions. These pull-backs are in general non-linear maps between the alg
ebras of functions which are so-called "non-linear homomorphisms". By defi
nition\, this means that their differentials are algebra homomorphisms in
the usual sense. The following conjecture was formulated: an arbitrary non
-linear homomorphism of algebras of smooth functions is generated by some
thick morphism. We prove here this conjecture in the class of formal funct
ionals. In this way\, we extend the well-known result for smooth maps of m
anifolds and algebra homomorphisms of $C^{\\infty}$ functions and\, more g
enerally\, provide an analog of classical "functional-algebraic duality" i
n the non-linear setting.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20200622T120000Z
DTEND:20200622T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/9
DESCRIPTION:Title: Lax representations via extensions and deformations of Lie symmetry al
gebras\nby Oleg Morozov as part of Geometry of differential equations
seminar\n\n\nAbstract\nThe challenging problem in the theory of integrable
partial differential equations is to find conditions that are formulated
in inherent terms of a PDE under study and ensure existence of a Lax repre
sentation. The talk will present the technique for constructing Lax repres
entations via extensions of the contact symmetry algebras of PDEs. Also I
will show examples that use deformations of infinite-dimensional Lie alge
bras for searching new integrable PDEs.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20200629T120000Z
DTEND:20200629T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/10
DESCRIPTION:Title: Extendable symplectic structures and the inverse problem of the calcu
lus of variations for systems of equations written in an extended Kovalevs
kaya form\nby Konstantin Druzhkov as part of Geometry of differential
equations seminar\n\n\nAbstract\nThe talk is devoted to extendable symplec
tic structures for systems of equations written in an extended Kovalevskay
a form.\n\nIt is shown\, that each extension of a symplectic structure to
jets is related to an extension of a special form.\n\nComplete description
of all extendable symplectic structures is obtained. Relation of this res
ult with the inverse problem of the calculus of variations is discussed.\n
\nIt is shown\, that each variational formulation for a system of evolutio
n equations is related to a two-sided invertible variational operator of a
special form.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev (Lebedev Physical Institute\, Institute for Theore
tical and Mathematical Physics of Moscow State University)
DTSTART:20200706T120000Z
DTEND:20200706T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/11
DESCRIPTION:Title: Presymplectic structures and intrinsic Lagrangians\nby Maxim Grig
oriev (Lebedev Physical Institute\, Institute for Theoretical and Mathemat
ical Physics of Moscow State University) as part of Geometry of differenti
al equations seminar\n\n\nAbstract\nIt is well-known that a Lagrangian ind
uces a compatible presymplectic form on the equation manifold (stationary
surface\, understood as a submanifold of the respective jet-space). Given
an equation manifold and a compatible presymplectic form therein\, we defi
ne the first-order Lagrangian system which is formulated in terms of the i
ntrinsic geometry of the equation manifold. It has a structure of a presym
plectic AKSZ sigma model for which the equation manifold\, equipped with t
he presymplectic form and the horizontal differential\, serves as the targ
et space. For a wide class of systems (but not all) we show that if the pr
esymplectic structure originates from a given Lagrangian\, the proposed fi
rst-order Lagrangian is equivalent to the initial one and hence the Lagran
gian per se can be entirely encoded in terms of the intrinsic geometry of
its stationary surface. If the compatible presymplectic structure is gener
ic\, the proposed Lagrangian is only a partial one in the sense that its s
tationary surface contains the initial equation manifold but does not nece
ssarily coincide with it. I also plan to briefly discuss extension of this
construction to gauge PDEs (gauge theories in BV framework).\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20200713T120000Z
DTEND:20200713T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/12
DESCRIPTION:Title: Polynomial Poisson algebras associated with elliptic curves. Part 1\nby Vladimir Rubtsov (Université d'Angers) as part of Geometry of diff
erential equations seminar\n\n\nAbstract\nI shall give an introduction in
a study of Poisson algebras which are quasi classical limit of Sklyanin-Od
esskii-Feigin elliptic algebras. I will restrict my description to the alg
ebras with a "small" number of generators (n = 3\,4\,5).\n\nThe results ar
e (almost) not new. The talk is based on my old papers with A. Odesskii\,
G. Ortenzi and S. Tagne Pelap.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20200720T120000Z
DTEND:20200720T140000Z
DTSTAMP:20260404T111417Z
UID:GDEq/13
DESCRIPTION:Title: Polynomial Poisson algebras associated with elliptic curves. Part 2\nby Vladimir Rubtsov (Université d'Angers) as part of Geometry of diff
erential equations seminar\n\n\nAbstract\nI shall give an introduction in
a study of Poisson algebras which are quasi classical limit of Sklyanin-Od
esskii-Feigin elliptic algebras. I will restrict my description to the alg
ebras with a "small" number of generators (n = 3\,4\,5).\n\nThe results ar
e (almost) not new. The talk is based on my old papers with A. Odesskii\,
G. Ortenzi and S. Tagne Pelap.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Krasil'shchik (Independent University of Moscow)
DTSTART:20200930T162000Z
DTEND:20200930T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/14
DESCRIPTION:Title: Nonlocal conservation laws of PDEs possessing differential coverings<
/a>\nby Joseph Krasil'shchik (Independent University of Moscow) as part of
Geometry of differential equations seminar\n\nLecture held in room 303 of
the Independent University of Moscow.\n\nAbstract\nIn his 1892 paper "Su
lla trasformazione di Bäcklund per le superfici pseudosferiche" (Rend. Ma
t. Acc. Lincei\, s. 5\, v. 1 (1892) 2\, pp. 3-12\; Opere\, vol. 5\, pp. 16
3-173) Luigi Bianchi noticed\, among other things\, that quite simple tran
sformations of the formulas that describe the Bäcklund transformation of
the sine-Gordon equation lead to what is called a nonlocal conservation la
w in modern language. Using the techniques of differential coverings [I.S.
Krasil'shchik\, A.M. Vinogradov\, Nonlocal trends in the geometry of diff
erential equations: symmetries\, conservation laws\, and Bäcklund transfo
rmations\, Acta Appl. Math. 15 (1989) 161-209]\, we show that this observa
tion is of a quite general nature. We describe the procedures to construct
such conservation laws and present a number of illustrative examples.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Markus Dafinger (University of Jena\, Germany)
DTSTART:20201021T162000Z
DTEND:20201021T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/15
DESCRIPTION:Title: A converse to Noether's theorem\nby Markus Dafinger (University o
f Jena\, Germany) as part of Geometry of differential equations seminar\n\
n\nAbstract\nThe classical Noether's theorem states that symmetries of a v
ariational functional lead to conservation laws of the corresponding Euler
-Lagrange equation. It is a well-known statement to physicists with many a
pplications. In the talk we investigate a reverse statement\, namely that
a differential equation which satisfies sufficiently many symmetries and c
orresponding conservation laws leads to a variational functional whose Eul
er-Lagrange equation is the given differential equation. The aim of the ta
lk is to provide some background of the so-called inverse problem of the c
alculus of variations and then to discuss some new results\, for example\,
how to prove the reverse statement.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Sheftel
DTSTART:20201104T162000Z
DTEND:20201104T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/17
DESCRIPTION:Title: Nonlocal symmetry of CMA generates ASD Ricci-flat metric with no Kill
ing vectors\nby Mikhail Sheftel as part of Geometry of differential eq
uations seminar\n\n\nAbstract\nThe complex Monge-Ampère equation (CMA) in
a two-component form is treated as a bi-Hamiltonian system. I present exp
licitly the first nonlocal symmetry flow in each of the two hierarchies of
this system. An invariant solution of CMA with respect to these nonlocal
symmetries is constructed which\, being a noninvariant solution in the usu
al sense\, does not undergo symmetry reduction in the number of independen
t variables. I also construct the corresponding 4-dimensional anti-self-du
al (ASD) Ricci-flat metric with either Euclidean or neutral signature. It
admits no Killing vectors which is one of characteristic features of the f
amous gravitational instanton K3.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierandrea Vergallo (University of Salento)
DTSTART:20201111T162000Z
DTEND:20201111T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/18
DESCRIPTION:Title: Hydrodynamic-type systems and homogeneous Hamiltonian operators: a ne
cessary condition of compatibility\nby Pierandrea Vergallo (University
of Salento) as part of Geometry of differential equations seminar\n\n\nAb
stract\nUsing the theory of coverings\, it is presented a necessary condit
ion to write a hydrodynamic-type system in Hamiltonian formulation. Explic
it conditions for first\, second and third order homogeneous Hamiltonian o
perators are shown. In particular\, an alternative proof of Tsarev's theor
em about compatibility conditions for first order operators is obtained b
y using this method.\n\nThen\, analogous conditions are presented for non
local homogeneous Hamiltonian operators of first and third order.\n\nFinal
ly\, it is discussed the projective invariance for second and third order
operators.\n\nThe talk is based on a joint work with Raffaele Vitolo.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrey Losev
DTSTART:20201118T162000Z
DTEND:20201118T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/19
DESCRIPTION:Title: Tau theory\, d=10 N=1 SUSY and BV\nby Andrey Losev as part of Geo
metry of differential equations seminar\n\n\nAbstract\nPlease\, see https:
//gdeq.org/Losev for the abstract.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20201125T162000Z
DTEND:20201125T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/20
DESCRIPTION:Title: Differential equations\, their symmetries\, invariants and quotients
\nby Valentin Lychagin as part of Geometry of differential equations s
eminar\n\n\nAbstract\nWe'll discuss quotients of PDEs by their symmetry al
gebras and show their applications for integrations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20201202T162000Z
DTEND:20201202T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/21
DESCRIPTION:Title: Real Monge-Ampère operators and (almost) complex structures\nby
Vladimir Rubtsov (Université d'Angers) as part of Geometry of differentia
l equations seminar\n\n\nAbstract\nWe observe some interesting geometric s
tructures which are naturally linked with the geometric approach to Monge-
Ampère operators developed by Lychagin in late 70th.\n\nAmong them are:
(almost) complex\, (almost) product\, generalized complex\, hyperkahler\,
hypersymplectic and many other geometric structures.\n\nI hope (if I have
time) to show few interesting examples of its applications.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Mikhailov (University of Leeds)
DTSTART:20201209T162000Z
DTEND:20201209T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/22
DESCRIPTION:Title: Quantisation ideals of nonabelian integrable systems\nby Alexande
r Mikhailov (University of Leeds) as part of Geometry of differential equa
tions seminar\n\n\nAbstract\nIn my talk I'll discuss a new approach to the
problem of quantisation of dynamical systems\, introduce the concept of q
uantisation ideals and show meaningful examples. Traditional quantisation
theories start with classical Hamiltonian systems with dynamical variables
taking values in commutative algebras and then study their non-commutativ
e deformations\, such that the commutators of observables tend to the corr
esponding Poisson brackets as the (Planck) constant of deformation goes to
zero. I am proposing to depart from systems defined on a free associative
algebra. In this approach the quantisation problem is reduced to a descri
ption of two-sided ideals which define the commutation relations (the quan
tisation ideals) in the quotient algebras and which are invariant with res
pect to the dynamics of the system. Surprisingly this idea works rather ef
ficiently and in a number of cases I have been able to quantise the system
\, i.e. to find consistent commutation relations for the system. To illus
trate this approach I'll consider the quantisation problem for the non-abe
lian Bogoyavlensky N-chains and other examples\, including quantisation of
nonabelian integrable ODEs on free associative algebras.\n\nThe talk is b
ased on: AVM\, Quantisation ideals of nonabelian integrable systems\, arXi
v preprint arXiv:2009.01838
\, 2020 (Published in Russ. Math. Surv. v.75:5\, pp 199-200\, 2020).\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Khavkine (Czech Academy of Sciences)
DTSTART:20201216T162000Z
DTEND:20201216T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/23
DESCRIPTION:Title: Killing compatibility complex on Kerr spacetime\nby Igor Khavkine
(Czech Academy of Sciences) as part of Geometry of differential equations
seminar\n\n\nAbstract\nThe Killing operator $K_{ab}[v] = \\nabla_a v_b +
\\nabla_b v_a$ on a Lorentzian spacetime $(M\,g)$ plays an important role
in General Relativity (GR): it generates infinitesimal gauge symmetries of
the theory. Gauge symmetry invariants play the role of physical observabl
es. In PDE language\, this translates to the following: the components of
a compatibility operator for $K_{ab}$ generate all local observables for
linearized GR on the background $(M\,g)$. In arXiv:1910.08756 we have explicitly constructed such a c
ompatibility operator (indeed\, a full compatibility complex) on the astro
physically interesting Kerr spacetime of a rotating black hole. I will mot
ivate and explain our approach and describe the complexity of the construc
tion.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20201223T162000Z
DTEND:20201223T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/24
DESCRIPTION:Title: Dispersionless integrable hierarchies and GL(2) geometry\nby Bori
s Kruglikov (UiT the Arctic University of Norway) as part of Geometry of d
ifferential equations seminar\n\n\nAbstract\n(joint work with Evgeny Ferap
ontov)\n\nParaconformal or GL(2) geometry on an n-dimensional manifold M i
s defined by a field of rational normal curves of degree n - 1 in the proj
ectivized cotangent bundle $\\mathbb{P}T^*M$. In dimension n=3 this is not
hing but a Lorentzian metric. GL(2) geometry is known to arise on solution
spaces of ODEs with vanishing Wünschmann invariants.\n\nWe show that GL(
2) structures also arise on solutions of dispersionless integrable hierarc
hies of PDEs such as the dispersionless Kadomtsev-Petviashvili (dKP) hiera
rchy. In fact\, they coincide with the characteristic variety (principal s
ymbol) of the hierarchy. GL(2) structures arising in this way possess the
property of involutivity. For n=3 this gives the Einstein-Weyl geometry.\n
\nThus we are dealing with a natural generalization of the Einstein-Weyl g
eometry. Our main result states that involutive GL(2) structures are gover
ned by a dispersionless integrable system whose general local solution dep
ends on 2n - 4 arbitrary functions of 3 variables. This establishes integr
ability of the system of Wünschmann conditions.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Chetverikov
DTSTART:20210203T162000Z
DTEND:20210203T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/25
DESCRIPTION:Title: Coverings and multivector pseudosymmetries of differential equations<
/a>\nby Vladimir Chetverikov as part of Geometry of differential equations
seminar\n\n\nAbstract\nFinite-dimensional coverings from systems of diffe
rential equations are investigated. This problem is of interest in view of
its relationship with the computation of differential substitution\, nonl
ocal symmetries\, recursion operators\, and Backlund transformations. We s
how that the distribution specified by the fibers of a covering is determi
ned by an integrable pseudosymmetry of the system. Conversely\, every inte
grable pseudosymmetry of a system defines a covering from this system. The
vertical component of the pseudosymmetry is a matrix analog of the evolut
ion differentiation. The corresponding generating matrix satisfies a matri
x analog of the linearization of the equation. We consider also the exteri
or product of vector fields defining a pseudosymmetry. The definition of p
seudosymmetry is rewritten in the language of the Schouten bracket of mult
ivector fields and total derivatives with respect to the independent varia
bles of the system. A method for constructing coverings is given and demon
strated by the examples of the Laplace equation and the Kapitsa pendulum s
ystem.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Samokhin
DTSTART:20210210T162000Z
DTEND:20210210T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/26
DESCRIPTION:Title: On monotonic pattern in periodic boundary solutions of cylindrical an
d spherical Kortweg-de Vries-Burgers equations\nby Alexey Samokhin as
part of Geometry of differential equations seminar\n\n\nAbstract\nWe studi
ed\, for the Kortweg-de Vries Burgers equations on cylindrical and spheric
al waves\, the development of a regular profile starting from an equilibri
um under a periodic perturbation at the boundary.\n\nThe regular profile a
t the vicinity of perturbation looks like a periodical chain of shock fron
ts with decreasing amplitudes. Further on\, shock fronts become decaying s
mooth quasi periodic oscillations. After the oscillations cease\, the wave
develops as a monotonic convex wave\, terminated by a head shock of a con
stant height and equal velocity. This velocity depends on integral charact
eristics of a boundary condition and on spatial dimensions.\n\nThe explici
t asymptotic formulas for the monotonic part\, the head shock and a median
of the oscillating part are found.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petr Pushkar
DTSTART:20210217T162000Z
DTEND:20210217T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/27
DESCRIPTION:Title: Morse theory\, Bruhat cells and Unitriangular geometry\nby Petr P
ushkar as part of Geometry of differential equations seminar\n\n\nAbstract
\nStrong Morse function is a Morse function with pairwise different critic
al values. For such a function we construct a collection of numbers\, whic
h is a (smooth) topological invariant of the strong Morse function.\n\nAlg
ebraically our construction is a close relative of the construction of Bru
hat cells and belongs to Unitriangular geometry. We will present a general
ization of determinant of any linear map between finite dimensional vector
spaces.\n\nTalk based on a joint work with Misha Temkin.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Sokolov (Landau Institute for Theoretical Physics\, Chern
ogolovka\, Russia)
DTSTART:20210224T162000Z
DTEND:20210224T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/28
DESCRIPTION:Title: Non-Abelian generalizations of integrable PDEs and ODEs\nby Vladi
mir Sokolov (Landau Institute for Theoretical Physics\, Chernogolovka\, Ru
ssia) as part of Geometry of differential equations seminar\n\n\nAbstract\
nA general procedure for nonabelinization of given integrable polynomial d
ifferential equation is described. We consider NLS type equations as an ex
ample. We also find nonabelinizations of the Euler top and of the Painleve
-2 equation.\n\nAlthough the talk will be in Russian\, the slides will be
in English and the discussion will be in both languages.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20210303T162000Z
DTEND:20210303T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/29
DESCRIPTION:Title: Real Monge-Ampère operators and (almost) complex structures. Part 2<
/a>\nby Vladimir Rubtsov (Université d'Angers) as part of Geometry of dif
ferential equations seminar\n\n\nAbstract\nWe observe some interesting geo
metric structures which are naturally linked with the geometric approach t
o Monge-Ampère operators developed by Lychagin in late 70th. I shall conc
entrate my attention on the Hitchin generalized complex structure\, hyper-
Kahler/symplectic and hope to show few interesting examples of its relatio
ns with the Monge-Ampère operators and applications.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Pavlov
DTSTART:20210310T162000Z
DTEND:20210310T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/30
DESCRIPTION:Title: New variational principles for one-dimensional gas dynamics and for E
gorov hydrodynamic type systems\nby Maxim Pavlov as part of Geometry o
f differential equations seminar\n\n\nAbstract\nThe Statement. If some Ego
rov hydrodynamic type system has one local Hamiltonian structure of Dubrov
in-Novikov type\, then such a system possesses infinitely many: local Hami
ltonian structures of all odd orders\, and infinitely many local Lagrangia
n representations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladislav Zhvick
DTSTART:20210317T162000Z
DTEND:20210317T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/31
DESCRIPTION:Title: Nonlocal conservation law in a submerged jet\nby Vladislav Zhvick
as part of Geometry of differential equations seminar\n\nLecture held in
room 303 of the Independent University of Moscow.\n\nAbstract\nLandau was
the first to obtain the exact solution of Navier-Stokes equations for an a
xisymmetric submerged jet generated by a point momentum source. The Landau
jet is the main term of a coordinate expansion of the flow far field in t
he case when the flow is generated by a finite size source (for example\,
a tube with flow). The next term of the expansion was calculated by Rumer.
This term has an indefinite coefficient. To determine this coefficient we
need a conservation law connecting the jet far field with the source. Wel
l-known conservation laws of mass\, momentum\, and angular momentum fail t
o calculate the coefficient. In my talk\, I will solve this problem for lo
w viscosity. In this case\, the flow satisfies the boundary layer equation
s that possess a nonlocal conservation law closing the problem. The proble
m for an arbitrary viscosity remains open.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anatolij Prykarpatski
DTSTART:20210324T162000Z
DTEND:20210324T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/32
DESCRIPTION:Title: On integrability of some Riemann type hydrodynamical systems and Dubr
ovin integrability classification of perturbed Korteweg-de Vries type equa
tions\nby Anatolij Prykarpatski as part of Geometry of differential eq
uations seminar\n\n\nAbstract\nIn our report we will stop on two closely r
elated to each other integrability theory aspects. The first one concerns
the obtained integrability results\, based on the gradient-holonomic integ
rability scheme\, devised and applied by me jointly with Maxim Pavlov and
collaborators to a virtually new important Riemann type hierarchy $D_{t}^{
N-1}u=z_{x}^{s}$\, $D_{t}z=0$\, where $s$\, \;$N\\in N$ are arbitrary
natural numbers\, and proposed in our work (M. Pavlov\, A. Prykarpatsky\,
at al.\, arXiv:1108.0878) as
a nontrivial generalization of the infinite hierarchy of the Riemann type
flows\, suggested before by M. Pavlov and D. Holm in the form of dynamica
l systems $D_{t}^{N}u=0$\, defined on a $2\\pi$-periodic functional manifo
ld $M^{N}\\subset C^{\\infty}( R/2\\pi Z\; R^{N})$\, the vector $(u\,D_{t}
u\,D_{t}²u\,...\,D_{t}^{N-1}u\,z)^{\\intercal}\\in M^{N}$\, the different
iations $D_{x}:=\\partial/\\partial x$\, $D_{t}:=\\partial/\\partial t+u\\
partial/\\partial x$ satisfy as above the Lie-algebraic commutator relatio
nship $[D_{x}\,D_{t}]=u_{x}D_{x}$ and t\\in R is an evolution parameter. T
he second aspect of our report concerns the integrability results obtained
by B. Dubrovin jointly with Y. Zhang and collaborators\, devoted to class
ification of a special perturbation of the Korteweg-de Vries equation in t
he form $u_{t}=uu_{x}+\\epsilon^2[f_{31}(u)u_{xxx}+f_{32}(u)u_{xx}u_{x}+f_
{33}(u)u_{x}^3]$\, where $f_{jk}(u)\,~j=3\,~k=1\,~3$\, are some smooth fun
ctions and \\epsiln\\in R is a real parameter. We will deal with classific
ation scheme of evolution equations of a special type suspicious on being
integrable which was devised some years ago by untimely passed away Prof.
Boris Dubrovin (19 March 2019) and developed with his collaborators\, main
ly with Youjin Zhang. We have reanalyzed in detail their interesting resul
ts on integrability classification of a suitably perturbed KdV type equati
on within our gradient-holonomic integrability scheme\, devised many years
ago and developed by me jointly with Maxim Pavlov and collaborators\, and
found out that the Dubrovin's scheme has missed at least a one very inter
esting integrable equation\, whose natural reduction became similar to the
well-known Krichever-Novikov equation\, yet different from it. As a conse
quence of the analysis\, we presented one can firmly claim that the Dubrov
in-Zhang integrability criterion inherits some important part of the menti
oned above gradient-holonomic integrability scheme properties\, coinciding
with the statement about the necessary existence of suitably ordered redu
ction expansions with coefficients to be strongly homogeneous differential
polynomials.\n\nJoint with Alex A. Balinsky\, Radoslaw Kycia and Yarema A
. Prykarpatsky.\n\nAlthough the talk will be in Russian\, the slides will
be in English and the discussion will be in both languages.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hovhannes Khudaverdian
DTSTART:20210331T162000Z
DTEND:20210331T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/33
DESCRIPTION:Title: Odd symplectic geometry in the BV-formalism\nby Hovhannes Khudave
rdian as part of Geometry of differential equations seminar\n\nLecture hel
d in room 304 of the Independent University of Moscow.\n\nAbstract\nOdd sy
mplectic geometry was considered by physicists as an exotic counterpart of
even symplectic geometry. Batalin and Vilkovisky changed this\npoint of v
iew by the seminal work considering the quantisation of general theory in
Lagrangian framework\, where they considered odd symplectic superspace of
fields and antifields. [In the case of Lie group of symmetries BV receipt
is reduced to the standard Faddeev-Popov method.]\n\nThe main ingredient o
f the theory\, the exponent of the master action\, is defined by the funct
ion $f$ such that $\\Delta f=0$\, where $\\Delta$ is second order differen
tial operator of the second order: $\\Delta=\\frac{\\partial^2}{\\partial
x^i \\partial\\theta_i}$\, ($x^i\,\\theta_j$ are the Darboux coordinates o
f an odd symplectic superspace.) This operator has no analogy in the stand
ard symplectic geometry.\n\nI consider in this talk the main properties of
the BV-formalism geometry.\n\nThe $\\Delta$-operator is defined in geomet
rical clear way\, and this operator depends on the volume form.\n\nIt is s
uggested the canonical operator $\\Delta$ on half-densities. This operator
is the proper framework for BV geometry. We also study the groupoid prope
rty of BV master-equation\; this leads us to the notion of BV groupoid. We
also discuss some constructions of invariants for odd symplectic structur
e.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20210407T162000Z
DTEND:20210407T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/34
DESCRIPTION:Title: On dynamics of molecular media and generalization of Navier-Stokes eq
uations\nby Valentin Lychagin as part of Geometry of differential equa
tions seminar\n\nLecture held in room 304 of the Independent University of
Moscow.\n\nAbstract\nThis talk is a prolongation of my previous talk that
was devoted to continuum mechanics of media possessing inner structure.\n
\nHere we'll consider molecular media\, its geometry and thermodynamics.\n
\nThe main goal of this talk is to present in the explicit form necessary
geometrical structures and to give the explicit form of the Navier-Stokes
equations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20210421T162000Z
DTEND:20210421T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/35
DESCRIPTION:Title: Lax representations via twisted extensions of infinite-dimensional Li
e algebras: some new results\nby Oleg Morozov as part of Geometry of d
ifferential equations seminar\n\nLecture held in room 304 of the Independe
nt University of Moscow.\n\nAbstract\nI will discuss the technique for con
structing integrable differential equations via twisted extensions of infi
nite-dimensional Lie algebras. Examples will include a 3D generalization o
f the Hunter-Saxton equation with the special value of the parameter and t
he "degenerate heavenly equation".\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Taras Skrypnyk
DTSTART:20210428T162000Z
DTEND:20210428T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/36
DESCRIPTION:Title: Asymmetric variable separation for the Clebsch model\nby Taras Sk
rypnyk as part of Geometry of differential equations seminar\n\n\nAbstract
\nIn the present talk we present our result on separation of variables (So
V) for the Clebsch model.\n\nIn particular\, we report on the development
of two methods in the variable separation theory:\n\n - the method
of the differential separability conditions\;
\n - the method of th
e vector fields $Z$.
\n
\nUsing these two methods we construct an
asymmetric variable separation for the Clebsch model. Our SoV is unusual:
it is characterized by two different curves of separation. We explicitly c
onstruct coordinates and momenta of separation\, the reconstruction formul
ae and the Abel-type quadratures for the Clebsch system. The solution of t
he non-standard Abel-Jacobi inversion problem is briefly discussed.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Sachkov
DTSTART:20210414T162000Z
DTEND:20210414T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/37
DESCRIPTION:Title: Sub-Riemannian geometry on the group of motions of the plane\nby
Yuri Sachkov as part of Geometry of differential equations seminar\n\n\nAb
stract\nWe will discuss the unique\, up to local isometries\, contact sub-
Riemannian struc\nture on the group SE(2) of proper motions of the plane (
aka group of rototransla\ntions).\nThe following questions will be address
ed:\n\n - \; geodesics\,
\n
- \; their local and global optimality\,
\n - \; cut ti
me\, cut locus\, and spheres\,
\n - \; infinite geodesics\,\n
- \; bicycle transform and relation of geodesics with Euler e
lasticae\,
\n - \; group of isometries and homogeneous geodesi
cs\,
\n - \; applications to imaging and robotics.
\n
\nJoint work with Andrei Ardentov.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugene Ferapontov (Loughborough University)
DTSTART:20210505T162000Z
DTEND:20210505T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/38
DESCRIPTION:Title: Second-order PDEs in 3D with Einstein-Weyl conformal structure\nb
y Eugene Ferapontov (Loughborough University) as part of Geometry of diffe
rential equations seminar\n\n\nAbstract\nI will discuss a general class of
second-order PDEs in 3D whose characteristic conformal structure satisfie
s the Einstein-Weyl conditions on every solution.\n\nThis property is know
n to be equivalent to the existence of a dispersionless Lax pair\, as well
as to other equivalent definitions of dispersionless integrability.\n\nI
will demonstrate that (a) the Einstein-Weyl conditions can be viewed as an
efficient contact-invariant test of dispersionless integrability\, (b) sh
ow some partial classification results\, and (c) formulate a rigidity conj
ecture according to which any second-order PDE with Einstein-Weyl conforma
l structure can be reduced to a dispersionless Hirota form via a suitable
contact transformation.\n\nBased on joint work with S. Berjawi\, B. Krugli
kov\, V. Novikov.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Zabrodin
DTSTART:20210519T162000Z
DTEND:20210519T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/39
DESCRIPTION:Title: Kadomtsev-Petviashvili hierarchies of types B and C\nby Anton Zab
rodin as part of Geometry of differential equations seminar\n\nLecture hel
d in room 303 or 304 of the Independent University of Moscow.\n\nAbstract\
nThis is a short review of the Kadomtsev-Petviashvili hierarchies of types
B and C. The main objects are the $L$-operator\, the wave operator\, the
auxiliary linear problems for the wave function\, the bilinear identity fo
r the wave function and the tau-function. All of them are discussed in the
paper. The connections with the usual Kadomtsev-Petviashvili hierarchy (o
f the type A) are clarified. Examples of soliton solutions and the dispers
ionless limit of the hierarchies are also considered.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20210512T162000Z
DTEND:20210512T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/40
DESCRIPTION:Title: Operations on universal enveloping algebra and the "argument shift" m
ethod\nby Georgy Sharygin as part of Geometry of differential equation
s seminar\n\nLecture held in room 303 or 304 of the Independent University
of Moscow.\n\nAbstract\nIf a vector field X is given on a Poisson manifol
d M such that the square of the Lie derivative in the X direction "kills"
the Poisson bivector\, then there is a well-known simple method of "shifti
ng the argument" (along X) to construct a commutative subalgebra (with res
pect to the Poisson bracket) inside the algebra of functions on M. In a pa
rticular case\, this method can be applied to the Poisson-Lie bracket on t
he symmetric algebra of an arbitrary Lie algebra and gives (according to a
well-known result\, the proven Mishchenko-Fomenko conjecture) maximal com
mutative subalgebras in the symmetric algebra. However\, the lifting of th
ese algebras to commutative subalgebras in the universal enveloping algebr
a\, although possible\, is based on very nontrivial results from the theor
y of infinite-dimensional Lie algebras. In my talk\, I will describe parti
al results that allow one to construct on the universal enveloping algebra
of the algebra $gl_{n}$ the operators of "quasidifferentiation" and wi
th their help\, in some cases\, construct a commutative subalgebra in $Ugl
_{n}$. I will also describe how\, in the general case\, this question i
s reduced to the combinatorial question of commuting a certain set of oper
ators in tensor powers $\\mathbb {R} ^{n}$. The story is based on colla
borations with Dmitry Gurevich\, Pavel Saponov and Ikeda Yasushi.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20210922T162000Z
DTEND:20210922T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/41
DESCRIPTION:Title: On metric invariants of spherical harmonics\nby Valentin Lychagin
as part of Geometry of differential equations seminar\n\nLecture held in
room 303 of the Independent University of Moscow.\n\nAbstract\nWe'll discu
ss the algebraic and differential SO(3)-invariants of spherical harmonics
and give a description of fields of rational algebraic and rational differ
ential invariants together with their application to the description of re
gular SO(3)-orbits of spherical harmonics.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20211006T162000Z
DTEND:20211006T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/43
DESCRIPTION:Title: WDVV equations and invariant bi-Hamiltonian formalism\nby Raffael
e Vitolo (Università del Salento) as part of Geometry of differential equ
ations seminar\n\n\nAbstract\nThe WDVV equations are central in Topologica
l Field Theory and Integrable Systems. We prove that in low dimensions the
WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian f
ormalism is proved for N = 3. More examples in higher dimensions show that
the result might hold in general. The invariance group of the bi-Hamilton
ian pairs is the group of projective reciprocal transformations. The signi
ficance of projective invariance of WDVV equations is discussed in detail.
Computer algebra programs that were used for calculations throughout the
paper are provided at https://github.com/Jakub-Vasicek/WDVV-computations/.
\n\nBased on a joint work with Jakub Vašíček.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Khavkine (Czech Academy of Sciences)
DTSTART:20211020T162000Z
DTEND:20211020T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/44
DESCRIPTION:Title: Triangular decoupling of systems of differential equations\, with app
lication to separation of variables on Schwarzschild spacetime\nby Igo
r Khavkine (Czech Academy of Sciences) as part of Geometry of differential
equations seminar\n\n\nAbstract\nCertain tensor wave equations admit a co
mplete separation of variables on the Schwarzschild spacetime (asymptotica
lly flat\, static\, spherically symmetric black hole in 4d)\, resulting in
complicated systems of radial mode ODEs. Almost none of the important que
stions about these radial mode equations can be answered in their original
form. I will discuss a drastic simplification of these ODE systems to spa
rse upper triangular form\, which uncovers their general properties. Essen
tial to this simplification are geometric properties of the original tenso
r wave equations\, ideas from homological algebra and from the theory of O
DEs with rational coefficients. Based on https://arxiv.org/abs/1711.00585
\, https://arxiv.org/abs/1801.09800 \, https://arxiv.org/abs/2004.09651\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20211027T162000Z
DTEND:20211027T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/45
DESCRIPTION:Title: Integrable PDEs and extensions of Lie-Rinehart algebras\nby Oleg
Morozov as part of Geometry of differential equations seminar\n\nLecture h
eld in room 303 of the Independent University of Moscow.\n\nAbstract\nI wi
ll discuss extensions of Lie-Rinehart algebras and their application to th
e problem of recognizing whether a given PDE admits a Lax representation.\
n
LOCATION:https://stable.researchseminars.org/talk/GDEq/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugene Ferapontov (Loughborough University)
DTSTART:20211201T162000Z
DTEND:20211201T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/46
DESCRIPTION:Title: Second order integrable Lagrangians and WDVV equations\nby Eugene
Ferapontov (Loughborough University) as part of Geometry of differential
equations seminar\n\n\nAbstract\nI will discuss integrability of 2D and 3D
Euler-Lagrange equations for second-order Lagrangians. A link to WDVV equ
ations will be established. Based on joint work with Maxim Pavlov and Ling
ling Xue.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kotov
DTSTART:20211013T162000Z
DTEND:20211013T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/48
DESCRIPTION:Title: Riemannian Cartan-Lie algebroids and groupoids and curved Yang-Mills-
Higgs models\nby Alexei Kotov as part of Geometry of differential equa
tions seminar\n\n\nAbstract\nIn this talk the generalization of the Yang-M
ills-Higgs model will be presented\, based upon the notion of Cartan struc
tures and compatible metrics on Lie algebroids and groupoids.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20211124T162000Z
DTEND:20211124T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/49
DESCRIPTION:Title: On Cartan's C-class differential equations\nby Boris Doubrov as p
art of Geometry of differential equations seminar\n\nLecture held in room
303 of the Independent University of Moscow.\n\nAbstract\nWe consider a ve
ry special class of differential equations\, which is characterized by the
condition that all its local differential invariants (under the action of
a suitable Lie pseudogroup) become first integrals when restricted to the
equation manifold. Such differential equations were introduced in a short
note of Elie Cartan (Les espaces généralisés et l'intégration de cert
aines classes d'équations différentielles\, C.R.\, 1938\, V.206\, N.23\,
1689-1693)\, who characterized them in two simplest cases: scalar 2nd ord
er ODEs viewed under the pseudogroup of point transformations and scalar 3
rd order ODEs under the group of contact transformations. We show how thes
e results generalize to any systems of ODEs and\, more generally\, differe
ntial equations of finite type. The same question for arbitrary systems of
PDEs still remains open.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eivind Schneider
DTSTART:20211110T162000Z
DTEND:20211110T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/50
DESCRIPTION:Title: Differential invariants of Kundt spacetimes\nby Eivind Schneider
as part of Geometry of differential equations seminar\n\n\nAbstract\nWe co
mpute generators for the algebra of rational scalar differential invariant
s of general and degenerate Kundt spacetimes. Special attention is given t
o dimensions 3 and 4 since in those dimensions the degenerate Kundt metric
s are known to be exactly the Lorentzian metrics that can not be distingui
shed by polynomial curvature invariants constructed from the Riemann tenso
r and its covariant derivatives.\n\nThe talk is based on joint work with B
oris Kruglikov.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Duyunova and Sergey Tychkov
DTSTART:20211103T162000Z
DTEND:20211103T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/51
DESCRIPTION:Title: The Euler system on a space curve\nby Anna Duyunova and Sergey Ty
chkov as part of Geometry of differential equations seminar\n\n\nAbstract\
nWe consider flows of an inviscid medium on a space curve in a constant gr
avitational field (the Euler system). We discuss symmetries and differenti
al invariants of the Euler system\, and give their classification based on
symmetries group of the system. Using differential invariants for this sy
stem\, we obtain its quotient. The solutions of the quotient equation that
are constant along characteristic vector field provide some solutions of
the Euler system.\n\nJoint work with Valentin Lychagin.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergei Igonin
DTSTART:20211229T162000Z
DTEND:20211229T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/52
DESCRIPTION:Title: Algebra and geometry of Lax representations and Bäcklund transformat
ions for (1+1)-dimensional partial differential and differential-differenc
e equations\nby Sergei Igonin as part of Geometry of differential equa
tions seminar\n\n\nAbstract\nSee IgoninSeminar20211229abstract.pdf\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hynek Baran
DTSTART:20211208T162000Z
DTEND:20211208T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/53
DESCRIPTION:Title: Jets\, a computer algebra on diffieties\nby Hynek Baran as part o
f Geometry of differential equations seminar\n\n\nAbstract\nJets is a set
of Maple procedures to facilitate solution of differential equations in to
tal derivatives on diffieties. Otherwise said\, Jets is a tool to compute
symmetries\, conservation laws\, zero-curvature representations\, recursio
n operators\, any many other invariants of systems of partial differential
equations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20211117T162000Z
DTEND:20211117T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/54
DESCRIPTION:Title: Presymplectic gauge PDEs and Batalin-Vilkovisky formalism\nby Max
im Grigoriev as part of Geometry of differential equations seminar\n\nLect
ure held in room 303 of the Independent University of Moscow.\n\nAbstract\
nGauge PDE is a geometrical object underlying what physicists call a local
gauge field theory defined at the level of equations of motion (i.e. wit
hout specifying Lagranian) in terms of BV-BRST formalism. Although gauge P
DE can be defined as a PDE equipped with extra structures\, the generaliza
tion is not entirely straightforward as\, for instance\, two gauge PDEs ca
n be equivalent even if the underlying PDEs are not. As far as Lagrangian
gauge systems are concerned the powerful framework is provided by the BV f
ormalism on jet-bundles. However\, just like in the case of usual PDEs it
is difficult to encode the BV extension of the Lagrangian in terms of the
intrinsic geometry of the equation manifold while working on jet-bundles i
s often very restrictive\, especially in analyzing boundary behaviour\, e.
g.\, in the context of AdS/CFT correspondence. We show that BV Lagrangian
(or its weaker analogs) can be encoded in the compatible graded presymplec
tic structure on the gauge PDE. In the case of genuine Lagrangian systems
this presymplectic structure is related to a certain completion of the can
onical BV symplectic structure. A presymplectic gauge PDE gives rise to a
BV formulation of the underlying system through an appropriate generalizat
ion of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) sigma-model con
struction followed by taking the symplectic quotient. The construction is
illustrated on the standard examples of gauge theories with particular emp
hasis on the Einstein gravity\, where this naturally leads to an elegant p
resymplectic AKSZ representation of the BV extension of the Cartan-Weyl fo
rmulation of gravity.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Petr Vojčák
DTSTART:20211208T162000Z
DTEND:20211208T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/55
DESCRIPTION:Title: On the algebras of nonlocal symmetries for the (modified) 4D Martı́
nez Alonso-Shabat equation\nby Petr Vojčák as part of Geometry of di
fferential equations seminar\n\n\nAbstract\nWe consider two four-dimension
al Lax-integrable equations known as the 4D Martı́nez Alonso-Shabat equa
tion and the modified Martı́nez Alonso-Shabat equation\, respectively. W
e construct two differential coverings for both of them and describe the a
lgebras of nonlocal symmetries in these coverings. We also analyze the act
ions of the known recursion operators on these nonlocal symmetries.\n\nPar
tially based on a joint work with Joseph Krasil'shchik.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Agafonov
DTSTART:20220209T162000Z
DTEND:20220209T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/56
DESCRIPTION:Title: Darboux integrability for diagonal systems of hydrodynamic type\n
by Sergey Agafonov as part of Geometry of differential equations seminar\n
\nLecture held in room 303 of the Independent University of Moscow.\n\nAbs
tract\nWe prove that diagonal systems of hydrodynamic type are Darboux int
egrable if and only if the Laplace transformation sequences of the system
for commuting flows terminate\, give geometric interpretation for Darboux
integrability of such systems in terms of congruences of lines and in term
s of solution orbits with respect to symmetry subalgebras\, show that Darb
oux integrable systems are necessarily semihamiltonian\, and discuss known
and new examples.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20220223T162000Z
DTEND:20220223T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/57
DESCRIPTION:Title: ODEs with essential contact or point symmetries\nby Boris Kruglik
ov (UiT the Arctic University of Norway) as part of Geometry of differenti
al equations seminar\n\nLecture held in room 303 of the Independent Univer
sity of Moscow.\n\nAbstract\n(joint work with Eivind Schneider)\n\nWe obse
rve that\, up to conjugation\, a majority of higher order ODEs and ODE sys
tems have only point fiber-preserving symmetries (surprisingly this is als
o true for "most interesting" ODEs). We describe all the exceptions in the
case of scal ar ODEs and systems of pairs of ODEs on a pair of functions.
We exploit classifications of Lie algebras of vector fields in 2 and 3 di
mensions.\n\nWhile we can express scalar ODEs with essentially contact or
point symmetry algebras via absolute and relative differential invariants\
, we have to invoke also conditional differential invariants in the case o
f ODE systems to deal with singular orbits of the action. In the scalar ca
se the result is partially due to Lie\, but we consider the global classif
ication and discuss the algebra of relative invariants. For systems the re
sult is new.\n\nInvestigating prolongations of the actions\, we observe so
me interesting relations between different realizations of Lie algebras. W
e also note that prolongation of a finite-dimensional Lie algebra acting o
n a differential equation may not eventually become free. An example of un
derdetermined ODE with this phenomenon shows limitations of the method of
moving frames.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitri Alekseevsky
DTSTART:20220316T162000Z
DTEND:20220316T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/59
DESCRIPTION:Title: Special Vinberg cones and their applications\nby Dmitri Alekseevs
ky as part of Geometry of differential equations seminar\n\nLecture held i
n room 303 of the Independent University of Moscow.\n\nAbstract\nThe talk
is based on joint works with Vicete Cortes\, Andrea Spiro and Alessio Marr
ani.\n\nA short survey of the Vinberg theory of convex cones (including it
s informational geometric interpretation) and homogeneous convex cones wil
l be presented. Then we concentrate on the theory of rank 3 special Vinber
g cones\, associated to metric Clifford $Cl({\\mathbb R}^n)$ modules.\n\nA
generalization of the theory to the indefinite special Vinberg cones\, as
sociated to indefinite metric Clifford $Cl({\\mathbb R}^{p\,q})$ modules i
s indicated. An application of special Vinberg cones to $N=2 \, \\\, d=5\,
4\,3$ Supergravity will be considered.\n\nWe will discuss also application
s of theory of homogeneous convex cones to convex programming\, informatio
n geometry and Frobenius manifolds.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20220309T162000Z
DTEND:20220309T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/60
DESCRIPTION:Title: Multiplicative kernels\, Non-abelian Abel theorem\, Kontsevich polyno
mials and around. Part 1\nby Vladimir Rubtsov (Université d'Angers) a
s part of Geometry of differential equations seminar\n\n\nAbstract\nWe dis
cuss recent progress (published and unpublished yet ) in studies of multip
licative kernels\, initiated by M. Konstevich. We will try to explain vari
ous links and applications of this notion in geometry\, differential equat
ions and integrable systems. My talk is based on the paper arXiv:2102.09511 and on two ongoing projec
ts with I. Gaiur and D. Van Straten.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20220413T162000Z
DTEND:20220413T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/61
DESCRIPTION:Title: Natural invariants and classification of quasilinear second-order dif
ferential operators\nby Valentin Lychagin as part of Geometry of diffe
rential equations seminar\n\nLecture held in room 303 of the Independent U
niversity of Moscow.\n\nAbstract\nThis talk is based on joint research wit
h Valery Yumaguzhin.\n\nIn the first part\, we outline the method of findi
ng rational natural differential invariants of a class of quasilinear seco
nd-order differential operators\, and then we show how these invariants co
uld be used to get local as well as global classification of such type ope
rators with respect to the diffeomorphism group.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dimitri Gurevich
DTSTART:20220330T162000Z
DTEND:20220330T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/62
DESCRIPTION:Title: Quantum Matrix Algebras and their applications\nby Dimitri Gurevi
ch as part of Geometry of differential equations seminar\n\n\nAbstract\nQu
antum Matrix Algebras are very interesting objects from algebraic viewpoin
t. Particular examples of these algebras are related to Drinfeld-Jimbo Qua
ntum Groups. Some of these QMA admit defining analogs of partial derivativ
es. In a limit it is possible to develop a new calculus on the enveloping
algebras $U(gl(N))$.\n\nOther applications will be discussed.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20220406T162000Z
DTEND:20220406T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/63
DESCRIPTION:Title: Multiplicative kernels\, Non-abelian Abel theorem\, Kontsevich polyno
mials and around. Part 2\nby Vladimir Rubtsov (Université d'Angers) a
s part of Geometry of differential equations seminar\n\n\nAbstract\nA cont
inuation of the talk
on 9 March.\n\nWe discuss recent progress (published and unpublished y
et) in studies of multiplicative kernels\, initiated by M. Konstevich. We
will try to explain various links and applications of this notion in geome
try\, differential equations and integrable systems. My talk is based on t
he paper arXiv:2102.09511 a
nd on two ongoing projects with I. Gaiur and D. Van Straten.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Marshall
DTSTART:20220504T162000Z
DTEND:20220504T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/64
DESCRIPTION:Title: On action-angle duality\nby Ian Marshall as part of Geometry of d
ifferential equations seminar\n\nLecture held in room 303 of the Independe
nt University of Moscow.\n\nAbstract\nAction-angle duality is a property e
njoyed by systems of Ruijsenaars type - many body systems\; relativistic a
nalogues of Calogero-Moser-Sutherland systems - whereby families of integr
able systems come in natural pairs: the canonical coordinates of one syste
m are the action-angle variables of the other\, and together they generate
the whole phase space. I will explain this property\, and why it is speci
al. When transported to quantum systems\, the action-angle duality propert
y is represented in the form of bispectral operators.\n\nI hope also to de
scribe results obtained with László Fehér in which Hamiltonian reductio
n is used to obtain systems in action-angle duality relation with one an o
ther.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilia Gaiur (University of Birmingham)
DTSTART:20220511T162000Z
DTEND:20220511T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/65
DESCRIPTION:Title: Hamiltonian reduction and rational Calogero system\nby Ilia Gaiur
(University of Birmingham) as part of Geometry of differential equations
seminar\n\n\nAbstract\nIn my talk I am going to give an introduction to th
e theory of the moment map for the Hamiltonian group action on the symplec
tic manifolds with the focus on Hamiltonian reduction and integrable syste
ms. In particular\, I will show how to translate symmetries of the Hamilto
nian system to the first integrals using the moment map and what kind of s
ystems we may obtain by performing such reduction. As the main example\, I
will demonstrate how to obtain a rational Calogero system from the free p
article system on the cotangent bundle to the Lie algebra $su(n)$.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrei Smilga
DTSTART:20220518T162000Z
DTEND:20220518T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/66
DESCRIPTION:Title: Noncommutative quantum mechanical systems associated with Lie algebra
s\nby Andrei Smilga as part of Geometry of differential equations semi
nar\n\n\nAbstract\nWe consider quantum mechanics on the noncommutative spa
ces characterized by the commutation relations\n$$ [x_a\, x_b] \\ =\\ i\\t
heta f_{abc} x_c\\\,\, $$\nwhere $f_{abc}$ are the structure constants of
a Lie algebra. We note that this problem can be reformulated as an ordinar
y quantum problem in a commuting {\\it momentum} space. The coordinates ar
e then represented as linear differential operators $\\hat x_a = -i \\hat
D_a = -iR_{ab} (p)\\\, \\partial /\\partial p_b $. Generically\, the matri
x $R_{ab}(p)$ represents a certain infinite series over the deformation pa
rameter $\\theta$: $R_{ab} = \\delta_{ab} + \\ldots$. The deformed Hamilto
nian\, $\\hat H \\ =\\ - \\frac 12 \\hat D_a^2\\\,\, $ describes the moti
on along the corresponding group manifolds with the characteristic size of
order $\\theta^{-1}$. Their metrics are also expressed into certain infi
nite series in $\\theta$.\n\nFor the algebras $su(2)$ and $u(2)$\, it has
been possible to represent the operators $\\hat x_a$ in a simple finite fo
rm. A byproduct of our study are new nonstandard formulas for the metrics
on $SU(2) \\equiv S^3$ and on $SO(3)$.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Talalaev (Moscow State University)
DTSTART:20221005T162000Z
DTEND:20221005T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/67
DESCRIPTION:Title: Category of braided sets\, extensions and 2-analogues\nby Dmitry
Talalaev (Moscow State University) as part of Geometry of differential equ
ations seminar\n\nLecture held in room 303 of the Independent University o
f Moscow.\n\nAbstract\nA braided set is the same thing as a solution of th
e set-theoretic Yang-Baxter equation. It is important to rephrase this in
a categorical language from the point of view of natural questions of morp
hisms\, extensions and simple objects in this family. I will tell about se
veral results in the problem of constructing extensions of braided sets an
d how this problem can be generalized to 2-braided categories\, how to bui
ld extensions of sets with solutions of the Zamolodchikov tetrahedron equa
tion.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kushner
DTSTART:20221109T162000Z
DTEND:20221109T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/68
DESCRIPTION:Title: On the integration of suspension filtration equations and thrombus fo
rmation\nby Alexei Kushner as part of Geometry of differential equatio
ns seminar\n\nLecture held in room 303 of the Independent University of Mo
scow.\n\nAbstract\nThe problem of one-dimensional filtration of a suspensi
on in a porous medium is considered. The process is described by a hyperbo
lic system of two first-order differential equations. This system is reduc
ed by a change of variables to the symplectic equation of the Monge-Ampèr
e type. It is noteworthy that this symplectic equation cannot be reduced t
o a linear wave equation by a symplectic transformation (the Lychagin-Rubt
sov theorem works here)\, but it can be done by a contact transformation.
This made it possible to find its exact general solution and exact solutio
ns of the original system. The solution of the initial-boundary value prob
lem and the Cauchy problem are constructed.\n\nJoint work with Svetlana Mu
khina.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20221012T162000Z
DTEND:20221012T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/69
DESCRIPTION:Title: On interplay between jet and information geometries\nby Valentin
Lychagin as part of Geometry of differential equations seminar\n\nLecture
held in room 303 of the Independent University of Moscow.\n\nAbstract\nWe
will consider the procedure of measurement of random vectors\, operators a
nd tensors from the double point of view: pure probabilistic and geometric
al. Using the principle of minimum information gain\, we reformulate the p
robabilistic approach as studies in the geometry of jet spaces over the ma
nifolds of extreme measures. Moreover\, the procedure of a measurement its
elf becomes equivalent to study various geometrical structures on integral
manifolds of the Cartan distribution. We will illustrate all of this for
the case of thermodynamics of real gases and phase transitions of the firs
t and second orders.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20221019T162000Z
DTEND:20221019T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/70
DESCRIPTION:Title: Homogeneous Hamiltonian operators\, projective geometry and integrabl
e systems\nby Raffaele Vitolo (Università del Salento) as part of Geo
metry of differential equations seminar\n\n\nAbstract\nFirst-order homogen
eous Hamiltonian operators play a central role in the Hamiltonian formulat
ion of quasilinear systems of PDEs. They have well-known differential-geom
etric invariance properties which find application in the theory of Froben
ius manifolds. In this talk we will show that second and third order homog
eneous Hamiltonian operators are invariant under reciprocal transformation
s of projective type\, thus allowing for a projective classification of th
e operators. Then\, we will describe how the above operators generate know
n and new integrable systems\, and discuss the invariance properties of th
e systems under projective transformations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roberto D'Onofrio
DTSTART:20221116T162000Z
DTEND:20221116T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/73
DESCRIPTION:Title: Monge-Ampère geometry and semigeostrophic equations\nby Roberto
D'Onofrio as part of Geometry of differential equations seminar\n\n\nAbstr
act\nSemigeostrophic equations are a central model in geophysical fluid dy
namics designed to represent large-scale atmospheric flows. Their remarkab
le duality structure allows for a geometric approach through Lychagin's th
eory of Monge-Ampère equations. We extend seminal earlier work on the sub
ject by studying the properties of an induced metric on solutions\, unders
tood as Lagrangian submanifolds of the phase space. We show the interplay
between singularities\, elliptic-hyperbolic transitions\, and the metric s
ignature through a few visual examples.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Talalaev
DTSTART:20221130T162000Z
DTEND:20221130T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/74
DESCRIPTION:Title: Zamolodchikov Tetrahedron equation\nby Dmitry Talalaev as part of
Geometry of differential equations seminar\n\nLecture held in room 303 of
the Independent University of Moscow.\n\nAbstract\nThe main subject of th
e talk is the Zamolodchikov tetrahedron equation\, which is the next n-sim
plex equation after the Yang-Baxter equation. This equation finds its embo
diments in the theory of cluster manifolds\, exactly-solvable models of st
atistical physics in dimension 3\, as well as the theory of invariants of
2-knots\, that is\, classes of isotopies of embeddings of a two-dimensiona
l surface in a 4-dimensional space.\n\nThe main focus of the report will b
e on the definition of this class of equations in terms of the hypercube f
ace coloring problem\, the cohomology complex associated with each solutio
n of the n-simplex equation. We will discuss how these definitions are rea
lized in the case of n=3\, that is\, in the case of the tetrahedron equati
on\, and some interesting classes of solutions to this equation arising in
modern mathematics.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-Pierre Magnot
DTSTART:20221123T162000Z
DTEND:20221123T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/75
DESCRIPTION:Title: New perspectives for generalized Kadomtsev-Petviashvili hierarchies\nby Jean-Pierre Magnot as part of Geometry of differential equations se
minar\n\n\nAbstract\nIn the setting of diffeological differential algebras
\, we first expose step by step how the classical algebraic construction o
f the solution of the (classical) Kadomtsev-Petviashvili hierarchy can be
extended in order to get well-posedness for Kadomtsev-Petviashvili hierarc
hies in this generalized setting. Of course\, we give a short exposition o
f the necessary notions in diffeologies for non-specialists of this topic.
\n\nThen\, we discuss the Hamiltonian formulation in a refreshed way. Fina
lly\, we deduce the corresponding Kadomtsev-Petviashvili equations\, first
in an abstract formulation\, and in a series of examples.\n\nReferences:<
/br>\narXiv:1007.3543\n
https://dx.doi.
org/10.1080/14029251.2017.1418057\narXiv:1608.03994\narXiv:2101.04523\, Mi tmf10046\narXiv:2
203.07062\narXiv:2212.
07583\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20221221T162000Z
DTEND:20221221T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/76
DESCRIPTION:Title: Chopping integrals of the full symmetric Toda system\, a new approach
\nby Georgy Sharygin as part of Geometry of differential equations sem
inar\n\nLecture held in room 303 of the Independent University of Moscow.\
n\nAbstract\nIn my talk I will try to answer the questions that has been c
ausing my anxiety for a rather long time: where do the additional integral
s of the full symmetric Toda system come from\, why they are rational and
what does all this have to do with "chopping". Even if we can use the AKS
method there remains the question\, why do the initial functions actually
commute (and whether it is possible to find other with the same property).
The known answers were concerned either with rather hard straightforward
computations\, or with the properties of a Gaudin system\; they look prett
y complicated. In my talk I will show how one can obtain these integrals w
ith the help of some simple differential operators (in the manner of the a
rgument shift method). Besides this\, we will discuss some other possible
integrals as well as the method to solve the corresponding flows by QR dec
omposition.\n\nThe talk is based on a common work with Yu. Chernyakov and
D. Talalaev.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20221207T162000Z
DTEND:20221207T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/77
DESCRIPTION:Title: On normal forms of differential operators\nby Valentin Lychagin a
s part of Geometry of differential equations seminar\n\nLecture held in ro
om 303 of the Independent University of Moscow.\n\nAbstract\nIn this talk\
, we classify linear (as well as some special nonlinear) scalar diff\neren
tial operators of order $k$ on $n$-dimensional manifolds with respect to t
he diffeomorphism pseudogroup.\n
Cases\, when $k = 2$\, $\\forall n$\
, and $k = 3$\, $n = 2$\, were discussed before\, and now we consider case
s $k\\ge5$\, $n = 2$ and $k\\ge4$\, $n = 3$ and $k\\ge3$\, $n\\ge4$. In al
l these cases\, the fields of rational differential invariants are generat
ed by the 0-order invariants of symbols.\n\nThus\, at first\, we consider
the classical problem of Gl-invariants of $n$-ary forms. We'll illustrate
here the power of the differential algebra approach to this problem and sh
ow how to find the rational Gl-invariants of $n$-are forms in a constructi
ve way.\n\nAfter all\, we apply the $n$ invariants principle in order to g
et (local as well as global) normal forms of linear operators with respect
to the diffeomorphism pseudogroup.\n\nDepending on available time\, we sh
ow how to extend all these results to some classes of nonlinear operators.
\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Tsarev\, Folkert Müller-Hoissen\, Dmitry Millionschikov\,
Boris Konopelchenko
DTSTART:20221214T140000Z
DTEND:20221214T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/78
DESCRIPTION:Title: One day workshop in honor of Maxim Pavlov's 60th birthday\nby Ser
gey Tsarev\, Folkert Müller-Hoissen\, Dmitry Millionschikov\, Boris Konop
elchenko as part of Geometry of differential equations seminar\n\nLecture
held in room 303 of the Independent University of Moscow.\n\nAbstract\n\n\nSpeaker: Sergey Tsarev (Krasnoyarsk)\n\nTitle: Hydrodynamic type systems and beyond: a long way towards
integrability with Maxim Pavlov\n\nSpeaker: Folkert Mül
ler-Hoissen (Göttingen)\n\nTitle: A relative of the NLS
equation revisited\n\nSpeaker: Dmitry Millionschikov (Mos
cow)\n\nTitle: Growth of Lie algebras and integrability\n
\nSpeaker: Boris Konopelchenko\n\nTitle:
Multi-dimensional MAS-Pavlov-Jordan chain and its reduct\nions\n\nThe abs
tracts\, slides\, and videos can be found on the page https://gdeq.org/Pav
lov60\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Orlov
DTSTART:20230208T162000Z
DTEND:20230208T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/79
DESCRIPTION:Title: Hurwitz numbers\, matrix models\, commuting operators\nby Alexand
er Orlov as part of Geometry of differential equations seminar\n\nLecture
held in room 303 of the Independent University of Moscow.\n\nAbstract\nWe
will analyze how matrix models are related to arbitrary Hurwitz numbers. T
here are equivalent descriptions using\n\n(a) differential operators\n\n(b
) oscillatory algebra and bosonic Fock space.\n\nCommuting sets of such op
erators will be presented. This is a modification of Calogero's quantum Ha
miltonians at a special point.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexey Samokhin
DTSTART:20230222T162000Z
DTEND:20230222T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/80
DESCRIPTION:Title: On perturbations retaining conservation laws of differential equation
s\nby Alexey Samokhin as part of Geometry of differential equations se
minar\n\n\nAbstract\nThe talk deals with perturbations of the equation tha
t have a number of conservation laws. When a small term is added to the eq
uation its conserved quantities usually decay at individual rates\, a phen
omenon known as a selective decay. These rates are described by the simple
law using the conservation laws' generating functions and the added term.
Yet some perturbation may retain a specific quantity(s)\, such as energy\
, momentum and other physically important characteristics of solutions. We
introduce a procedure for finding such perturbations and demonstrate it b
y examples including the KdV-Burgers equation and a system from magnetodyn
amics.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20230215T162000Z
DTEND:20230215T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/81
DESCRIPTION:Title: On invariants and equivalence differential operators under algebraic
Lie pseudogroups actions\nby Valentin Lychagin as part of Geometry of
differential equations seminar\n\nLecture held in room 303 of the Independ
ent University of Moscow.\n\nAbstract\nIt is the concluding talk on invari
ants and the equivalence of differential operators under actions of Lie ps
eudogroups. We'll show\, that under some natural algebraic restrictions on
Lie pseudogroups and nonlinearities of differential operators under consi
deration\, there is a reasonable description of their orbits under the Lie
pseudogroups\, as well as local model forms. Then\, the general approach
will be applied to the Cartan list of primitive Lie pseudogroups.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henrik Winther
DTSTART:20230301T162000Z
DTEND:20230301T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/82
DESCRIPTION:Title: Jet functors in noncommutative geometry\nby Henrik Winther as par
t of Geometry of differential equations seminar\n\n\nAbstract\nWe construc
t an infinite family of endofunctors $J_d^n$ on the category of left $A$-m
odules\, where $A$ is a unital associative algebra over a commutative ring
$k$\, equipped with an exterior algebra $\\Omega^\\bullet_d$. We prove th
at these functors generalize the corresponding classical notion of jet fun
ctors. The functor $J_d^n$ comes equipped with a natural transformation fr
om the identity functor to itself\, which plays the rôle of the classical
prolongation map. This allows us to define the notion of linear different
ial operator with respect to $\\Omega^{\\bullet}_d$. These retain most cla
ssical properties of differential operators\, and operators such as partia
l derivatives and connections belong to this class. Moreover\, we construc
t a functor of quantum symmetric forms $S^n_d$ associated to $\\Omega^\\bu
llet_d$\, and proceed to introduce the corresponding noncommutative analog
ue of the Spencer $\\delta$-complex. We give necessary and sufficient cond
itions under which the jet functor $J_d^n$ satisfies the jet exact sequenc
e\, $0\\rightarrow S^n_d \\rightarrow J_d^n \\rightarrow J_d^{n-1} \\right
arrow 0$. This involves imposing mild homological conditions on the exteri
or algebra\, in particular on the Spencer cohomology $H^{\\bullet\,2}$.\n\
nThis is a joint work with K. Flood and M. Mantegazza.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eugene Ferapontov (Loughborough University)
DTSTART:20230322T162000Z
DTEND:20230322T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/83
DESCRIPTION:Title: Quasilinear systems of Jordan block type\nby Eugene Ferapontov (L
oughborough University) as part of Geometry of differential equations semi
nar\n\n\nAbstract\nI will discuss integrability aspects of quasilinear sys
tems whose velocity matrix has a nontrivial Jordan block structure. I plan
to cover the following topics:\n\n- Integrable systems of Jordan bl
ock type and modified KP hierarchy\;
\n- Hamiltonian aspects of quas
ilinear systems of Jordan block type\;
\n- Example: delta-functional
reductions of the soliton gas equation.
\n
\nThe talk will be bas
ed on joint work with Lingling Xue\, Maxim Pavlov and Pierandrea Vergallo.
\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Dobrokhotov and Vladimir Nazaikinskii
DTSTART:20230329T162000Z
DTEND:20230329T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/84
DESCRIPTION:Title: Exact and asymptotic solutions of a system of nonlinear shallow water
equations in basins with gentle shores\nby Sergey Dobrokhotov and Vla
dimir Nazaikinskii as part of Geometry of differential equations seminar\n
\nLecture held in room 303 of the Independent University of Moscow.\n\nAbs
tract\nWe suggest an effective approximate method for constructing solutio
ns to problems with a free boundary for 1-D and 2-D-systems of nonlinear s
hallow water equations in basins with gentle shores. The method is a modif
ication (and pragmatic simplification) of the Carrier-Greenspan transforma
tion in the theory of 1-D shallow water over a flat sloping bottom. The re
sult is as follows: approximate solutions of nonlinear equations are expre
ssed through solutions of naively linearized equations via parametrically
defined functions. This allows us to describe the effects of waves run-up
on a shore and their splash. Among the applications we can mention tsunami
waves\, seiches and coastal waves. We also present a comparison of the ob
tained formulas with the V.A. Kalinichenko (Institute for Problems in Mech
anics RAS) experiment with standing Faraday waves in an extended basin wit
h gently sloping shores.\n\nJoint work with D. Minenkov.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20230315T162000Z
DTEND:20230315T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/85
DESCRIPTION:Title: Quasiderivations and commutative subalgebras of the algebra $U\\mathf
rak{gl}_n$\nby Georgy Sharygin as part of Geometry of differential equ
ations seminar\n\nLecture held in room 303 of the Independent University o
f Moscow.\n\nAbstract\nLet $\\mathfrak{gl}_n$ be the Lie algebra of $n\\ti
mes n$ matrices over a characteristic zero field $\\Bbbk$ (one can take $\
\Bbbk=\\mathbb R$ or $\\mathbb C$)\; let $S(\\mathfrak{gl}_n)$ be the Pois
son algebra of polynomial functions on $\\mathfrak{gl}_n^*$\, and $U\\math
frak{gl}_n$ the universal enveloping algebra of $\\mathfrak{gl}_n$. By Poi
ncaré-Birkhoff-Witt theorem $S(\\mathfrak{gl}_n)$ is isomorphic to the gr
aded algebra $gr(U\\mathfrak{gl}_n)$\, associated with the order filtratio
n on $U\\mathfrak{gl}_n$. Let $A\\subseteq S(\\mathfrak{gl}_n)$ be a Poiss
on-commutative subalgebra\; one says that a commutative subalgebra $\\hat
A\\subseteq U\\mathfrak{gl}_n$ is a $\\textit{quantisation}$ of $A$\, if i
ts image under the natural projection $U\\mathfrak{gl}_n\\to gr(U\\mathfra
k{gl}_n)\\cong S(\\mathfrak{gl}_n)$ is equal to $A$.\n\nIn my talk I will
speak about the so-called "argument shift" subalgebras $A=A_\\xi$ in $S(\\
mathfrak{gl}_n)$\, generated by the iterated derivations of central elemen
ts in $S(\\mathfrak{gl}_n)$ by a constant vector field $\\xi$. There exist
several ways to define a quantisation of $A_\\xi$\, most of them are rela
ted with the considerations of some infinite-dimensional Lie algebras. In
my talk I will explain\, how one can construct such quantisation of $A_\\x
i$ using as its generators iterated $\\textit{quasi-derivations}$ $\\hat\\
xi$ of $U\\mathfrak{gl}_n$. These operations are "quantisations" of the de
rivations on $S(\\mathfrak{gl}_n)$ and verify an analog of the Leibniz rul
e. In fact\, I will show that iterated quasiderivation of certain generati
ng elements in $U\\mathfrak{gl}_n$ are equal to the linear combinations of
the elements\, earlier constructed by Tarasov.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20230405T162000Z
DTEND:20230405T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/86
DESCRIPTION:Title: Lagrangian formalism and the intrinsic geometry of PDEs\nby Konst
antin Druzhkov as part of Geometry of differential equations seminar\n\n\n
Abstract\nThis report is an attempt to answer the following question. Wher
e exactly does a differential equation contain information about its varia
tional nature? Apparently\, in the general case\, the concept of a presymp
lectic structure as a closed variational 2-form may not be sufficient to d
escribe variational principles in terms of intrinsic geometry. I will intr
oduce the concept of an internal Lagrangian and relate it to the Vinogrado
v C-spectral sequence.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexei Kushner
DTSTART:20230426T162000Z
DTEND:20230426T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/87
DESCRIPTION:Title: Finite-dimensional dynamics of systems of evolutionary differential e
quations with many spatial variables\nby Alexei Kushner as part of Geo
metry of differential equations seminar\n\n\nAbstract\nThe main ideas of t
he theory of finite-dimensional dynamics were formulated in the 2000s in t
he works of B.S. Kruglikov\, V.V. Lychagin and O.V. Lychagina. These paper
s also found finite-dimensional dynamics of the Kolmogorov-Petrovsky-Pisku
nov and Korteweg-de Vries equations. This theory is a natural development
of the theory of dynamical systems. Finite-dimensional dynamics make it po
ssible to find families of solutions depending on a finite number of param
eters among all solutions of evolutionary differential equations. Namely\,
finite-dimensional submanifolds are constructed in the space of smooth fu
nctions that are invariant under the flow given by the evolution equation.
This removes the question of the existence of solutions\, since such subm
anifolds consist of solutions to ordinary differential equations\, and\, m
oreover\, gives a constructive method for finding them. Note that finite-d
imensional dynamics can exist for equations that do not have symmetries. T
he talk will present the results obtained by us for systems of evolutionar
y equations\, including those with many spatial variables.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Fels
DTSTART:20230510T162000Z
DTEND:20230510T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/88
DESCRIPTION:Title: Variational/Symplectic and Hamiltonian Operators\nby Mark Fels as
part of Geometry of differential equations seminar\n\n\nAbstract\nGiven a
differential equation (or system) $\\Delta$ = 0 the inverse problem in th
e calculus of variations asks if there is a multiplier function $Q$ such t
hat\n\\[Q\\Delta=E(L)\,\\]\nwhere $E(L)=0$ is the Euler-Lagrange equation
for a Lagrangian $L$. A solution to this problem can be found in principle
and expressed in terms of invariants of the equation $\\Delta$. The varia
tional operator problem asks the same question but $Q$ can now be a differ
ential operator as the following simple example demonstrates for the evolu
tion equation $u_t - u_{xxx} = 0$\,\n\\[D_x(u_t - u_{xxx}) = u_{tx}-u_{xxx
x}=E(-\\frac12(u_tu_x+u_{xx}^2)).\\]\nHere $D_x$ is a variational operator
for $u_t - u_{xxx} = 0$.\n\nThis talk will discuss how the variational op
erator problem can be solved in the case of scalar evolution equations and
how variational operators are related to symplectic and Hamiltonian opera
tors.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Svetlana Mukhina
DTSTART:20230607T162000Z
DTEND:20230607T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/89
DESCRIPTION:Title: Contact vs symplectic geometry\nby Svetlana Mukhina as part of Ge
ometry of differential equations seminar\n\nLecture held in room 303 of th
e Independent University of Moscow.\n\nAbstract\nThe report will show how
some symplectic Monge-Ampère type equations can be solved by applying con
tact transformations to them.\n\nAs is known\, symplectic Monge-Ampère eq
uations with two independent variables are locally symplectic equivalent t
o linear equations with constant coefficients if and only if the correspon
ding Nijenhuis bracket is zero (the Lychagin-Rubtsov theorem). Necessary a
nd sufficient conditions for the contact equivalence of the general (not n
ecessarily symplectic) Monge-Ampère linear equations were found by Kushne
r.\n\nUsing these results\, we consider the problem of constructing exact
solutions to some equations arising in filtration theory. We will consider
a model of unsteady displacement of oil by a solution of active reagents.
This model describes the process of oil extraction from hard-to-recover d
eposits. This model is described by a hyperbolic system of partial differe
ntial equations of the first order of the Jacobi type. Unknown functions a
re the water saturation and concentration of reagents in an aqueous soluti
on\, and independent variables are time and linear coordinate.\n\nWith the
help of symplectic and contact transformations\, it is possible to reduce
the model equations to a linear wave equation. The exact solution of this
system is obtained and the Cauchy problem is solved.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20230517T162000Z
DTEND:20230517T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/90
DESCRIPTION:Title: Overdetermined systems of PDEs related to representations of semi-sim
ple Lie algebras\nby Boris Doubrov as part of Geometry of differential
equations seminar\n\n\nAbstract\nWe explore a class of finite-type system
s of PDEs whose symbol is determined by an (arbitrary) irreducible represe
ntation of a graded semisimple Lie algebra.\n\nWe show that trivial equati
ons with such symbol correspond to rational homogeneous varieties\, non-tr
ivial linear equations define symbol-preserving deformations of such varie
ties. In particular\, we determine when such deformations exist. In terms
of the corresponding PDE system this corresponds to the question when comp
atibility conditions imply that the system is equivalent to trivial. The a
nswer to this question is given in terms of certain Lie algebra cohomology
\, which can be effectively computed using the results for the theory of s
emisimple Lie algebras.\n\nWe solve local equivalence problem for such sys
tems under fiber+symbol preserving transformations and show how this is re
lated to the projective geometry of submanifolds. Finally\, we discuss the
case of non-linear systems with the same symbol and show that under certa
in additional conditions their solution spaces admit remarkable geometric
structures.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maksim Gadzhiev and Alexander Kuleshov
DTSTART:20230531T162000Z
DTEND:20230531T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/91
DESCRIPTION:Title: Integrability of the problem of motion of a body with a fixed point i
n a flow of particles\nby Maksim Gadzhiev and Alexander Kuleshov as pa
rt of Geometry of differential equations seminar\n\nLecture held in room 3
03 of the Independent University of Moscow.\n\nAbstract\nThe problem of th
e motion\, in the free molecular flow of particles\, of a rigid body with
a fixed point is considered. The molecular flow is assumed to be sufficien
tly sparse\, there is no interaction between the particles. Based on the a
pproach proposed by V.V. Beletsky\, an expression is obtained for the mome
nt of forces acting on a body with a fixed point. It is shown that the equ
ations of motion of a body are similar to the classical Euler-Poisson equa
tions of motion of a heavy rigid body with a fixed point and are presented
in the form of classical Euler-Poisson equations in the case when the sur
face of a body is a sphere. The existence of the first integrals is discus
sed. Constraints on the system parameters are obtained under which there a
re integrable cases corresponding to the classical Euler-Poinsot\, Lagrang
e and Hess cases of integrability of the equations of motion of a heavy ri
gid body with a fixed point. The case when the surface of the body is an e
llipsoid is considered. Using the methods developed in the works of V.V. K
ozlov\, proved the absence of an integrable case in this problem\, similar
to the Kovalevskaya case.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20230913T162000Z
DTEND:20230913T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/92
DESCRIPTION:Title: On equivalence of planar webs\nby Valentin Lychagin as part of Ge
ometry of differential equations seminar\n\nLecture held in room 303 of th
e Independent University of Moscow.\n\nAbstract\nIn this talk\, I'll discu
ss the equivalence problem for planar d-webs.\n\nTo this end\, the fields
of rational differential invariants will be found\, and natural geometric
objects related to planar webs will be discussed.\n\nThe cases of d-webs w
ith d<6 will be discussed in detail.\n\nPlease download the formula file <
a href="https://gdeq.org/files/pl.pdf" title="pl.pdf">pl.pdf and keep
it handy during the talk so that the speaker can refer to it.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20230920T162000Z
DTEND:20230920T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/93
DESCRIPTION:Title: Internal Lagrangians as variational principles\nby Konstantin Dru
zhkov as part of Geometry of differential equations seminar\n\n\nAbstract\
nThe principle of stationary action deals with Lagrangians defined on jets
. However\, for some reason\, the intrinsic geometry of the corresponding
equations knows about their variational nature. It turns out that the expl
anation is quite simple: each stationary-action principle reproduces itsel
f in terms of the intrinsic geometry. More precisely\, each admissible Lag
rangian gives rise to a unique integral functional defined on some particu
lar class of submanifolds of the corresponding equations. Such submanifold
s can be treated as almost solutions since (informally speaking) they are
composed of initial-boundary conditions lifted to infinitely prolonged equ
ations. Intrinsic integral functionals produced by variational principles
are related to so-called internal Lagrangians. This relation allows us to
introduce the notion of stationary point of an internal Lagrangian\, formu
late the corresponding intrinsic version of Noether's theorem\, and discus
s the nondegeneracy of presymplectic structures of differential equations.
Despite the generality of the approach\, its application to gauge theorie
s proves to be challenging. Perhaps the construction needs some modificati
on in this case.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andronick Arutyunov
DTSTART:20230927T162000Z
DTEND:20230927T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/94
DESCRIPTION:Title: Derivations in group algebra bimodules\nby Andronick Arutyunov a
s part of Geometry of differential equations seminar\n\nLecture held in ro
om 303 of the Independent University of Moscow.\n\nAbstract\nIf one introd
uces a norm in a group algebra which is understood as a vector space and c
onsiders a closure over this norm\, a natural structure of a free bimodule
over a group ring arises. The most natural example is $\\ell_p(G)$\, for
$p \\geq 1$. This structure makes it natural to consider the problem of de
scribing derivations with values in such bimodules\, which I will talk abo
ut. A "character" approach will be used\, which consists in identifying th
e derivations with characters on a suitable category (in our case\, the gr
oupoid of adjoint action of a group on itself)\, and further study is alre
ady underway with the active use of combinatorial methods.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20231011T162000Z
DTEND:20231011T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/95
DESCRIPTION:Title: Extrinsic geometry and linear differential equations of SL(3)-type\nby Boris Doubrov as part of Geometry of differential equations seminar\
n\n\nAbstract\nAs an application of the general theory on extrinsic geomet
ry\, we investigate extrinsic geometry of submanifolds in flag varieties a
nd systems of linear PDEs for a class of special interest associated with
the adjoint representation of SL(3). It may be seen as a contact generaliz
ation of the classical description of surfaces in P^3 in terms of two line
ar PDEs of second order.\n\nWe carry out a complete local classification o
f the homogeneous structures in this class. As a result\, we find 7 kinds
of new systems of linear PDE's of second order on a 3-dimensional contact
manifold each of which has a solution space of dimension 8. Among them the
re are included a system of PDE's called contact Cayley's surface and one
which has SL(2) symmetry.\n\nJoint work with Tohru Morimoto.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuri Sachkov
DTSTART:20231025T162000Z
DTEND:20231025T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/96
DESCRIPTION:Title: Lorentzian geometry in the Lobachevsky plane\nby Yuri Sachkov as
part of Geometry of differential equations seminar\n\n\nAbstract\nWe consi
der left-invariant Lorentzian problems on the group of proper affine funct
ions on the line. These problems have constant sectional curvature\, thus
are locally isometric to standard constant curvature Lorentzian manifolds
(Minkowski space\, de Sitter space\, and anti-de Sitter space).\n\nFor the
se problems\, the attainability set is described\, existence of optimal tr
ajectories is studied\, a parameterization of Lorentzian length maximizers
is obtained\, and Lorentzian distance and spheres are described.\n\nFor z
ero curvature problem a global isometry into a half-plane of Minkowski pla
ne is constructed.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Pavlov
DTSTART:20231018T162000Z
DTEND:20231018T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/97
DESCRIPTION:Title: Dubrovin paradigm and beyond\nby Maxim Pavlov as part of Geometry
of differential equations seminar\n\nLecture held in room 303 of the Inde
pendent University of Moscow.\n\nAbstract\nThe paradigm proposed by Boris
Dubrovin\, consisted of two parts: description of Frobenius manifolds + "r
ecovery" of an infinite set of dispersion corrections with the requirement
of preservation of integrability in the sense of existence of the Lax rep
resentation.\n\nThe talk will propose infinitely many alternatives to the
Frobenius manifolds.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20231108T162000Z
DTEND:20231108T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/98
DESCRIPTION:Title: Dispersionless integrable systems in dimension 5\nby Boris Krugli
kov (UiT the Arctic University of Norway) as part of Geometry of different
ial equations seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20231101T162000Z
DTEND:20231101T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/99
DESCRIPTION:Title: Extensions of Lie algebras and integrability of some equations of flu
id dynamics and magnetohydrodynamics.\nby Oleg Morozov as part of Geom
etry of differential equations seminar\n\nLecture held in room 303 of the
Independent University of Moscow.\n\nAbstract\nWe find the twisted extensi
on of the symmetry algebra of the 2D Euler equation in the vorticity form
and use it to construct new Lax representation for this equation. Then we
consider the transformation Lie-Rinehart algebras generated by finite-dime
nsional subalgebras of the symmetry algebra and employ them to derive a fa
mily of Lax representations for the Euler equation. The family depends on
functional parameters and contains a non-removable spectral parameter. Fur
thermore we exhibit Lax representations for the reduced magnetohydrodynami
cs equations (or the Kadomtsev-Pogutse equations)\, the ideal magnetohydro
dynamics equations\, the quasigeostrophic two-layer model equations\, and
the Charney-Obukhov equation for the ocean.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sergey Agafonov
DTSTART:20231206T162000Z
DTEND:20231206T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/100
DESCRIPTION:Title: Hexagonal circular 3-webs with polar curves of degree three\nby
Sergey Agafonov as part of Geometry of differential equations seminar\n\nL
ecture held in room 303 of the Independent University of Moscow.\n\nAbstra
ct\nLie sphere geometry describes circles on the unit sphere by polar poin
ts of these circles. Therefore a one parameter family of circles correspon
ds to a curve and a 3-web of circles\, i.e.\, 3 foliations by circles\, is
fixed by 3 curves. We call the union of these curves the polar curve and
show how analysis of the singular set of hexagonal 3-webs helps to classif
y circular hexagonal 3-webs with polar curves of degree 3. Many of the fou
nd webs are new. The presented results mark the progress in the Blaschke-B
ol problem posed almost one hundred years ago. More detail in arXiv:2306.11707.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irina Bobrova
DTSTART:20231122T162000Z
DTEND:20231122T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/101
DESCRIPTION:Title: On a classification of non-Abelian Painlevé equations\nby Irina
Bobrova as part of Geometry of differential equations seminar\n\n\nAbstra
ct\nThe famous Painlevé equations define the most general special functio
ns and appear ubiquitously in integrable models. Since the latter have bee
n intensively studied in the matrix or\, more general\, non-Abelian case\,
examples of non-Abelian Painlevé equations arise.\n\nWe will discuss the
problem of classifying such equations. This talk is based on a series of
papers joint with Vladimir Sokolov and an ongoing project with Vladimir Re
takh\, Vladimir Rubtsov\, and Georgy Sharygin.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20231129T162000Z
DTEND:20231129T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/102
DESCRIPTION:Title: Presymplectic minimal models of local gauge theories\nby Maxim G
rigoriev as part of Geometry of differential equations seminar\n\nLecture
held in room 303 of the Independent University of Moscow.\n\nAbstract\nWe
describe how the BV-AKSZ construction (or\, more generally\, finite dimens
ional symplectic gauge PDE) can be extended to generic local gauge field t
heories including non-topological and non-diffeomorphism-invariant ones. T
he minimal formulation of this sort has a finite-dimensional target space
which is a pre Q-manifold equipped with a compatible presymplectic structu
re. The nilpotency condition for the homological vector field is replaced
with a presymplectic version of the classical BV master equation. Given su
ch a presymplectic BV-AKSZ formulation\, it defines a standard jet-bundle
BV formulation by taking a symplectic quotient of the respective super jet
-bundle. In other words all the information about the underlying PDE\, its
Lagrangian\, and the corresponding BV formulation turns out to be encoded
in the finite dimensional graded geometrical object. Standard examples in
clude Yang-Mills\, Einstein gravity\, conformal gravity etc.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yagub Aliyev
DTSTART:20231213T162000Z
DTEND:20231213T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/103
DESCRIPTION:Title: Apollonius problem and caustics of an ellipsoid\nby Yagub Aliyev
as part of Geometry of differential equations seminar\n\n\nAbstract\nIn t
he talk we discuss Apollonius Problem on the number of normals of an ellip
se passing through a given point. It is known that the number is dependent
on the position of the given point with respect to a certain astroida. Th
e intersection points of the astroida and the ellipse are used to study th
e case when the given point is on the ellipse. The problem is then general
ized for 3-dimensional space\, namely for Ellipsoids. The number of concur
rent normals in this case is known to be dependent on the position of the
given point with respect to caustics of the ellipsoid. If the given point
is on the ellipsoid then the number of normals is dependent on position of
the point with respect to the intersections of the ellipsoid with its cau
stics. The main motivation of this talk is to find parametrizations and cl
assify all possible cases of these intersections.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20240214T162000Z
DTEND:20240214T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/104
DESCRIPTION:Title: On flows and filtration in the presence of thermodynamic processes:
generalized Navier-Stokes equations\nby Valentin Lychagin as part of G
eometry of differential equations seminar\n\nLecture held in room 303 of t
he Independent University of Moscow.\n\nAbstract\nWe plan to present a gen
eralization of the Navier-Stokes equations that describes the flows of hom
ogeneous multicomponent media in the presence of various thermodynamic pro
cesses\, especially chemical reactions. To achieve this\, we discuss the c
lassical thermodynamics of homogeneous multicomponent media and related th
ermodynamic processes (especially chemical reactions) from the contact geo
metry perspective.\n\nIt makes it possible to work with thermodynamic proc
esses as contact vector fields on a contact manifold and easily include in
the standard scheme of continuous mechanics. At the end\, we outline meth
ods of solving resulting equations and discuss possible singularities aris
ing in solutions.\n\nPlease download the formula file fl.pdf and keep it handy during the talk so that t
he speaker can refer to it.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20240221T162000Z
DTEND:20240221T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/105
DESCRIPTION:Title: Bi-Hamiltonian systems and projective geometry\nby Raffaele Vito
lo (Università del Salento) as part of Geometry of differential equations
seminar\n\n\nAbstract\nWe introduce the problem of classification of bi-H
amiltonian structures of KdV type under projective reciprocal transformati
ons. This problem leads naturally to studying the compatibility of a first
order localizable homogeneous Hamiltonian operator with a higher order ho
mogeneous Hamiltonian operator. We study the simplest second-order and thi
rd-order case where the orbit contains a constant operator. Computations w
ith weakly non local Hamiltonian operators have been made by techniques de
veloped in a previous paper.\n\nJoint work with P. Lorenzoni.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20240306T162000Z
DTEND:20240306T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/106
DESCRIPTION:Title: Deformation quantisation of the argument shift on $U\\mathfrak{gl}(n
)$\nby Georgy Sharygin as part of Geometry of differential equations s
eminar\n\nLecture held in room 303 of the Independent University of Moscow
.\n\nAbstract\nArgument shift algebras are the commutative subalgebras in
the symmetric algebras of a Lie algebra\, generated by the iterated deriva
tions (in direction of a constant vector field) of Casimir elements in $S\
\mathfrak{gl}(n)$. In particular all these quasiderivations do mutually co
mmute. In my talk I will show that a similar statement holds for the algeb
ra $U\\mathfrak{gl}(n)$ and its quasiderivations: namely\, I will show tha
t iterated quasiderivations of the central elements of $U\\mathfrak{gl}(n)
$ with respect to a constant quasiderivation do mutually commute. Our proo
f is based on the existence and properties of "Quantum Mischenko-Fomenko"
algebras\, and (which is worse) cannot be extended to other Lie algebras\,
but we believe that the fact that the "shift operator" can be raised to $
U\\mathfrak{gl}(n)$ is an interesting fact.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20240320T162000Z
DTEND:20240320T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/107
DESCRIPTION:Title: Besselland: autour and beyond. Part 1\nby Vladimir Rubtsov (Univ
ersité d'Angers) as part of Geometry of differential equations seminar\n\
n\nAbstract\nI shall try to explain – why it is interesting to study and
to generalize analytic solutions of modified Bessel equation. My talk is
based on ongoing projects in progress with V. Buchstaber\, I. Gaiur and D.
Van Straten.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20240327T162000Z
DTEND:20240327T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/108
DESCRIPTION:Title: Besselland: autour and beyond. Part 2\nby Vladimir Rubtsov (Univ
ersité d'Angers) as part of Geometry of differential equations seminar\n\
n\nAbstract\nI shall try to explain – why it is interesting to study and
to generalize analytic solutions of modified Bessel equation. My talk is
based on ongoing projects in progress with V. Buchstaber\, I. Gaiur and D.
Van Straten.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gerard Helminck
DTSTART:20240417T162000Z
DTEND:20240417T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/109
DESCRIPTION:Title: A construction of solutions of an integrable deformation of a commut
ative Lie algebra of skew Hermitian $\\mathbb{Z}\\times\\mathbb{Z}$-matric
es\nby Gerard Helminck as part of Geometry of differential equations s
eminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20240403T162000Z
DTEND:20240403T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/110
DESCRIPTION:Title: Besselland: autour and beyond. Part 3\nby Vladimir Rubtsov (Univ
ersité d'Angers) as part of Geometry of differential equations seminar\n\
n\nAbstract\nContinuation of the talks held on 20 and 27 March. \n\nI shal
l try to explain – why it is interesting to study and to generalize anal
ytic solutions of modified Bessel equation. My talk is based on ongoing pr
ojects in progress with V. Buchstaber\, I. Gaiur and D. Van Straten.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20240501T162000Z
DTEND:20240501T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/111
DESCRIPTION:Title: Internal Lagrangians and gauge systems\nby Konstantin Druzhkov a
s part of Geometry of differential equations seminar\n\n\nAbstract\nIn cla
ssical mechanics\, the Hamiltonian formalism is given in terms of instanta
neous phase spaces of mechanical systems. This explains why it can be inte
rpreted as an encapsulation of the Lagrangian formalism into the intrinsic
geometry of equations of motion. This observation can be generalized to t
he case of arbitrary variational equations. To do this\, we describe insta
ntaneous phase spaces using the intrinsic geometry of PDEs. The descriptio
n is given by the lifts of involutive codim-1 distributions from the base
of a differential equation viewed as a bundle with a flat connection (Cart
an distribution). Such lifts can be considered differential equations\, wh
ich one can regard as gauge systems. They encode instantaneous phase space
s. In addition\, each Lagrangian of a variational system generates a uniqu
e element of a certain cohomology of the system. We call such elements int
ernal Lagrangians. Internal Lagrangians can be varied within classes of pa
ths in the instantaneous phase spaces. This fact yields a direct (non-cova
riant) reformulation of the Hamiltonian formalism in terms of the intrinsi
c geometry of PDEs. Finally\, the non-covariant internal variational princ
iple gives rise to its covariant child.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eivind Schneider
DTSTART:20240515T162000Z
DTEND:20240515T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/112
DESCRIPTION:Title: Invariant divisors and equivariant line bundles\nby Eivind Schne
ider as part of Geometry of differential equations seminar\n\n\nAbstract\n
Scalar relative invariants play an important role in the theory of group a
ctions on a manifold as their zero sets are invariant hypersurfaces. Relat
ive invariants are central in many applications\, where they often are tre
ated locally\, since an invariant hypersurface is not necessarily the locu
s of a single function. Our aim is to outline a global theory of relative
invariants in the complex analytic setting. For a Lie algebra $\\mathfrak{
g}$ of holomorphic vector fields on a complex manifold $M$\, any holomorph
ic $\\mathfrak{g}$-invariant hypersurface is given in terms of a $\\mathfr
ak{g}$-invariant divisor. This generalizes the classical notion of scalar
relative $\\mathfrak{g}$-invariant. Since any $\\mathfrak{g}$-invariant di
visor gives rise to a $\\mathfrak{g}$-equivariant line bundle\, we investi
gate the group $\\mathrm{Pic}_{\\mathfrak{g}}(M)$ of $\\mathfrak{g}$-equiv
ariant line bundles. A cohomological description of $\\mathrm{Pic}_{\\math
frak{g}}(M)$ is given in terms of a double complex interpolating the Cheva
lley-Eilenberg complex for $\\mathfrak{g}$ with the Čech complex of the s
heaf of holomorphic functions on $M$. In the end we will discuss applicati
ons of the theory to jet spaces and differential invariants.\n\nThe talk i
s based on joint work with Boris Kruglikov (arXiv:2404.19439).\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20240522T162000Z
DTEND:20240522T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/113
DESCRIPTION:Title: Gauge PDEs on manifolds with boundaries and asymptotic symmetries\nby Maxim Grigoriev as part of Geometry of differential equations semina
r\n\n\nAbstract\nWe propose a framework to study local gauge theories on m
anifolds with boundaries and their asymptotic symmetries\, which is based
on representing them as so-called gauge PDEs. These objects extend the con
ventional BV-AKSZ sigma-models to the case of not necessarily topological
and diffeomorphism invariant systems and are known to behave well when res
tricted to submanifolds and boundaries. We introduce the notion of gauge P
DE with boundaries\, which takes into account generic boundary conditions\
, and apply the framework to asymptotically flat gravity. In so doing\, we
start with a suitable representation of gravity as a gauge PDE with bound
aries\, which implements the Penrose description of asymptotically simple
spacetimes. We then derive the minimal model of the gauge PDE induced on t
he boundary and observe that it provides the Cartan (frame-like) descripti
on of a (curved) conformal Carollian structure on the boundary. Furthermor
e\, imposing a version of the familiar boundary conditions in the induced
boundary gauge PDE\, leads immediately to the conventional BMS algebra of
asymptotic symmetries.\n\nMore references: arXiv:2212.11350\;\narXiv:1207.3439\, arXiv:
1305.0162\, arXiv:1903.0282
0\, arXiv:1009.0190\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20240911T162000Z
DTEND:20240911T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/114
DESCRIPTION:Title: On a structure of the first order differential operators\nby Val
entin Lychagin as part of Geometry of differential equations seminar\n\nLe
cture held in room 303 of the Independent University of Moscow.\n\nAbstrac
t\nThe various geometrical structures associated with differential operato
rs of the first order shall be discussed as well as notions of singular an
d regular points. At the end normal forms of operators at regular points w
ill be presented.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pavel Bedrikovetsky (University of Adelaide)
DTSTART:20240918T162000Z
DTEND:20240918T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/115
DESCRIPTION:Title: Exact solutions and upscaling in conservation law systems\nby Pa
vel Bedrikovetsky (University of Adelaide) as part of Geometry of differen
tial equations seminar\n\n\nAbstract\nNumerous transport processes in natu
re and industry are described by $n\\times n$ conservation law systems $u_
t+f(u)_x=0$\, $u=(u^1\,\\dots\,u^n)$. This corresponds to upper scale\, li
ke rock or core scale in porous media\, column length in chemical engineer
ing\, or multi-block scale in city transport. The micro heterogeneity at l
ower scales introduces $x$- or $t$-dependencies into the large-scale conse
rvation law system\, like $f=f(u\,x)$ or $f(u\,t)$. Often\, numerical micr
o-scale modelling highly exceeds the available computational facilities in
terms of calculation time or memory. The problem is a proper upscaling: h
ow to "average" the micro-scale $x$-dependent $f(u\,x)$ to calculate the u
pper-scale flux $f(u)$?\n\nWe present general case for $n=1$ and several s
ystems for $n=2$ and $3$. The key is that the Riemann invariant at the mic
roscale is the "flux" rather than "density". It allows for exact solutions
of several 1D problems: "smoothing" of shocks and "sharpening" of rarefac
tion waves into shocks due to microscale $x$- and $t$-dependencies\, flows
in piecewise homogeneous media. It also allows formulating an upscaling a
lgorithm based on the analytical solutions and its invariant properties.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Morozov
DTSTART:20241009T162000Z
DTEND:20241009T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/116
DESCRIPTION:Title: Lax representations for Euler equations of ideal hydrodynamics\n
by Oleg Morozov as part of Geometry of differential equations seminar\n\n\
nAbstract\nI will discuss Lax representations with non-removable parameter
s for the Euler equations of ideal hydrodynamics on a 2D Riemannian manifo
ld and for the 3D Euler-Helmholtz equations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20241106T162000Z
DTEND:20241106T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/117
DESCRIPTION:Title: Remarkable properties of the full symmetric Toda system\nby Geor
gy Sharygin as part of Geometry of differential equations seminar\n\nLectu
re held in room 303 of the Independent University of Moscow.\n\nAbstract\n
A full symmetric Toda system is a Hamiltonian dynamical system on the spac
e of symmetric real matrices with zero trace\, generalizing the usual open
Toda chain. This system is given by the Lax equation $\\dot L=[L\,M(L)]$\
, where $M(L)$ is the (naive) antisymmetrization of the symmetric matrix $
L$: the difference of its super and subdiagonal parts (with zeros on the d
iagonal). The Hamiltonianity of this system comes from the identification
of the space of symmetric matrices with the space dual to the algebra of u
pper triangular matrices\, with the Hamilton function being $1/2Tr(L^2)$.
This system can be further generalized to obtain systems on the spaces of
"generalized symmetric matrices"\, the symmetric components of the Cartan
expansion of the semi-simple real Lie algebras. In a somewhat unexpected w
ay\, all these systems turn out to be integrable (in the sense of having a
sufficiently large commutative algebra of first integrals) and possess a
number of remarkable properties which I will discuss: their trajectories a
lways connect fixed points corresponding to the elements of the Weyl group
of the original Lie algebra\, and two such points are connected if and on
ly if the elements of the Weyl group are comparable in Bruhat order\; in t
he case of a system on spaces of generalized symmetric matrices\, this pro
perty allows one to describe the intersections of the real Bruhat cells\;
this system has a large set of symmetries (sufficient for it to be Lie-Bia
nchi integrable)\; its additional first integrals can be obtained by a "cu
t" procedure\, and the trajectories of the corresponding Hamiltonian field
s can be obtained by the QR decomposition\; if time permits\, I will descr
ibe alternative families of first integrals (commutative and non-commutati
ve)\; finally\, I will describe a way to lift the extra first integrals of
the "cut" into the universal enveloping algebra with commutativity preser
ved.\n\nThe talk is based on a series of works by the author jointly with
Yu. Chernyakov\, A. Sorin and D. Talalaev.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Zheglov
DTSTART:20241113T162000Z
DTEND:20241113T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/118
DESCRIPTION:Title: Normal forms for differential operators\nby Alexander Zheglov as
part of Geometry of differential equations seminar\n\nLecture held in roo
m 303 of the Independent University of Moscow.\n\nAbstract\nIn my talk I'l
l give an overview of the results obtained by me\, as well as jointly with
co-authors\, related to the problem of classifying commuting (scalar) d
ifferential\, or more generally\, differential-difference or integral-di
fferential operators in several variables.\n\nConsidering such rings as su
brings of a certain complete non-commutative ring $\\hat{D}_n^{sym}$ (not
the known ring of formal pseudo-differential operators!)\, the normal fo
rms of differential operators mentioned in the title are obtained after co
njugation by some invertible operator ("Schur operator")\, calculated with
the help of one of the operators in a ring. Normal forms of commuting
operators are polynomials with constant coefficients in the differe
ntiation\, integration and shift operators\, which have a finite order in
each variable\, and can be effectively calculated for any given commuting
operators.\n\nI'll talk about some recent applications of the theory of no
rmal forms: an effective parametrisation of torsion free sheaves with va
nishing cohomologies on a projective curve\, and a correspondence between
solutions to the string equation and pairs of commuting ordinary differen
tial operators of rank one.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yasushi Ikeda
DTSTART:20241120T130000Z
DTEND:20241120T144000Z
DTSTAMP:20260404T111417Z
UID:GDEq/119
DESCRIPTION:Title: Quantum argument shifts in general linear Lie algebras\nby Yasus
hi Ikeda as part of Geometry of differential equations seminar\n\n\nAbstra
ct\nArgument shift algebras in $S(g)$ (where $g$ is a Lie algebra) are Poi
sson commutative subalgebras (with respect to the Lie-Poisson bracket)\, g
enerated by iterated argument shifts of Poisson central elements. Inspired
by the quantum partial derivatives on $U(gl_d)$ proposed by Gurevich\, Py
atov\, and Saponov\, I and Georgy Sharygin showed that the quantum argumen
t shift algebras are generated by iterated quantum argument shifts of cent
ral elements in $U(gl_d)$. In this talk\, I will introduce a formula for c
alculating iterated quantum argument shifts and generators of the quantum
argument shift algebras up to the second order\, recalling the main theore
m.\n\nNote the non-standard start time!\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ekaterina Shemyakova
DTSTART:20241127T162000Z
DTEND:20241127T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/120
DESCRIPTION:Title: On differential operators generating higher brackets\nby Ekateri
na Shemyakova as part of Geometry of differential equations seminar\n\n\nA
bstract\nOn supermanifolds\, a Poisson structure can be either even\, corr
esponding to a Poisson bivector\, or odd\, corresponding to an odd Hamilto
nian quadratic in momenta. An odd Poisson bracket can also be defined by a
n odd second-order differential operator that squares to zero\, known as a
"BV-type" operator.\n\nA higher analog\, $P_\\infty$ or $S_\\infty$\, is
a series of brackets of alternating parities or all odd\, respectively\, t
hat satisfy relations that are higher homotopy analogs of the Jacobi ident
ity. These brackets are generated by arbitrary multivector fields or Hamil
tonians. However\, generating an $S_\\infty$-structure by a higher-order d
ifferential operator is not straightforward\, as this would violate the Le
ibniz identities. Kravchenko and others studied these structures\, and Vor
onov addressed the Leibniz identity issue by introducing formal $\\hbar$-d
ifferential operators.\n\nIn this talk\, we revisit the construction of an
$\\hbar$-differential operator that generates higher Koszul brackets on d
ifferential forms on a $P_\\infty$-manifold.\n\nIt is well known that a ch
ain map between the de Rham and Poisson complexes on a Poisson manifold at
the same time maps the Koszul bracket of differential forms to the Schout
en bracket of multivector fields. In the $P_\\infty$-case\, however\, the
chain map is also known\, but it does not connect the corresponding bracke
t structures. An $L_\\infty$-morphism from the higher Koszul brackets to t
he Schouten bracket has been constructed recently\, using Voronov's thick
morphism technique. In this talk\, we will show how to lift this morphism
to the level of operators.\n\nThe talk is partly based on joint work with
Yagmur Yilmaz.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20241218T162000Z
DTEND:20241218T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/122
DESCRIPTION:Title: Towards a theory of homotopy structures for differential equations:
First definitions and examples. Part 1\nby Vladimir Rubtsov (Universit
é d'Angers) as part of Geometry of differential equations seminar\n\n\nAb
stract\nWe define $A_\\infty$-algebra structures on horizontal and vertic
al cohomologies of (formally integrable) partial differential equations.\n
\nSince higher order $A_\\infty$-algebra operations are related to Massey
products\, our observation implies the existence of invariants for differe
ntial equations that go beyond conservation laws.\n\nWe also propose noti
ons of formality for PDEs\, and we present examples of formal equations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20241211T162000Z
DTEND:20241211T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/123
DESCRIPTION:Title: Invariant reduction for PDEs. I: Conservation laws of 1+1 systems of
evolution equations\nby Konstantin Druzhkov as part of Geometry of di
fferential equations seminar\n\n\nAbstract\nAmong various methods for cons
tructing exact solutions of partial differential equations\, the symmetry-
invariant approach is particularly noteworthy. This method is especially e
ffective in the case of point symmetries\, but when it comes to higher sym
metries\, additional steps are required to obtain invariant solutions. It
turns out that systems that describe symmetry-invariant solutions inherit
symmetry-invariant geometric structures even in the case of higher symmetr
ies. Moreover\, the reduction of invariant conservation laws of 1+1 system
s of evolution equations can be described as an algorithm and implemented
in Maple. Starting from invariant conservation laws\, we get constants of
invariant motion. They are analogs of first integrals of ODEs\, and one ca
n use them in the same way. In particular\, a sufficient number of indepen
dent constants of invariant motion allows one to integrate the correspondi
ng system for invariant solutions.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20250108T162000Z
DTEND:20250108T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/124
DESCRIPTION:Title: Towards a theory of homotopy structures for differential equations:
First definitions and examples. Part 2\nby Vladimir Rubtsov (Universit
é d'Angers) as part of Geometry of differential equations seminar\n\n\nAb
stract\nWe define $A_\\infty$-algebra structures on horizontal and vertic
al cohomologies of (formally integrable) partial differential equations.\n
\nSince higher order $A_\\infty$-algebra operations are related to Massey
products\, our observation implies the existence of invariants for differe
ntial equations that go beyond conservation laws.\n\nWe also propose noti
ons of formality for PDEs\, and we present examples of formal equations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacob Kryczka
DTSTART:20250219T162000Z
DTEND:20250219T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/125
DESCRIPTION:Title: Singularities and Bi-complexes for PDEs\nby Jacob Kryczka as par
t of Geometry of differential equations seminar\n\n\nAbstract\nMany moduli
spaces in geometry and physics\, like those appearing in symplectic topol
ogy\, quantum gauge field theory (e.g. in homological mirror symmetry and
Donaldson-Thomas theory) are constructed as parametrizing spaces of soluti
ons to non-linear partial differential equations modulo symmetries of the
underlying theory. These spaces are often non-smooth and possess multi non
-equidimensional components. Moreover\, when they may be written as inters
ections of higher dimensional components they typically exhibit singularit
ies due to non-transverse intersections. To account for symmetries and pro
vide a suitable geometric model for non-transverse intersection loci\, one
should enhance our mathematical tools to include higher and derived stack
s. Secondary Calculus\, due to A. Vinogradov\, is a formal replacement for
the differential calculus on the typically infinite dimensional space of
solutions to a non-linear partial differential equation and is centered ar
ound the study of the Variational Bi-complex of a system of equations. I
n my talk I will discuss a generalization in the setting of (relative) hom
otopical algebraic geometry for possibly singular PDEs.\n\nThis is based o
n a series of joint works with Artan Sheshmani and Shing-Tung Yau.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dmitry Rudinsky
DTSTART:20250226T162000Z
DTEND:20250226T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/126
DESCRIPTION:Title: Weak gauge PDEs\nby Dmitry Rudinsky as part of Geometry of diffe
rential equations seminar\n\nLecture held in room 303 of the Independent U
niversity of Moscow.\n\nAbstract\nGauge PDEs are flexible graded geometric
al objects that generalise AKSZ sigma models to the case of local gauge th
eories. However\, aside from specific cases - such as PDEs of finite type
or topological field theories - gauge PDEs are inherently infinite-dimensi
onal. It turns out that these objects can be replaced by finite dimensiona
l objects called weak gauge PDEs. Weak gauge PDEs are equipped with a vert
ical involutive distribution satisfying certain properties\, and the nilpo
tency condition for the homological vector field is relaxed so that it hol
ds modulo this distribution. Moreover\, given a weak gauge PDE\, it induce
s a standard jet-bundle BV formulation at the level of equations of motion
. In other words\, all the information about PDE and its corresponding BV
formulation turns out to be encoded in the finite-dimensional graded geome
trical object. Examples include scalar field theory and self-dual Yang-Mil
ls theory.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20250305T162000Z
DTEND:20250305T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/127
DESCRIPTION:Title: Invariant reduction for PDEs. II: The general mechanism\nby Kons
tantin Druzhkov as part of Geometry of differential equations seminar\n\n\
nAbstract\nGiven a local (point\, contact\, or higher) symmetry of a syste
m of partial differential equations\, one can consider the system that des
cribes the invariant solutions (the invariant system). It seems natural to
expect that the invariant system inherits symmetry-invariant geometric st
ructures in a specific way. We propose a mechanism of reduction of symmetr
y-invariant geometric structures\, which relates them to their counterpart
s on the respective invariant systems. This mechanism is homological and c
overs the stationary action principle and all terms of the first page of t
he Vinogradov C-spectral sequence. In particular\, it applies to invariant
conservation laws\, presymplectic structures\, and internal Lagrangians.
A version of Noether's theorem naturally arises for systems that describe
invariant solutions. Furthermore\, we explore the relationship between the
C-spectral sequences of a system of PDEs and systems that are satisfied b
y its symmetry-invariant solutions. Challenges associated with multi-reduc
tion under non-commutative symmetry algebras are also clarified.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20250312T162000Z
DTEND:20250312T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/128
DESCRIPTION:Title: Turbulence geometry and Navier-Stokes equations\nby Valentin Lyc
hagin as part of Geometry of differential equations seminar\n\nLecture hel
d in room 303 of the Independent University of Moscow.\n\nAbstract\nIt is
proposed to consider turbulent media and\, in particular\, random vector f
ields from a geometric point of view. This leads to a geometry similar to\
, but not identical to\, Finsler's.\n\nWe show that a turbulence generates
a horizontal differential symmetric 2-form on the tangent bundle\, which
we call the Mahalanobis metric.\n\nThus\, vector fields on the underlying
manifold generate Riemannian structures on it by the restriction of the Ma
halanobis metric on the graphs of vector fields.\n\nIn the case of so-call
ed Gaussian turbulences\, these Riemannian structures coincide and generat
e a unique Riemannian structure.\n\nMoreover\, similar to Finsler geometry
\, turbulence generates a nonlinear connection in the tangent bundle but t
he Mahalanobis metric generates an affine connection in the tangent bundle
.\n\nThis affine connection and the Mahalanobis metric give us everything
we need to construct the Navier-Stokes equations for turbulent media.\n\nW
e will present two implementations of this scheme: for the flow of ideal g
ases and plasma\, where turbulence is described by the Maxwell-Boltzmann d
istribution law\, and compare them with the standard Navier-Stokes equatio
ns.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20250319T162000Z
DTEND:20250319T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/129
DESCRIPTION:Title: Geometry of the full symmetric Toda system\nby Georgy Sharygin a
s part of Geometry of differential equations seminar\n\nLecture held in ro
om 303 of the Independent University of Moscow.\n\nAbstract\nFull symmetri
c Toda system is the Lax-type system $\\dot L=[M(L)\,L]$\, where the varia
ble $L$ is a real symmetric $n\\times n$ matrix and $M(L)=L_+-L_-$ denote
s its "naive" anti-symmetrisation\, i.e. the matrix constructed by taking
the difference of strictly upper- and lower-triangular parts $L_+$ and $L_
-$ of $L$. This system has lots of interesting properties: it is a Liouv
ille-integrable Hamiltonian system (with rational first integrals)\, it i
s also super-integrable (in the sense of Nekhoroshev)\, its singular point
s and trajectories represent the Hasse diagram of Bruhat order on permutat
ions group. Its generalizations to other semisimple real Lie algebras have
similar properties. In my talk I will sketch the proof of some of these p
roperties and will describe a construction of infinitesimal symmetries of
the Toda system. It turns out that there are many such symmetries\, their
construction depends on representations of $\\mathfrak{sl}_n$. As a bypr
oduct we prove that the full symmetric Toda system is integrable in the se
nse of Lie-Bianchi criterion.\n\nThe talk is based on a series of papers j
oint with Yu.Chernyakov\, D.Talalaev and A.Sorin.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20250402T162000Z
DTEND:20250402T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/131
DESCRIPTION:Title: The quantum argument shift method on $U\\mathfrak{gl}_n$\nby Geo
rgy Sharygin as part of Geometry of differential equations seminar\n\nLect
ure held in room 303 of the Independent University of Moscow.\n\nAbstract\
nThe term "argument shift method" is used for a simple and efficient metho
d to construct commutative subalgebras in Poisson algebras by deforming t
he Casimir elements in them. This method is primarily used to search for
Poisson commutative subalgebras in symmetric algebras of various Lie algeb
ras\; it is closely related with the bi-Hamiltonian induction (Lenard-Magr
i scheme). However little is known about the possible extension of this m
ethod to the quantum algebras associated with given Poisson algebras\; thi
s is true even for the symmetric algebra of a Lie algebra\, where the qua
ntization is well known (it is equal to the universal enveloping algebra).
I will tell about a particular case\, the algebra $U\\mathfrak{gl}_n$\,
for which one can find a shifting operator raising to this algebra the sh
ift on $S(\\mathfrak{gl}_n)$\, and prove that this operator verifies the
same condition as before: when used to deform the elements in the center o
f $U\\mathfrak{gl}_n$\, it yields a set of commuting elements.\n\nThe tal
k is partially based on joint works with Y.Ikeda.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20250409T162000Z
DTEND:20250409T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/132
DESCRIPTION:Title: On inverse variational problem: examples\nby Boris Kruglikov (Ui
T the Arctic University of Norway) as part of Geometry of differential equ
ations seminar\n\n\nAbstract\nInverse problem of the calculus of variation
s is a vast subject with many results. I will discuss some examples relate
d to ODEs\, making an emphasis on parametrized vs unparametrized problems.
\n\nThe simplest and most studied case is about systems of second order di
fferential equations\, also known as path geometries. Here I will mention
some results joint with Vladimir Matveev arXiv:2203.15029.\n\nThen I will discuss recent results join
t with Vladimir Matveev and Wijnand Steneker arXiv:2412.04890 about variationality of so-called confo
rmal geodesics. This system is given by third order differential equations
\, which makes it rather unconventional for traditional approaches. I will
also mention an on-going project using the invariant variational bicomple
x.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20250326T162000Z
DTEND:20250326T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/133
DESCRIPTION:Title: A non-trivial conservation law with a trivial characteristic\nby
Konstantin Druzhkov as part of Geometry of differential equations seminar
\n\n\nAbstract\nAs far as I am aware\, no nontrivial conservation laws sur
viving to the second page of Vinogradov's C-spectral sequence have been es
tablished. It turns out that presymplectic structures that cannot be descr
ibed in terms of cosymmetries produce such conservation laws for closely r
elated overdetermined systems. In particular\, the presymplectic structure
$D_x$ of the potential mKdV equation gives rise to such a conservation la
w for the overdetermined system $u_t = 4u_x^3 + u_{xxx}$\, $u_y = 0$. Whil
e this example is somewhat degenerate\, it may be one of the simplest syst
ems exhibiting this phenomenon.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/133/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evgeny Ferapontov (Loughborough University)
DTSTART:20250416T162000Z
DTEND:20250416T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/134
DESCRIPTION:Title: Lagrangian multiforms and dispersionless integrable systems\nby
Evgeny Ferapontov (Loughborough University) as part of Geometry of differe
ntial equations seminar\n\nLecture held in room 303 of the Independent Uni
versity of Moscow.\n\nAbstract\nWe demonstrate that interesting examples o
f Lagrangian multiforms appear naturally in the theory of multidimensional
dispersionless integrable systems as (a) higher-order conservation laws o
f linearly degenerate PDEs in 3D\, and (b) in the context of Gibbons-Tsare
v equations governing hydrodynamic reductions of heavenly type equations i
n 4D.\n\nBased on joint work with Mats Vermeeren.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/134/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20250423T162000Z
DTEND:20250423T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/135
DESCRIPTION:Title: Bi-Hamiltonian structures of WDVV-type\nby Raffaele Vitolo (Univ
ersità del Salento) as part of Geometry of differential equations seminar
\n\n\nAbstract\nWe study a class of nonlinear partial differential equatio
ns (PDEs) that admit the same bi-Hamiltonian structure as the Witten-Dijkg
raaf-Verlinde-Verlinde (WDVV) equations: a Ferapontov-type first-order Ham
iltonian operator and a homogeneous third-order Hamiltonian operator in a
canonical Doyle-Potëmin form\, which are compatible. Properties of these
systems and their classification in low dimension will be discussed.\n\nJo
int work with S. Opanasenko.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/135/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20250514T162000Z
DTEND:20250514T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/136
DESCRIPTION:Title: Equivalence of scalar ODEs under contact\, point and fiber-preservin
g transformations\nby Boris Doubrov as part of Geometry of differentia
l equations seminar\n\n\nAbstract\nIt is known that the equivalence proble
m for scalar ODEs of order 3 and higher can be solved via the construction
of a canonical Cartan connection. The invariants then appear as part of t
he curvature of this connection. This allows to describe explicitly all sc
alar ODEs of order 3 and higher that can be brought to the trivial equatio
n by contact transformations. The goal of this talk is to show how most of
this story can be extended to the equivalence of scalar ODEs under point
and fiber-preserving transformations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/136/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20250521T162000Z
DTEND:20250521T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/137
DESCRIPTION:Title: Background fields and symmetries in the gauge PDE approach\nby M
axim Grigoriev as part of Geometry of differential equations seminar\n\nLe
cture held in room 303 of the Independent University of Moscow.\n\nAbstrac
t\nWe develop an extension of the (presympletic) gauge PDE approach to des
cribe local gauge theories with background fields. It turns out that such
theories correspond to (presymplectic) gauge PDEs whose base spaces are ag
ain gauge PDEs describing background fields. As such\, the geometric struc
ture is that of a bundle over a bundle over a given spacetime. Gauge PDEs
over backgrounds arise naturally when studying linearisation\, coupling (g
auge) fields to background geometry\, gauging global symmetries\, etc. Les
s obvious examples involve parameterised systems\, Fedosov equations\, and
the so-called homogeneous (presymplectic) gauge PDEs. The latter are the
gauge-invariant generalisations of the familiar homogeneous PDEs and they
provide a very concise description of gauge fields on homogeneous spaces s
uch as higher spin gauge fields on Minkowski\, (A)dS\, and conformal space
s. Finally\, we briefly discuss how the higher-form symmetries and their g
auging fit into the framework using the simplest example of the Maxwell fi
eld.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/137/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20250924T162000Z
DTEND:20250924T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/138
DESCRIPTION:Title: Information geometry of turbulent media\nby Valentin Lychagin as
part of Geometry of differential equations seminar\n\nLecture held in roo
m 303 of the Independent University of Moscow.\n\nAbstract\nIn the talk\,
we plan to discuss a method of geometrization of statistics on the example
of random vectors and its application to turbulent media\, by which we un
derstand random vector fields on base manifolds.\n\nWe show that this appr
oach gives rise to various geometric structures on the tangent as well as
cotangent bundles.\n\nAmong these\, the most important is the Mahalanobis
metric on the tangent bundle\, which allows us to obtain all the necessary
ingredients for the description of flows in turbulent media.\n\nAs an ill
ustration of the method\, we consider its applications to flows of real ga
ses based on Maxwell-Boltzmann-Gibbs statistics.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/138/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20251112T162000Z
DTEND:20251112T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/139
DESCRIPTION:Title: Invariant reduction for PDEs. III: Poisson brackets\nby Konstant
in Druzhkov as part of Geometry of differential equations seminar\n\n\nAbs
tract\nI will show that\, under suitable conditions\, finite-dimensional s
ystems describing invariant solutions of PDEs inherit local Hamiltonian op
erators through the mechanism of invariant reduction\, which applies unifo
rmly to point\, contact\, and higher symmetries. The inherited operators e
ndow the reduced systems with Poisson bivectors that relate constants of i
nvariant motion to symmetries. The induced Poisson brackets agree with tho
se of the original systems\, up to sign. At the core of this construction
lies the interpretation of Hamiltonian operators as degree-2 conservation
laws of degree-shifted cotangent equations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Doubrov
DTSTART:20251001T162000Z
DTEND:20251001T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/140
DESCRIPTION:Title: Geometry of CR manifolds via finite type systems of PDEs\nby Bor
is Doubrov as part of Geometry of differential equations seminar\n\n\nAbst
ract\nWe show how complexification of CR manifolds leads to systems of PDE
s with finite-dimensional solution space. Applications of this approach in
clude classification of homogeneous 5D CR manifolds and identification mod
els with large symmetry in other dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20251015T162000Z
DTEND:20251015T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/141
DESCRIPTION:Title: On the geometry of WDVV equations and their Hamiltonian formalism in
arbitrary dimension\nby Raffaele Vitolo (Università del Salento) as
part of Geometry of differential equations seminar\n\n\nAbstract\nIt is kn
own that in low dimensions WDVV equations can be rewritten as commuting qu
asilinear bi-Hamiltonian systems. We extend some of these results to arbit
rary dimension N and arbitrary scalar product $\\eta$. In particular\, we
show that WDVV equations can be interpreted as a set of linear line congru
ences in suitable Plücker embeddings. This form leads to their representa
tion as Hamiltonian systems of conservation laws. Moreover\, we show that
in low dimensions and for an arbitrary $\\eta$ WDVV equations can be reduc
ed to passive orthonomic form. This leads to the commutativity of the Hami
ltonian systems of conservation laws. We conjecture that such a result hol
ds in all dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20251022T162000Z
DTEND:20251022T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/142
DESCRIPTION:Title: Källén function and around. Part 1\nby Vladimir Rubtsov (Unive
rsité d'Angers) as part of Geometry of differential equations seminar\n\n
Lecture held in room 303 of the Independent University of Moscow.\n\nAbstr
act\nThere are elementary functions that turn out to be ubiquitous. The K
ällén function is one of them. Of course\, it is incomparably less well-
known than the exponential function (although\, in a certain sense\, it is
related to it).\n\nOriginally arising in "school-textbooks" mathematics\,
the Källén function was later "rediscovered" by physicists in the conte
xt of scattering amplitude calculations. It is directly connected with the
famous combinatorial generating functions (for the Catalan numbers)\, app
ears in the study of solutions of classical ODEs\, and is related to the g
eneralized hypergeometric functions of Appell and Kampé de Fériet as wel
l as to properties of discriminants.\n\nI will try to tell some stories ar
ound of this remarkable function.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Georgy Sharygin
DTSTART:20251029T162000Z
DTEND:20251029T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/143
DESCRIPTION:Title: Van den Berg double Poisson brackets on finite-dimensional algebras<
/a>\nby Georgy Sharygin as part of Geometry of differential equations semi
nar\n\nLecture held in room 303 of the Independent University of Moscow.\n
\nAbstract\nOne of the basic principles of algebraic noncommutative geomet
ry is the condition proposed by Kontsevich and Rosenberg that a "geometric
" structure on a noncommutative algebra A should generate a similar ordina
ry\, "commutative" structure on its representation spaces $Rep_d(A)=Hom(A\
,Mat_d(k))$. The concept of "double Poisson brackets" was introduced by va
n den Berg (and almost simultaneously\, in a slightly modified form\, by C
rowley-Bovey) in 2008 as an answer to the question of which noncommutative
structures correspond to Poisson brackets on representation spaces. The r
esulting construction turned out to be quite rich and interesting\, howeve
r\, the vast majority of examples of such structures now deal with algebra
s A that are (close to being) free. In my talk\, based on a joint work wit
h my master's student A. Hernandez-Rodriguez\, I will describe some simple
examples of how such structures look on finite-dimensional algebras.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Kruglikov (UiT the Arctic University of Norway)
DTSTART:20251008T162000Z
DTEND:20251008T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/144
DESCRIPTION:Title: Rational integrals of geodesic flows\nby Boris Kruglikov (UiT th
e Arctic University of Norway) as part of Geometry of differential equatio
ns seminar\n\n\nAbstract\nPolynomial (in momenta) integrals of geodesic fl
ows\, also known as Killing tensors of the metric\, play an important role
in finite-dimensional integrable systems. Recently\, rational integrals c
ame in focus of investigations. (These are natural\, especially for algebr
aic Hamiltonian actions.) I will discuss the problem of their computations
and count\, relation to relative Killing tensors and show some examples.\
n
LOCATION:https://stable.researchseminars.org/talk/GDEq/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Grigoriev
DTSTART:20251105T162000Z
DTEND:20251105T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/145
DESCRIPTION:Title: Gauge PDEs on spaces with asymptotic boundaries\nby Maxim Grigor
iev as part of Geometry of differential equations seminar\n\n\nAbstract\nI
plan to discuss a general setup for studying the boundary structure of
gauge fields on spaces with asymptotic boundaries. The main example of thi
s situation is asymptotically-anti-de-Sitter (AdS) or flat gravity and (op
tionally) gauge fields living on such a background. A suitable tool to stu
dy systems of this sort in a geometrical way is the so-called gauge PDE
on spaces with (asymptotic) boundaries. When applied to the case of asymp
totically-AdS gravity this gives the generalization of the familiar Feffe
rman-Graham construction that also takes the subleading boundary value int
o account. When additional (gauge) fields are present this generalizes the
known gauge PDE approach to boundary values of AdS gauge fields. An inte
resting feature is that the gauge PDE induced on the boundary is itself
a fibre bundle of gauge PDEs (also known as gauge PDE over background)\, w
here the base describes the leading (conformal geometry in the case of gra
vity) while the fiber correspond to the subleading (conserved currents).\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Evgeny Ferapontov (Loughborough University)
DTSTART:20251217T162000Z
DTEND:20251217T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/146
DESCRIPTION:Title: Involutive scroll structures on solutions of 4D dispersionless integ
rable hierarchies\nby Evgeny Ferapontov (Loughborough University) as p
art of Geometry of differential equations seminar\n\n\nAbstract\nA rationa
l normal scroll structure on an (n+1)-dimensional manifold M is defined
as a field of rational normal scrolls of degree n-1 in the projectivised c
otangent bundle $PT^*M$.\n\nWe show that geometry of this kind naturally a
rises on solutions of various 4D dispersionless integrable hierarchies o
f heavenly type equations. In this context\, rational normal scrolls c
oincide with the characteristic varieties (principal symbols) of the hiera
rchy. Furthermore\, such structures automatically satisfy an additional pr
operty of involutivity.\n\nOur main result states that involutive scroll s
tructures are themselves governed by a dispersionless integrable hierarc
hy\, namely\, the hierarchy of conformal self-duality equations.\n\nBased
on joint work with Boris Kruglikov.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maxim Pavlov
DTSTART:20251203T162000Z
DTEND:20251203T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/147
DESCRIPTION:Title: Isomonodromic deformations and the Tsarev generalized hodograph meth
od\nby Maxim Pavlov as part of Geometry of differential equations semi
nar\n\nLecture held in room 303 of the Independent University of Moscow.\n
\nAbstract\nGeneral and particular solutions of the so called semi-Hamilto
nian hydrodynamic type systems can be obtained by the Tsarev Generalized H
odograph Method. Here we show that a natural extension of this approach ap
plied to dispersive integrable systems is determined by isomonodromic defo
rmations.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20251210T162000Z
DTEND:20251210T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/148
DESCRIPTION:Title: Källén function and around. Part 2\nby Vladimir Rubtsov (Unive
rsité d'Angers) as part of Geometry of differential equations seminar\n\n
\nAbstract\nA continuation of the talk on 22 October.\n\nThere are elementary functions th
at turn out to be ubiquitous. The Källén function is one of them. Of co
urse\, it is incomparably less well-known than the exponential function (a
lthough\, in a certain sense\, it is related to it).\n\nOriginally arising
in "school-textbooks" mathematics\, the Källén function was later "redi
scovered" by physicists in the context of scattering amplitude calculation
s. It is directly connected with the famous combinatorial generating funct
ions (for the Catalan numbers)\, appears in the study of solutions of clas
sical ODEs\, and is related to the generalized hypergeometric functions of
Appell and Kampé de Fériet as well as to properties of discriminants.\n
\nI will try to tell some stories around of this remarkable function.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mikhail Markov
DTSTART:20251224T162000Z
DTEND:20251224T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/149
DESCRIPTION:Title: Boundary calculus for gauge fields on asymptotically AdS spaces\
nby Mikhail Markov as part of Geometry of differential equations seminar\n
\n\nAbstract\nI plan to discuss the applications of the gauge PDE approach
to the study of the boundary structure of gauge fields on the conformal b
oundary of asymptotically AdS (also known as Poincaré-Einstein) manifolds
.\n\nThe main result is the construction of an efficient calculus for the
gauge PDE induced on the boundary\, which allows one to systematically der
ive Weyl-invariant equations induced on the boundary. The so-called obstru
ction equations (e.g. Bach in dimension d=4)\, higher conformal Yang-Mills
equations\, and GJMS operators are derived systematically\, as the constr
aints on the leading boundary value of\, respectively\, the metric\, YM
field\, and the critical scalar field. In particular\, the higher conforma
l Yang-Mills equation in dimension d=8\, obtained within this framework ap
pears to be new. The Weyl-invariant equations on the subleading boundary d
ata for these fields are also derived.\n\nThe approach is very general and
can be considered as an extension of the Fefferman-Graham construction
that is applicable to generic gauge fields and explicitly takes into accou
nt both the leading and the subleading sector.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/149/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valentin Lychagin
DTSTART:20260211T162000Z
DTEND:20260211T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/150
DESCRIPTION:Title: On the management of thermodynamic processes\nby Valentin Lychag
in as part of Geometry of differential equations seminar\n\nLecture held i
n room 303 of the Independent University of Moscow.\n\nAbstract\nAt the be
ginning of the talk\, three geometric approaches to thermodynamics will be
discussed.\n\nThe first approach is the Gibbs energy approach\, which wil
l be reformulated in terms of contact geometry and where the description o
f substances (the so-called equations of state) is given in terms of Legen
dre submanifolds in thermodynamic contact phase spaces\, and thermodynamic
processes\, as well as controls\, will be given by contact vector fields.
\n\nThe second approach is based on information geometry and follows the p
rinciple of maximum entropy\, also known as the principle of minimum infor
mation gain or Occam's razor. Both of these approaches lead us to the same
model of thermodynamics\, but they also introduce important new concepts\
, such as the Gibbs-Duhem principle and Riemannian structures on Legendre
submanifolds.\n\nThe third approach is based on the geometry of jet spaces
(or the geometry of differential equations)\, and it provides a more conv
enient apparatus for the practical description and calculation of both equ
ations of state and thermodynamic processes\, taking into account phase tr
ansitions.\n\nThermodynamic process control will be understood as a thermo
dynamic process that does not destroy the process in question\, but allows
it to be accelerated or slowed down. \n\nThe set of controls forms a Lie
algebra\, in which the Lie algebra of symmetries is a Lie subalgebra. \n
\nWe will present equations that depend on the equations of state of the m
edium and allow us to find control processes\, as well as illustrate their
application in the case of adiabatic processes.\n\nIf time permits\, phas
e transitions of the first\, second and higher orders will be considered\,
both in thermodynamic processes and in controls\, as well as their connec
tion with Arnold's theory on the singularities of projections of Lagrangia
n manifolds.\n\nPlease download the formula file td.pdf and keep it handy during the talk
so that the speaker can refer to it.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vladimir Rubtsov (Université d'Angers)
DTSTART:20260218T162000Z
DTEND:20260218T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/151
DESCRIPTION:Title: 2-Valued algebraic groups\, the Chazy equation\, and quasimodular fo
rms\nby Vladimir Rubtsov (Université d'Angers) as part of Geometry of
differential equations seminar\n\nLecture held in room 303 of the Indepen
dent University of Moscow.\n\nAbstract\nI will discuss some (un)known rela
tions between the objects in the title.\n\nIn particular\, the celebrated
Chazy equation emerges as an associativity condition. \n\nThe talk is bas
ed on ongoing joint work with V. Buchstaber and M. Kornev (Steklov Mathematical Institute\, RAS
).\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anna Duyunova
DTSTART:20260225T162000Z
DTEND:20260225T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/152
DESCRIPTION:Title: Differential invariants and quotient of the Euler equations on a sph
ere\nby Anna Duyunova as part of Geometry of differential equations se
minar\n\nLecture held in room 303 of the Independent University of Moscow.
\n\nAbstract\nWe consider the Euler system on a sphere written in stereogr
aphic coordinates. Since the system is underdetermined we consider flow of
a medium taking into account thermodynamic equations of state.\n\nLie alg
ebras of symmetries of the Euler system are found and we give their classi
fication depending on possible equations of state. Among these Lie algebra
s there is one that preserves any thermodynamic equation. Such symmetries
and the corresponding rational differential invariants we call kinematic.
The field of kinematic differential invariants is described: basis differe
ntial invariants as well as invariant derivations are found. Then we find
relations (syzygies) between the second-order invariants\, from which we f
ind a quotient equation for the Euler system on a sphere.\n\nJoint work wi
th Valentin Lychagin and Sergey Tychkov.\n\nPlease download the formula fi
le di.pdf and k
eep it handy during the talk so that the speaker can refer to it.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Raffaele Vitolo (Università del Salento)
DTSTART:20260318T162000Z
DTEND:20260318T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/153
DESCRIPTION:Title: Bi-Hamiltonian systems from homogeneous operators\nby Raffaele V
itolo (Università del Salento) as part of Geometry of differential equati
ons seminar\n\n\nAbstract\nMany "famous" integrable systems (KdV\, AKNS\,
dispersive water waves etc.) have a bi-Hamiltonian pair of the following f
orm: $A_1 = P_1 + R_k$ and $A_2 = P_2$\, where $P_1$\, $P_2$ are homogeneo
us first-order Hamiltonian operators and $R_k$ is a homogeneous Hamiltonia
n operator of degree (order) $k$. The Hamiltonian property of $P_1$\, $P_2
$ and their compatibility were given an explicit analytic form and geometr
ic interpretation long ago (Dubrovin\, Novikov\, Ferapontov\, Mokhov). The
Hamiltonian property of $R_k$ was studied in the past (Doyle\, Potemin\;
$k=2\,3$) and recently revisited with interesting results.\n\nIn this talk
\, we illustrate the analytic form and some preliminary geometric interpre
tation of the compatibility conditions between $P_i$ and $R_k$\, $k=2\,3$.
\n\nSee the recent papers arXiv
:2602.14739\, arXiv:2407.17
189\, arXiv:2311.13932.
\n\nJoint work with P. Lorenzoni and S. Opanasenko.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Kuleshov
DTSTART:20260325T162000Z
DTEND:20260325T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/154
DESCRIPTION:Title: Exact solutions of some problems of rigid body dynamics\nby Alex
ander Kuleshov as part of Geometry of differential equations seminar\n\nLe
cture held in room 303 of the Independent University of Moscow.\n\nAbstrac
t\nInterest in integrable problems in mechanics has never waned. Finding n
ew integrable cases of differential equations of motion for various mechan
ical systems\, as well as finding solutions in quadratures for these cases
\, is one of the main problem of theoretical mechanics. The problem of exa
ct integration of differential equations of motion has several aspects. Th
e geometric aspect is associated with the qualitative study of the regular
behavior of the trajectories of integrable systems. The constructive aspe
ct is associated with finding the conditions under which an algorithm for
explicit solving differential equations using quadratures can be specified
. In this regard\, another important aspect of the range of issues under c
onsideration arises: the explicit solution of systems of differential equa
tions. For certain classes of differential equations\, relying on their sp
ecific structure\, special methods can be used. An example here is the bro
ad and important class of linear differential equations. The study of many
problems in mechanics and mathematical physics reduces to solving a secon
d-order linear homogeneous differential equation. If\, by changing the ind
ependent variable\, it is possible to reduce the corresponding second-orde
r linear differential equation to an equation with rational coefficients\,
then the necessary and sufficient for solvability by quadratures for such
an equation are determined by the so-called Kovacic algorithm. In 1986\,
the American mathematician J. Kovacic presented an algorithm for finding L
iouvillian solutions of a second-order linear homogeneous differential equ
ation with rational coefficients. If the differential equation has no Liou
villian solution\, the algorithm also allows one to establish this fact.\n
\nThis talk will discuss the application of the Kovacic algorithm to inves
tigate the existence of Liouvillian solutions in the problem of motion of
a rotationally symmetric rigid body on a perfectly rough plane and on a pe
rfectly rough sphere. It will also discuss the application of the algorith
m to investigate the existence of Liouvillian solutions in the problem of
motion of a heavy homogeneous ball on a fixed perfectly rough surface of r
evolution. The existence of Liouvillian solutions in the Hess case of the
problem of motion of a heavy rigid body with a fixed point is also analyze
d.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wijnand Steneker
DTSTART:20260408T162000Z
DTEND:20260408T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/155
DESCRIPTION:Title: On globally invariant Euler-Lagrange equations for curves\nby Wi
jnand Steneker as part of Geometry of differential equations seminar\n\n\n
Abstract\nInvariant Lagrangians yield invariant Euler-Lagrange equations a
nd local methods for computing these are well-established\, starting with
Anderson and Griffiths. We focus on global algebraic invariants\, using a
n invariant version of variational bicomplex or\, more generally\, C-spect
ral sequence. One motivation is the question\, posed by Kogan and Olver\,
whether invariant variational problems with only singular extremals can ex
ist. We show that the example of conformal geodesics answers this question
positively and motivates the need for global invariant methods. We then d
iscuss how to compute invariant Euler-Lagrange equations using global inva
riants and how this can be applied in practice\, both as a supplementary t
ool for existing local methods\, as well as in a purely global setting. We
demonstrate these principles with some examples\, all for systems of ODEs
(unparametrized curves). \n\nThis talk is based on joint work with Boris
Kruglikov and Eivind Schneider (Tromsø).\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Druzhkov
DTSTART:20260415T162000Z
DTEND:20260415T180000Z
DTSTAMP:20260404T111417Z
UID:GDEq/156
DESCRIPTION:Title: Invariant reduction for PDEs. IV: Symmetries that rescale geometric
structures\nby Konstantin Druzhkov as part of Geometry of differential
equations seminar\n\n\nAbstract\nFor a system of differential equations a
nd a symmetry\, the framework of invariant reduction systematically comput
es how invariant geometric structures are inherited by the subsystem gove
rning invariant solutions. In this setting\, the reduction of structures i
nvariant under a two-dimensional Lie algebra requires its commutativity.
We extend this mechanism to the case where geometric structures are invari
ant under one symmetry $X$ and are rescaled\, by a factor of $-a$\, by ano
ther symmetry $X_s$ satisfying $[X_s\, X] = aX$. As an application\, we de
scribe a class of exact solutions to systems possessing sufficiently many
symmetries and conservation laws subject to certain compatibility conditi
ons. These solutions are invariant under pairs of symmetries and are compl
etely determined by explicitly constructed functions that are constant on
them\; the description is geometric and does not require any integrabilit
y-related structures such as Lax pairs.\n
LOCATION:https://stable.researchseminars.org/talk/GDEq/156/
END:VEVENT
END:VCALENDAR