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BEGIN:VEVENT
SUMMARY:Jake Fillman (Texas State University)
DTSTART:20200417T185000Z
DTEND:20200417T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/1
DESCRIPTION:Title: Spectra of Fibonacci Hamiltonians\nby Jake Fillman (Texas State Un
iversity) as part of TAMU: Mathematical Physics and Harmonic Analysis Semi
nar\n\n\nAbstract\nThe Fibonacci sequence is a prominent model of a 1D qua
sicrystal. We will talk about some properties of continuum Schr\\"odinger
operators with potentials that are determined by the Fibonacci sequence. W
e show that the spectrum is an (unbounded) Cantor set of zero Lebesgue mea
sure and that the local Hausdorff dimension of the spectrum tends to one i
n the regimes of high energy and small coupling. We also show that multidi
mensional Schr\\"odinger operators patterned on the Fibonacci sequence can
exhibit the coexistence of two phenomena: (1) Cantor structure near the b
ottom of the spectrum and (2) an absence of gaps in the spectrum at high e
nergies. To prove (2)\, we develop an "abstract" Bethe--Sommerfeld criteri
on for sums of extended Cantor sets\, which may be of independent interest
. [Based on joint projects with David Damanik\, Anton Gorodetski\, and May
Mei]\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Stephen Shipman (LSU)
DTSTART:20200424T185000Z
DTEND:20200424T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/2
DESCRIPTION:Title: Reducible and irreducible Fermi surfaces for periodic operators\nb
y Stephen Shipman (LSU) as part of TAMU: Mathematical Physics and Harmonic
Analysis Seminar\n\n\nAbstract\nI will discuss new theorems concerning re
ducibility of the Fermi surface for periodic Schrödinger operators. (1)
Irreducibility for a class of planar discrete graph operators\; (2) Reduci
bility of multilayer graphs due to compatible asymmetries of the connectin
g edges\; (3) Reducibility of multilayer graphs due to separability or bip
artiteness of the layers. Parts of this work are in collaboration with We
i Li\, Lee Fisher\, Karl-Michael Schmidt\, Ian Wood\, and Malcolm Brown.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Milivoje Lukic (Rice University)
DTSTART:20200501T185000Z
DTEND:20200501T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/3
DESCRIPTION:Title: Stahl--Totik regularity for continuum Schr\\"odinger operators\nby
Milivoje Lukic (Rice University) as part of TAMU: Mathematical Physics an
d Harmonic Analysis Seminar\n\n\nAbstract\nThis talk describes joint work
with Benjamin Eichinger: a\ntheory of regularity for one-dimensional conti
nuum Schr\\"odinger\noperators\, based on the Martin compactification of t
he complement of\nthe essential spectrum. For a half-line Schr\\"odinger o
perator\n$-\\partial_x^2+V$ with a bounded potential $V$\, it was previous
ly\nknown that the spectrum can have zero Lebesgue measure and even zero\n
Hausdorff dimension\; however\, we obtain universal thickness statements\n
in the language of potential theory.\nNamely\, we prove that the essential
spectrum is not polar\, it obeys\nthe Akhiezer--Levin condition\, and mor
eover\, the Martin function at\n$\\infty$ obeys the two-term asymptotic ex
pansion $\\sqrt{-z} +\n\\frac{a}{2\\sqrt{-z}} + o(\\frac 1{\\sqrt{-z}})$ a
s $z \\to -\\infty$. The\nconstant $a$ in its asymptotic expansion plays t
he role of a\nrenormalized Robin constant suited for Schr\\"odinger operat
ors and\nenters a universal inequality $a \\le \\liminf_{x\\to\\infty} \\f
rac 1x\n\\int_0^x V(t) dt$. This leads to a notion of regularity\, with\nc
onnections to the exponential growth rate of Dirichlet solutions and\nthe
zero counting measures for finite restrictions of the operator. We\nalso p
resent applications to decaying and ergodic potentials.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wei Li (LSU)
DTSTART:20200508T185000Z
DTEND:20200508T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/4
DESCRIPTION:Title: Embedded eigenvalues of the Neumann Poincaré operator\nby Wei Li
(LSU) as part of TAMU: Mathematical Physics and Harmonic Analysis Seminar\
n\n\nAbstract\nThe Neumann-Poincaré (NP) operator arises in boundary valu
e problems\, and plays an important role in material design\, signal ampli
fication\, particle detection\, etc. The spectrum of the NP operator on do
mains with corners was studied by Carleman before tools for rigorous discu
ssion were created\, and received a lot of attention in the past ten years
. In this talk\, I will present our discovery and verification of eigenval
ues embedded in the continuous spectrum of this operator. The main ideas a
re decoupling of spaces by symmetry and construction of approximate eigenv
alues. This is based on two works with Stephen Shipman and Karl-Mikael Per
fekt.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Jauslin (Princeton University)
DTSTART:20200515T185000Z
DTEND:20200515T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/5
DESCRIPTION:Title: A simple equation to study interacting Bose gasses\nby Ian Jauslin
(Princeton University) as part of TAMU: Mathematical Physics and Harmonic
Analysis Seminar\n\n\nAbstract\nIn this talk\, I will discuss a partial d
ifferential equation introduced by\n Lieb in 1963 in the context of study
ing interacting Bose gasses. I will first\n discuss how this equation can
be used to accurately compute physically\n relevant quantities related t
o the Bose gas\, such as the ground state energy\n and condensate fractio
n. I will then present a construction of the solutions\n to the equation\
, and discuss some of their properties.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Kachkovskiy (MSU)
DTSTART:20200522T185000Z
DTEND:20200522T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/6
DESCRIPTION:Title: On spectral band edges of discrete periodic Schrodinger operators\
nby Ilya Kachkovskiy (MSU) as part of TAMU: Mathematical Physics and Harmo
nic Analysis Seminar\n\n\nAbstract\nWe consider discrete Schrodinger opera
tors on $\\ell^2(\\mathbb Z^d)$\, periodic with respect to some lattice $\
\Gamma$ in $\\mathbb Z^d$ of full rank. Our main goal is to study dimensio
ns of level sets of spectral band functions at the energies corresponding
to their extremal values (the edges of the bands).Suppose that $d\\ge 3$ a
nd the dual lattice $\\Gamma’$ does not contain the vector $(1/2\,…\,1
/2)$. Then the above mentioned level sets have dimension at most $d-2$.\n\
nSuppose that $d=2$ and the dual lattice does not contain vectors of the f
orm $(1/p\,1/p)$ and $(1/p\,-1/p)$ for all $p\\ge 2$. Then the same statem
ent holds (in other words\, the corresponding level sets are finite modulo
$\\mathbb Z^d$).For all lattices that do not satisfy the above assumption
s\, there are known counterexamples of level sets of dimensions $d-1$.\n\n
Part of the argument also implies a discrete Bethe-Sommerfeld property: if
$d\\ge 2$ and the dual lattice does not contain the vector $(1/2\,…\,1/
2)$\, then\, for sufficiently small potentials (depending on the lattice)\
, the spectrum of the periodic Schrodinger operator is an interval. Previo
usly\, this property was studied by Kruger\, Embree-Fillman\, Jitomirskaya
-Han\, and Fillman-Han. Our proof is different and implies some new cases.
\n\nThe talk is based on joint work with in progress with N. Filonov.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christoph Fischbacher (UC Irvine)
DTSTART:20200529T185000Z
DTEND:20200529T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/7
DESCRIPTION:Title: Logarithmic lower bounds for the entanglement entropy of droplet state
s for the XXZ model on the ring\nby Christoph Fischbacher (UC Irvine
) as part of TAMU: Mathematical Physics and Harmonic Analysis Seminar\n\n\
nAbstract\nWe study the free XXZ quantum spin model defined on a ring of s
ize L and \nshow that the bipartite entanglement entropy of eigenstates be
longing to \nthe first energy band above the vacuum ground state satisfy a
\nlogarithmically corrected area law. This is joint work with Ruth Schult
e \n(LMU).\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Kuchment (TAMU)
DTSTART:20200731T185000Z
DTEND:20200731T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/8
DESCRIPTION:Title: Spectral properties of periodically perforated spaces\nby Peter Ku
chment (TAMU) as part of TAMU: Mathematical Physics and Harmonic Analysis
Seminar\n\n\nAbstract\nWe study spectra of Schr\\"odinger operators with p
eriodic \npotentials in R^n with periodic perforations. We prove analytic
\ndependence on the shape of the perforation and absolute continuity of \n
the spectrum.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Casey Rodriguez (MIT)
DTSTART:20200807T203000Z
DTEND:20200807T213000Z
DTSTAMP:20260404T092653Z
UID:MPHA/9
DESCRIPTION:Title: The Radiative Uniqueness Conjecture for Bubbling Wave Maps\nby Cas
ey Rodriguez (MIT) as part of TAMU: Mathematical Physics and Harmonic Anal
ysis Seminar\n\n\nAbstract\nWe will discuss the finite time breakdown of s
olutions to a canonical example of a geometric wave equation: energy criti
cal wave maps. Breakthrough works of Krieger-Schlag–Tataru\, Rodnianski-
Sterbenz and Raphael–Rodnianski produced examples of wave maps that deve
lop singularities in finite time. These solutions break down by concentrat
ing energy at a point in space (via bubbling a harmonic map) but have a re
gular limit\, away from the singular point\, as time approaches the final
time of existence. The regular limit is referred to as the radiation. This
mechanism of breakdown occurs in many other PDE including energy critical
wave equations\, Schrodinger maps and Yang-Mills equations. A basic quest
ion is the following:\n\nCan we give a precise description of all bubbling
singularities for wave maps with the goal of finding the natural unique c
ontinuation of such solutions past the singularity?\n\nIn this talk\, we w
ill discuss recent work (joint with J. Jendrej and A. Lawrie) which is the
first to directly and explicitly connect the radiative component to the b
ubbling dynamics by constructing and classifying bubbling solutions with a
simple form of prescribed radiation. Our results serve as an important fi
rst step in formulating and proving the following Radiative Uniqueness Con
jecture for a large class of wave maps: every bubbling solution is uniquel
y characterized by its radiation\, and thus\, every bubbling solution can
be uniquely continued past blow-up time while conserving energy.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Avy Soffer (Rutgers)
DTSTART:20200814T185000Z
DTEND:20200814T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/10
DESCRIPTION:Title: Evolution of NLS with Bounded Data\nby Avy Soffer (Rutgers) as pa
rt of TAMU: Mathematical Physics and Harmonic Analysis Seminar\n\n\nAbstra
ct\nWe study the nonlinear Schrodinger equation (NLS) with bounded initial
data which does not vanish at infinity. Examples include periodic\, quasi
-periodic and random initial data. On the lattice we prove that solutions
are polynomially bounded in time for any bounded data. In the continuum\,
local existence is proved for real analytic data by a Newton iteration sch
eme. Global existence for NLS with a regularized nonlinearity follows by a
nalyzing a local energy norm (arXiv:2003.08849 [math.AP]\, J.Stat.Phys\,
2020).\nThis is a joint work with Ben Dodson and Tom Spencer.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Tovbis (University of Central Florida)
DTSTART:20200821T185000Z
DTEND:20200821T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/11
DESCRIPTION:Title: Soliton and breather gases for the focusing Nonlinear Schrödinger eq
uation (fNLS): spectral theory and possible applications\nby Alexander
Tovbis (University of Central Florida) as part of TAMU: Mathematical Phys
ics and Harmonic Analysis Seminar\n\n\nAbstract\nIn the talk we introduce
the idea of an "integrable gas" as a collection of large random ensembles
of special localized solutions (solitons\, breathers) of a given integrabl
e system. These special solutions can be treated as "particles". Known law
s of pairwise elastic collisions allow one to write the heuristic "equatio
n of state" for the gas of such particles.\n\nIn this talk we consider sol
iton and breather gases for the fNLS as special thermodynamic limits of fi
nite gap (nonlinear multi phase wave) fNLS solutions. In this limit the ra
te of growth of the number of bands is linked with the rate of (simultaneo
us) shrinkage of the size of individual bands. This approach leads to the
derivation of the equation of state for the gas and its certain limiting r
egimes (condensate\, ideal gas limits)\, as well as construction of variou
s interesting examples. We also discuss the recent progress and perspectiv
es of future work\, as well as some possible applications.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seonghyeon Jeong (MSU)
DTSTART:20200911T185000Z
DTEND:20200911T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/12
DESCRIPTION:Title: Strong MTW type condition to local Holder regularity in generated Jac
obian equations\nby Seonghyeon Jeong (MSU) as part of TAMU: Mathemati
cal Physics and Harmonic Analysis Seminar\n\n\nAbstract\nn this talk\, we
present a proof of local Holder regularity of solutions to generated Jacob
ian equations as a generalization of optimal transport case\, which is pro
ved by George Loeper. We compare structures of generated Jacobian equation
s with optimal transport\, and point out differences with difficulties whi
ch the differences can cause. For local Holder regularity theory\, we use
(G3s) condition and solution in Alexandrov sense. (G3s) is a strict positi
veness type condition on MTW tensor associated to the generating function
G\, and Alexandrov solution is a solution that satisfies pullback measure
condition. (G3s) is used to show a quantitative version of (glp)\, which g
ives some room for G-subdifferentials of solutions. Then the inequality fo
r Holder regularity is shown by comparing volumes of G-subdifferentials us
ing the fact that our solutions is in Alexandrov sense.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Breuer (Hebrew University of Jerusalem)
DTSTART:20201001T150000Z
DTEND:20201001T160000Z
DTSTAMP:20260404T092653Z
UID:MPHA/13
DESCRIPTION:Title: Periodic Jacobi Matrices on Trees\nby Jonathan Breuer (Hebrew Uni
versity of Jerusalem) as part of TAMU: Mathematical Physics and Harmonic A
nalysis Seminar\n\n\nAbstract\nThe theory of periodic Jacobi matrices on t
he line is extremely rich and very well studied. Viewing the line as a reg
ular tree of degree 2 leads to a natural generalization to periodic Jacobi
matrices on general trees. This family of operators\, which is at least a
s rich (by definition)\, but considerably less well understood\, is at the
center of this talk. We review some of the few known results\, present so
me examples\, and discuss open problems and directions for future research
. The talk is based on joint work with Nir Avni and Barry Simon.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Delio Mugnolo (University of Hagen)
DTSTART:20201008T150000Z
DTEND:20201008T160000Z
DTSTAMP:20260404T092653Z
UID:MPHA/14
DESCRIPTION:Title: Bi-Laplacians on graphs: self-adjoint extensions and parabolic theory
\nby Delio Mugnolo (University of Hagen) as part of TAMU: Mathematical
Physics and Harmonic Analysis Seminar\n\n\nAbstract\nElastic beams have b
een studied by means of hyperbolic equations driven by bi-Laplacian operat
ors since the early 18th century: several properties of the corresponding
parabolic equation on the whole Euclidean space have been discovered since
the 1960s by Krylov\, Hochberg\, and Davies\, among others. On a bounded
domain or a metric graph\, the bi-Laplacian is generally not anymore actin
g as a squared operator\, though: this strongly affects the features of as
sociated PDEs.\n\nI am going to present a full characterization of self-ad
joint extensions of the bi-Laplacian\, focusing on a class of realizations
that encode dynamic boundary conditions. Maximum principles of parabolic
equations will also be discussed: after a transient time\, I am going to s
how that solutions often display Markovian features.\n\nThis is joint work
with Federica Gregorio.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jiaqi Yang (GeorgiaTech)
DTSTART:20200925T185000Z
DTEND:20200925T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/15
DESCRIPTION:Title: Persistence of Invariant Objects in Functional Differential Equations
close to ODEs\nby Jiaqi Yang (GeorgiaTech) as part of TAMU: Mathemati
cal Physics and Harmonic Analysis Seminar\n\n\nAbstract\nWe consider funct
ional differential equations which are perturbations of ODEs in $\\mathbb{
R}^n$. This is a singular perturbation problem even for small perturbation
s. We prove that for small enough perturbations\, some invariant objects o
f the unperturbed ODEs persist and depend on the parameters with high regu
larity. We formulate a-posteriori type of results in the case when the unp
erturbed equations admit periodic orbits. The results apply to state-depen
dent delay equations and equations which arise in the study of electrodyna
mics. The proof is constructive and leads to an algorithm. This is a joint
work with Joan Gimeno and Rafael de la Llave.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rodrigo Matos (Texas A&M University)
DTSTART:20201002T185000Z
DTEND:20201002T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/16
DESCRIPTION:Title: Dynamical Contrast on Highly Correlated Anderson-type models\nby
Rodrigo Matos (Texas A&M University) as part of TAMU: Mathematical Physics
and Harmonic Analysis Seminar\n\n\nAbstract\nWe present examples of rando
m Schödinger operators obtained in a similar fashion but exhibiting disti
nct transport properties. The models are constructed by connecting\, in di
fferent ways\, infinitely many copies of the one dimensional Anderson mode
l. \nSpectral aspects of the models will also be presented. In particular\
, we obtain a physically motivated example of a random operator with purel
y absolutely continuous spectrum where the transient and recurrent compone
nts coexist. This can be interpreted as a sharp phase transition within th
e absolutely continuous spectrum. Time allowing\, I will discuss some tool
s related to harmonic analysis\, including a version of Boole's equality w
hich\, to the best of our knowledge\, is new. Based on joint work with Raj
inder Mavi and Jeffrey Schenker.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Farhan Abedin (Michigan State University)
DTSTART:20201030T185000Z
DTEND:20201030T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/18
DESCRIPTION:Title: Hele-Shaw Flow and Parabolic Integro-Differential Equations\nby F
arhan Abedin (Michigan State University) as part of TAMU: Mathematical Phy
sics and Harmonic Analysis Seminar\n\n\nAbstract\nI will present a regular
ization result for a special case of the two-phase Hele-Shaw free boundary
problem (a.k.a. interfacial Darcy flow)\, which models the evolution of t
wo immiscible fluids flowing in the narrow gap between two parallel plates
and subject to an external pressure source. Assuming that the fluid inter
face is given by the graph of a function\, recent work of Chang-Lara\, Gui
llen\, and Schwab establishes the equivalence between the Hele-Shaw free b
oundary problem and a first-order parabolic integro-differential equation.
By exploiting this equivalence and using available regularity theory for
nonlocal parabolic equations\, we show that if the gradient of the graph o
f the fluid interface has a Dini modulus of continuity for all times\, the
n the gradient must be Holder continuous. This is joint work with Russell
Schwab (MSU).\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jun Kitagawa (Michigan State University)
DTSTART:20201023T185000Z
DTEND:20201023T195000Z
DTSTAMP:20260404T092653Z
UID:MPHA/19
DESCRIPTION:Title: On free discontinuities in optimal transport\nby Jun Kitagawa (Mi
chigan State University) as part of TAMU: Mathematical Physics and Harmoni
c Analysis Seminar\n\n\nAbstract\nIt is well known that regularity results
for the optimal transport (Monge-Kantorovich) problem require rigid geome
tric restrictions. In this talk\, we consider the structure of the set of
``free discontinuities'' which arise when transporting mass from a connect
ed domain to a disconnected one\, and show regularity of this set\, along
with a stability result under suitable perturbations of the target measure
. These are based on a non-smooth implicit function theorem for convex fun
ctions\, which is of independent interest. This talk is based on joint wor
k with Robert McCann (Univ. of Toronto).\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sebastian Egger (Technion)
DTSTART:20201029T150000Z
DTEND:20201029T160000Z
DTSTAMP:20260404T092653Z
UID:MPHA/20
DESCRIPTION:Title: Well-defined spectral position for Neumann domains\nby Sebastian
Egger (Technion) as part of TAMU: Mathematical Physics and Harmonic Analys
is Seminar\n\n\nAbstract\nA Laplacian eigenfunction on a two-dimensional R
iemannian manifold provides a natural partition generated by specific grad
ient flow lines of the eigenfunction. The restricted eigenfunction onto th
e partition's components satisfies Neumann boundary conditions and the com
ponents are therefore coined 'Neumann domains'. Neumann domains represent
a complementary path to the famous nodal-domain partition to study ellipti
c eigenfunctions where the latter is associated with the Dirichlet Laplaci
an. A very basic but fundamental property of nodal domains is that the res
tricted eigenfunction onto a nodal domain always gives the ground-state of
the Dirichlet Laplacian. That feature becomes significantly more complex
for Neumann domains due to the presence of possible cusps and cracks. In t
his talk\, we focus on this problem and show that the spectral position fo
r Neumann domains is well-defined. Moreover\, we provide explicit examples
of Neumann domains displaying a fundamentally different behavior in their
spectral position than their nodal-domain counterparts.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James Kennedy (University of Lisbon)
DTSTART:20201119T160000Z
DTEND:20201119T170000Z
DTSTAMP:20260404T092653Z
UID:MPHA/21
DESCRIPTION:Title: Spectral partitions of metric graphs\nby James Kennedy (Universit
y of Lisbon) as part of TAMU: Mathematical Physics and Harmonic Analysis S
eminar\n\n\nAbstract\nWe introduce a theory of partitions of metric graphs
via spectral-type functionals\, inspired by the theory of spectral minima
l partitions of domains but also with a view to understanding how to detec
t "clusters" in metric graphs.\n\nThe goal is to associate with any given
partition a spectral energy built around eigenvalues of differential opera
tors like the Laplacian\, and then minimize (or maximize) this energy over
all admissible partitions. Since metric graphs are essentially one-dimens
ional manifolds with singularities (the vertices)\, the range of well-pose
d problems is much greater than on domains. We first sketch a general exis
tence theory for optimizers of such partition functionals\, and discuss a
number of natural functionals and optimization problems.\n\nWe also illust
rate how changing the functionals and the classes of partitions under cons
ideration -- for example\, imposing Dirichlet versus standard conditions a
t the cut vertices or considering min-max versus max-min type functionals
-- may lead to qualitatively different optimal partitions which seek out d
ifferent features of the graph.\n\nFinally\, we show how for many problems
the optimal energies behave very similarly to the eigenvalues of the Lapl
acian (with Dirichlet or standard vertex conditions)\, in terms of Weyl as
ymptotics\, upper and lower bounds\, and interlacing inequalities.\n\nThis
is based on joint works with Matthias Hofmann\, Pavel Kurasov\, Corentin
Léna\, Delio Mugnolo and Marvin Plümer.\n
LOCATION:https://stable.researchseminars.org/talk/MPHA/21/
END:VEVENT
END:VCALENDAR