BEGIN:VCALENDAR
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BEGIN:VEVENT
SUMMARY:Melvin Hochster (University of MIchigan)
DTSTART:20200721T130000Z
DTEND:20200721T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/1
DESCRIPTION:Title: Tight Closure\, lim Cohen-Maculay sequences\, content of local cohomol
ogy\, and related open questions - Part 1\nby Melvin Hochster (Univers
ity of MIchigan) as part of IIT Bombay Virtual Commutative Algebra Seminar
\n\n\nAbstract\nThe talks will give multiple characterizations of tight cl
osure\, discuss some of its applications\, indicate connections with the
existence of big and small Cohen-Macaulay algebras and modules\, as well a
s variant notions\, and also explain connections with the theory of conte
nt. There will be some discussion of the many open questions in the area\
, including the very long standing problem of proving that Serre intersect
ion multiplicities have the behavior one expects.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hai Long Dao (The University of Kansas)
DTSTART:20200724T130000Z
DTEND:20200724T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/2
DESCRIPTION:Title: Reflexive modules over curve singularities\nby Hai Long Dao (The U
niversity of Kansas) as part of IIT Bombay Virtual Commutative Algebra Sem
inar\n\n\nAbstract\nA finitely generated module $M$ over a commutative rin
g $R$ is called reflexive if the natural map from $M$ to $M^{**} = Hom(Hom
(M\,R)\, R)$ is an isomorphism. In understanding reflexive modules\, the c
ase of dimension one is crucial. If $R$ is Gorenstein\, then any maximal C
ohen-Macaulay module is reflexive\, but in general it is quite hard to und
erstand reflexive modules even over well-studied one-dimensional singulari
ties. In this work\, joint with Sarasij Maitra and Prashanth Sridhar\, we
will address this problem and give some partial answers.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melvin Hochster (University of Michigan)
DTSTART:20200728T130000Z
DTEND:20200728T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/3
DESCRIPTION:Title: Tight Closure\, lim Cohen-Maculay sequences\, content of local cohomol
ogy\, and related open questions - Part 2\nby Melvin Hochster (Univers
ity of Michigan) as part of IIT Bombay Virtual Commutative Algebra Seminar
\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linquan Ma (Purdue University)
DTSTART:20200804T130000Z
DTEND:20200804T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/4
DESCRIPTION:Title: The deformation problem for $F$-injective singularities\nby Linqua
n Ma (Purdue University) as part of IIT Bombay Virtual Commutative Algebra
Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudia Polini (University of Notre Dame)
DTSTART:20201103T130000Z
DTEND:20201103T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/5
DESCRIPTION:Title: The core of ideals\nby Claudia Polini (University of Notre Dame) a
s part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nLe
t I be an ideal in a Noetherian commutative ring. Among all the closures\n
of I\, the integral closure plays a central role. A reduction of I\nis a s
ub ideal with the same integral closure.\nWe can think of reductions as si
mplifications of the given ideal\,\nwhich carry most of the information ab
out I itself but\, in general\,\nwith fewer generators. Minimal reductions
\, reductions\nminimal with respect to inclusion\, are loosely speaking th
e\ncounterpart of the integral closure. However\,\nunlike the integral cl
osure\, minimal reductions are not unique.\nFor this reason\, we consider
their intersection\, called the core of\nI. The core is related to adjoin
t and\nmultiplier ideals. Motivation for studying\nthis object comes from
the Briancon-Skoda theorem. Furthermore\,\na better understanding of the c
ore could lead\nto solving Kawamata's conjecture on the non-vanishing of\n
sections of a certain line bundle. In this talk\, I will discuss the\nimpo
rtance of the core\, its ubiquity in algebra and geometry\,\nand some effe
ctive formulas for its computation.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudia Polini (University of Notre Dame)
DTSTART:20201106T130000Z
DTEND:20201106T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/6
DESCRIPTION:Title: The core of monomial ideals\nby Claudia Polini (University of Notr
e Dame) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbs
tract\nLet $I$ be a monomial ideal. Even though there may not exist any\np
roper reduction of $I$ which is monomial (or even homogeneous)\, the\ninte
rsection of all reductions\, the core\, is again a monomial ideal.\nThe in
tegral closure and the adjoint of a monomial ideal are again\nmonomial ide
als and can be described in terms of the Newton\npolyhedron of $I$. Such a
description cannot exist for the core\,\nsince the Newton polyhedron only
recovers the integral closure of\nthe ideal\, whereas the core may change
when passing from $I$ to\nits integral closure. When attempting to derive
any kind of combinatorial\ndescription for the core of a monomial ideal f
rom the known colon\nformulas\, one faces the problem that the colon formu
la involves\nnon-monomial ideals\, unless $I$ has a reduction $J$ generate
d by a\nmonomial regular sequence. Instead\, in joint work with Ulrich and
\nVitulli we exploit the existence of such non-monomial reductions to\ndev
ise an interpretation of the core in terms of monomial\noperations. This
algorithm provides a new interpretation of the\ncore as the largest monomi
al ideal contained in a general locally\nminimal reduction of $I$. In rece
nt joint work with Fouli Montano\, \nand Ulrich we extend this formula to
a large class of monomial ideals \nand we study the core of lex-segment mo
nomial ideals generated in one-degree.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Jeffries (University of Nebraska)
DTSTART:20201023T130000Z
DTEND:20201023T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/7
DESCRIPTION:Title: Faithfulness of top local cohomology modules in domains\nby Jack J
effries (University of Nebraska) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nInspired by a question of Lynch\, we consi
der the following question: under what conditions is the highest nonvanish
ing local cohomology module of a domain $R$ with support in an ideal $I$ f
aithful as an R-module? We will review some of what is known about this qu
estion\, and provide an affirmative answer in positive characteristic when
the cohomological dimension is equal to the number of generators of the i
deal. This is based on joint work with Mel Hochster.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ken-ichi Yoshida (Nihon University\, Japan)
DTSTART:20201222T120000Z
DTEND:20201222T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/8
DESCRIPTION:Title: Lower bound on Hilbert-Kunz multiplicities and some related results.\nby Ken-ichi Yoshida (Nihon University\, Japan) as part of IIT Bombay V
irtual Commutative Algebra Seminar\n\n\nAbstract\nIn my talk\, we introduc
e some results of lower bounds on Hilbert-Kunz multiplicities\nfor non-reg
ular local rings. In the later half\, we will discuss the upper bound\non
F-signature.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro De Stefani\, (University of Genoa)
DTSTART:20200811T120000Z
DTEND:20200811T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/9
DESCRIPTION:Title: Deformation and stability of F-injective singularities\nby Alessan
dro De Stefani\, (University of Genoa) as part of IIT Bombay Virtual Commu
tative Algebra Seminar\n\n\nAbstract\nPicking up from the talk given by Li
nquan Ma\, I will discuss some more cases where deformation of F-injectivi
ty is known to hold\, and I will discuss the related notion of m-adic stab
ility. The talk will be based on joint projects with Linquan Ma (deformati
on) and Ilya Smirnov (stability).\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Núñez Betancourt (CIMAT\, Mexico)
DTSTART:20200814T130000Z
DTEND:20200814T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/10
DESCRIPTION:Title: Splittings and symbolic powers of Ideals\nby Luis Núñez Betanco
urt (CIMAT\, Mexico) as part of IIT Bombay Virtual Commutative Algebra Sem
inar\n\n\nAbstract\nSplittings of Frobenius have been employed to study th
e singularities\nand cohomology of rings. In this talk we will employ idea
s inspired by\nthis technique to obtain results of symbolic powers of mono
mial and\ndeterminantal ideals. This is joint work with Jonathan Montaño.
\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pham Hung Quy (FPT University\, Hanoi -)
DTSTART:20200818T120000Z
DTEND:20200818T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/11
DESCRIPTION:Title: Frobenius closure of parameter ideals\nby Pham Hung Quy (FPT Univ
ersity\, Hanoi -) as part of IIT Bombay Virtual Commutative Algebra Semina
r\n\n\nAbstract\nWe discuss recent results about Frobenius closure of para
meter ideals and $F$-singularities as well as the Frobenius test exponent
of parameter ideals.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arindam Banerjee (RKM Vivekananda Institute\, Belur)
DTSTART:20200821T120000Z
DTEND:20200821T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/12
DESCRIPTION:Title: Lyubeznik numbers\nby Arindam Banerjee (RKM Vivekananda Institute
\, Belur) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nA
bstract\nLyubeznik numbers are certain Bass numbers of local cohomology mo
dules associated to local rings containing a field. This numerical invaria
nts are known to have many interesting homological\, geometric and topolog
ical properties and have been an active area of research. In this talk we
plan to give a brief overview of these.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:N. V. Trung (Hanoi Institute of Mathematics)
DTSTART:20201029T120000Z
DTEND:20201029T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/13
DESCRIPTION:Title: Multiplicity sequence and integral dependence\nby N. V. Trung (Ha
noi Institute of Mathematics) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\n\nAbstract\nThe first numerical criterion for integral de
pendence was proved by Rees in 1961 which states that two m-primary ideals
$I \\subset J$ in an equidimensional and universally catenary local ring
$(R\, m)$ have the same integral closure if and only if they have the same
Hilbert-Samuel multiplicity. This result plays an important role in Teiss
ier's work on the equisingularity of families of hypersurfaces with isolat
ed singularities. For hypersurfaces with non-isolated singularities\, one
needs a similar numerical criterion for integral dependence of non-$m$-pr
imary ideals. Since the Hilbert-Samuel multiplicity is no longer defined f
or non-$m$-primary ideals\, one has to use other notions of multiplicities
that can be used to check for integral dependence. A possibility is the m
ultiplicity sequence which was introduced by Achilles and Manaresi in 1997
and has its origin in the intersection numbers of the Stuckrad-Vogel algo
rithm. It was conjectured that two arbitrary ideals $I \\subset J$ in an e
quidimensional and universally catenary local ring have the same integral
closure if and only if they have the same multiplicity sequence. This talk
will present a recent solution of this conjecture by Polini\, Trung\, Ulr
ich and Validashti.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neena Gupta (Indian Statistical Institute\, Kolkata)
DTSTART:20200731T120000Z
DTEND:20200731T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/14
DESCRIPTION:Title: On the triviality of the affine threefold $x^my = F(x\, z\, t)$ - Par
t 2\nby Neena Gupta (Indian Statistical Institute\, Kolkata) as part o
f IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn this ta
lk we will discuss a theory for affine threefolds of the form $x^my = F(x\
, z\, t)$ which will yield several necessary and sufficient conditions for
the coordinate ring of such a threefold to be a polynomial ring. For ins
tance\, we will see that this problem of four variables reduces to the equ
ivalent but simpler two-variable question as to whether F(0\, z\, t) defin
es an embedded line in the affine plane. As one immediate consequence\, o
ne readily sees the non-triviality of the famous Russell-Koras threefold
$x^2y+x+z^2+t^3=0$ (which was an exciting open problem till the mid 1990s)
from the obvious fact that $z^2+t^3$ is not a coordinate. The theory on t
he above threefolds connects several central problems on Affine Algebraic
Geometry. It links the study of these threefolds with the famous Abhyanka
r-Moh “Epimorphism Theorem” in characteristic zero and the Segre-Nagat
a lines in positive characteristic. We will also see a simplified proof o
f the triviality of most of the Asanuma threefolds (to be defined in the t
alk) and an affirmative solution to a special case of the Abhyankar-Sathay
e Conjecture. Using the theory\, we will also give a recipe for constructi
ng infinitely many counterexample to the Zariski Cancellation Problem (ZCP
) in positive characteristic. This will give a simplified proof of the spe
aker's earlier result on the negative solution for the ZCP.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivek Mukundan (Indian Institute of Technoogy Delhi)
DTSTART:20200825T120000Z
DTEND:20200825T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/15
DESCRIPTION:Title: Reduction to characteristic p - Part 1\nby Vivek Mukundan (Indian
Institute of Technoogy Delhi) as part of IIT Bombay Virtual Commutative A
lgebra Seminar\n\n\nAbstract\nThis is an expository talk introducing the m
ethods of reducing to characteristic $p$. The main tools and general noti
ons necessary to reduce a problem to characteristic p will be discussed i
n this talk. It is based on chapter 2 of the excellent resource "Tight Cco
usres in Characterisitic zero" by Hochster and Huneke. We will be restrict
ing ourselves to the case of affine algebras since it is more accessible.\
n
LOCATION:https://stable.researchseminars.org/talk/VCAS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivek Mukundan (Indian Institute of Technoogy Delhi)
DTSTART:20200828T120000Z
DTEND:20200828T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/16
DESCRIPTION:Title: Reduction to characteristic p- Part 2\nby Vivek Mukundan (Indian
Institute of Technoogy Delhi) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\n\nAbstract\nThis talk presents problems solved by using t
he method of reduction to characteristic $p.$ Mainly\, we present two nice
problems which have been solved using the reduction to characteristic p m
ethods. We also present a recent result in the field of symbolic power whi
ch uses the reduction to characteristic $p.$\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Varbaro (University of Genoa)
DTSTART:20200901T120000Z
DTEND:20200901T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/17
DESCRIPTION:Title: F-splittings of the polynomial ring and compatibly split homogeneous
ideals\nby Matteo Varbaro (University of Genoa) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\n\nAbstract\nA polynomial ring R in
n variables over a field K of positive characteristic is F-split. It has m
any F-splittings. When K is a perfect field every F-splitting is given by
a polynomial g in R with the monomial u^{p-1} in its support (where u is t
he product of all the variables) occurring with coefficient 1\, plus a fur
ther condition\, which is not needed if g is homogeneous (w.r.t. any posit
ive grading). Fixed an F-splitting s : R -> R\, an ideal I of R such that
s(I) is contained in I is said compatibly split (w.r.t. the F-splitting s)
. In this case R/I is F-split. Furthermore\, by Fedder’s criterion when
I is a homogeneous ideal of R\, R/I is F-split if and only if I is compati
bly split for some F-splitting s : R -> R. If\, moreover\, u^{p-1} is the
initial monomial of the associated polynomial g of s w.r.t. some monomial
order\, then in(I) is a square-free monomial ideal… In this talk I will
survey these facts (some of them classical\, some not so classical)\, and
make some examples\, focusing especially on determinantal ideals.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mandira Mondal (Chennai Mathematical Institute)
DTSTART:20200904T120000Z
DTEND:20200904T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/18
DESCRIPTION:Title: Density functions for the coefficients of the Hilbert-Kunz function o
f polytopal monoid algebra\nby Mandira Mondal (Chennai Mathematical In
stitute) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAb
stract\nWe shall discuss Hilbert-Kunz density function of a\nNoetherian st
andard graded ring over a perfect field of characteristic $p>0$. We will
\nalso talk about the second coeffcient of the Hilbert-Kunz function and t
he possibility of existence\nof a $\\beta$-density function for this coeff
icient.\n\nWatanabe and Eto have shown that Hilbert-Kunz multiplicity of
affine monoid rings with respect to a monomial ideal of finite colength ca
n be expressed as relative\nvolume of certain nice set arising from the co
nvex geometry associated to the ring. In this talk\, we shall discuss simi
lar expression for the density functions of polytopal monoid algebra with
respect to the homogeneous maximal ideal in terms of the associated convex
geometric structure. This is a joint work with Prof. V. Trivedi. We shall
also discuss the existence of $\\beta$-density function for monomial prim
e ideals of hight one of these rings in this context.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irena Swanson (Purdue University)
DTSTART:20200908T130000Z
DTEND:20200908T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/19
DESCRIPTION:Title: Primary decomposition and powers of ideals\nby Irena Swanson (Pur
due University) as part of IIT Bombay Virtual Commutative Algebra Seminar\
n\n\nAbstract\nThis talk is about associated primes of powers of an ideal
in Noetherian\ncommutative rings. Brodmann proved that the set of associa
ted primes\nstabilizes for large powers. In general\, the number of assoc
iated primes can\ngo up or down as the exponent increases. This talk is a
bout sequences\n$\\{a_n\\}$ for which there exists an ideal $I$ in a Noeth
erian commutative\nring $R$ such that the number of associated primes of $
R/I^n$ is $a_n$. This\nis a report on my work with Sarah Weinstein\, with
Jesse Kim and ongoing\nwork with Roswitha Rissner.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Puthenpurakal (IIT Bombay)
DTSTART:20200911T120000Z
DTEND:20200911T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/20
DESCRIPTION:Title: Homological algebra over complete intersections\nby Tony Puthenpu
rakal (IIT Bombay) as part of IIT Bombay Virtual Commutative Algebra Semin
ar\n\n\nAbstract\nWe discuss Eisenbud operators over a complete intersecti
on. As an application we prove that if A is a strict complete intersection
of positive dimension and if M is\na maximal CM A-module with bounded bet
ti numbers then the Hilbert function of M is non-decreasing\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Briggs (University of Utah)
DTSTART:20200915T133000Z
DTEND:20200915T143000Z
DTSTAMP:20260404T110657Z
UID:VCAS/21
DESCRIPTION:Title: On a conjecture of Vasconcelos - Part 1\nby Ben Briggs (Universit
y of Utah) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\n
Abstract\nThese two talks are about the following theorem: If I is an idea
l of finite projective dimension in a ring $R\,$ and the conormal module $
I/I^2$ has finite projective dimension over R/I\, then I is locally genera
ted by a regular sequence. This was conjectured by Vasconcelos\, after he
and (separately) Ferrand established the case that the conormal module is
projective.\n\nThe key tool is the homotopy Lie algebra\, an object sittin
g at the centre of a bridge between commutative algebra and rational homot
opy theory. In the first part I will explain what the homotopy Lie algebra
is\, and how it can be constructed by differential graded algebra techniq
ues\, following the work of Avramov. In the second part I will bring all o
f the ingredients together and\, hopefully\, present the proof of Vasconce
los' conjecture.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Briggs (University of Utah)
DTSTART:20200918T133000Z
DTEND:20200918T143000Z
DTSTAMP:20260404T110657Z
UID:VCAS/22
DESCRIPTION:Title: On a conjecture of Vasconcelos - Part 2\nby Ben Briggs (Universit
y of Utah) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAb
stract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuke Takagi (University of Tokyo)
DTSTART:20200922T120000Z
DTEND:20200922T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/23
DESCRIPTION:Title: $F$-singularities and singularities in birational geometry - Part 1\nby Shunsuke Takagi (University of Tokyo) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\n$F$-singularities are singular
ities in positive characteristic defined using the Frobenius map and there
are four basic classes of $F$-singularities: $F$-regular\, $F$-pure\, $F$
-rational and $F$-injective singularities. They conjecturally correspond v
ia reduction modulo $p$ to singularities appearing in complex birational g
eometry. In the first talk\, I will survey basic properties of $F$-singul
arities. In the second talk\, I will explain what is known and what is not
known about the correspondence of $F$-singularities and singularities in
birational geometry. If the time permits\, I will also discuss its geometr
ic applications.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuke Takagi (University of Tokyo)
DTSTART:20200925T120000Z
DTEND:20200925T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/24
DESCRIPTION:by Shunsuke Takagi (University of Tokyo) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K. N. Raghavan (Institute of Mathematical Sciences\, Chennai)
DTSTART:20200929T120000Z
DTEND:20200929T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/25
DESCRIPTION:Title: Multiplicities of points on Schubert varieties in the Grassmannian-I<
/a>\nby K. N. Raghavan (Institute of Mathematical Sciences\, Chennai) as p
art of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K. N. Raghavan (Institute of Mathematical Sciences\, Chennai)
DTSTART:20201002T120000Z
DTEND:20201002T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/26
DESCRIPTION:Title: Multiplicities of points on Schubert varieties in the Grassmannian-II
\nby K. N. Raghavan (Institute of Mathematical Sciences\, Chennai) as
part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nGive
n an arbitrary point on a Schubert (sub)variety in a Grassmannian\, how
to compute the Hilbert function (and\, in particular\, the multiplicity) o
f the local ring at that point? A solution to this problem based on "st
andard monomial theory" was conjectured by Kreiman-Lakshmibai circa 2000 a
nd the conjecture was proved about a year or two later by them and indepen
dently also by Kodiyalam and the speaker. The two talks will be an expos
ition of this material aimed at non-experts in the sense that we will not
presume familiarity with Grassmannians (let alone flag varieties) or Schub
ert varieties. \n\nThere are two steps to the solution. The first tr
anslates the problem from geometry to algebra and in turn to combinatorics
. The second is a solution of the resulting combinatorial problem\, whi
ch involves establishing a bijection between two combinatorially defined s
ets. The two talks will roughly deal with these two steps respectively.
\n\nThree aspects of the combinatorial formulation of the problem (and its
solution) are noteworthy: (A) it shows that the natural determinantal
generators of the tangent cone (at the given point) form a Groebner basis
(in any "anti-diagonal" term order)\; (B) it leads to an interpretation of
the multiplicity as counting certain non-intersecting lattice paths\; an
d (C) as was observed by Kreiman some years later\, the combinatorial bi
jection is a kind of Robinson-Schensted-Knuth correspondence\, which he c
alls the "bounded RSK".\n\nDeterminantal varieties arise as tangent cones
at points on Schubert varieties (in the Grassmannian)\, and thus one recov
ers multiplicity formulas for these obtained earlier by Abhyankar and Herz
og-Trung. (The multiplicity part of the Kreiman-Lakshmibai conjecture was
also proved by Krattenthaler\, but by very different methods.)\n\nWhat a
bout Schubert varieties in other (full or partial) flag varieties (G/Q wit
h Q being a parabolic subgroup of a reductive algebraic group G)? The pr
oblem remains open in general\, even for the case of the full flag variety
GL(n)/B\, although there are several papers over the last two decades b
y various authors using various methods that solve the problem in various
special cases. Time permitting\, we will give some indication of these
results\, without however any attempt at comprehensiveness.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mrinal Das (ISI\, Kolkata)
DTSTART:20201006T120000Z
DTEND:20201006T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/27
DESCRIPTION:Title: Some open problems in projective modules and complete intersections\nby Mrinal Das (ISI\, Kolkata) as part of IIT Bombay Virtual Commutativ
e Algebra Seminar\n\n\nAbstract\nConsider a surjective $k$-algebra morphis
m\, where k is a field\, from a polynomial ring of \n$n$ variables to a p
olynomial ring of $m$ variables over $k.$ Is the kernel generated by \n$n
- m$ elements? Our discussion will primarily be around this question and i
ts variants.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarang Sane (IIT Madras)
DTSTART:20201009T120000Z
DTEND:20201009T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/28
DESCRIPTION:Title: $K_0$ and ideals\nby Sarang Sane (IIT Madras) as part of IIT Bomb
ay Virtual Commutative Algebra Seminar\n\n\nAbstract\nWe begin by discussi
ng $K_0$ and defining $K_1$ for a ring $R$ and the exact sequence connecti
ng them on localization with respect to a multiplicative set $S$. More gen
erally\, there is a similar localization exact sequence for an open set $V
(I)^c$ of Spec(R) connecting $K_0$ and $K_1$\, and we relate the propertie
s of the ideal $I$ with the intermediate term in the sequence.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kamran Divaani Aazar (IPM\, Tehran)
DTSTART:20201013T120000Z
DTEND:20201013T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/29
DESCRIPTION:Title: A survey on the finiteness properties of local cohomology modules - P
art 1\nby Kamran Divaani Aazar (IPM\, Tehran) as part of IIT Bombay Vi
rtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kamran Divaani Aazar (IPM\, tehran)
DTSTART:20201016T120000Z
DTEND:20201016T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/30
DESCRIPTION:Title: A survey on the finiteness properties of local cohomology modules - P
art 2\nby Kamran Divaani Aazar (IPM\, tehran) as part of IIT Bombay Vi
rtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Satya Mandal (The University of Kansas)
DTSTART:20201027T130000Z
DTEND:20201027T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/31
DESCRIPTION:Title: Quillen $K$-Theory: A reclamation in Commutative Algebra - Part 1
\nby Satya Mandal (The University of Kansas) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\nIn these two talks I take a pe
dagogic approach to Quillen $K$-theory. What it takes to teach (and learn)
Quillen $K$-theory? I am at the tail end of completing a book on this\, w
hich would eventually be available through some outlet. This is based on a
course I taught. Current version has nearly 400 pages\, in eleven chapter
s. I finish with Swan’s paper on quadrics. I tried to do it in a reader
friendly way\, and tried to avoid expressions like “left to the readers
”. I would give an overview and a road map. To justify the title\, le
t me remind you that $K$-theory used to be part of Commutative algebra. In
this endeavor\, I consolidate the background needed\, in about 100 pages\
, for a commutative algebraist to pick up the book and give a course\, or
learn. There is a huge research potential in this direction. This is becau
se\, with it\, topologists have done what they are good at. However\, thes
e higher $K$-groups have not been descried in a tangible manner. That woul
d be the job of commutative algebraist\, and would require such expertise.
\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Satya Mandal (The University of Kansas)
DTSTART:20201030T130000Z
DTEND:20201030T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/32
DESCRIPTION:Title: Quillen $K$-Theory: A reclamation in Commutative Algebra - Part 2
\nby Satya Mandal (The University of Kansas) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\nIn these two talks I take a pe
dagogic approach to Quillen $K$-theory. What it takes to teach (and learn)
Quillen $K$-theory? I am at the tail end of completing a book on this\, w
hich would eventually be available through some outlet. This is based on a
course I taught. Current version has nearly 400 pages\, in eleven chapter
s. I finish with Swan’s paper on quadrics. I tried to do it in a reader
friendly way\, and tried to avoid expressions like “left to the readers
”. I would give an overview and a road map. To justify the title\, le
t me remind you that $K$-theory used to be part of Commutative algebra. In
this endeavor\, I consolidate the background needed\, in about 100 pages\
, for a commutative algebraist to pick up the book and give a course\, or
learn. There is a huge research potential in this direction. This is becau
se\, with it\, topologists have done what they are good at. However\, thes
e higher $K$-groups have not been descried in a tangible manner. That woul
d be the job of commutative algebraist\, and would require such expertise.
\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amartya Datta (ISI\, Kolkata)
DTSTART:20201110T120000Z
DTEND:20201110T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/33
DESCRIPTION:Title: G_a-actions on Affine Varieties: Some Applications - Part 1\nby A
martya Datta (ISI\, Kolkata) as part of IIT Bombay Virtual Commutative Alg
ebra Seminar\n\n\nAbstract\nOne of the hardest problems that come up in af
fine algebraic geometry is to decide whether a certain d-dimensional facto
rial affine domain is ``trivial''\, i.e.\, isomorphic to the polynomial
ring in d variables. There are instances when the ring of invariants of a
suitably chosen G_a-action has been able to distinguish between two rings
(i.e.\, to prove they are non-isomorphic)\, when all other known invariant
s failed to make the distinction. It was using one such invariant that Ma
kar-Limanov proved the non-triviality of the Russell-Koras threefold\, lea
ding to the solution of the Linearization Problem\; and again\, it was us
ing an invariant of G_a-actions that Neena Gupta proved the nontriviality
of a large class of Asanuma threefolds leading to her solution of the Zar
iski Cancellation Problem in positive characteristic.\n\nG_a actions are a
lso involved in the algebraic characterisation of the affine plane by M. M
iyanishi and the algebraic characterisation of the affine 3-space.by Nikh
ilesh Dasgupta and Neena Gupta. Miyanishi's characterisation had led to th
e solution of Zariski's Cancellation Problem for the affine plane. Using
G_a-actions\, a simple algebraic proof for this cancellation theorem was
obtained three decades later by Makar-Limanov.\n\nIn this talk (in two par
ts)\, we will discuss the concept of G_a-actions along with the above appl
ications\, and the closely related theme of Invariant Theory. The concept
of G_a-action can be reformulated in the convenient ring-theoretic languag
e of ``locally nilpotent derivation'' (in characteristic zero) and ``expon
ential map'' (in arbitrary characteristic). The ring of invariants of a G_
a- action corresponds to the kernel of the corresponding locally nilpotent
derivation (in characteristic zero) and the ring of invariants of an expo
nential map. We will recall these concepts. We will also mention a theore
m on G_a actions on affine spaces (or polynomial rings) due to C.S. Sesha
dri. \n\nWe will also discuss the close alignment of the kernel of a lo
cally nilpotent derivation on a polynomial ring over a field of characteri
stic zero with Hilbert's fourteenth problem. While Hilbert Basis Theorem
had its genesis in a problem on Invariant Theory\, Hilbert's fourteenth p
roblem seeks a further generalisation: Zariski generalises it still furth
er. The connection with locally nilpotent derivations has helped construct
some low-dimensional counterexamples to Hilbert's problem. We will also m
ention an open problem about the kernel of a locally nilpotent derivation
on the polynomial ring in four variables\; and some partial results on it
due to Daigle-Freudenburg\, Bhatwadekar-Daigle\, Bhatwadekar-Gupta-Lokha
nde and Dasgupta-Gupta. Finally\, we will state a few technical results on
the ring of invariants of a G_a action on the polynomial ring over a Noet
herian normal domain\, obtained by Bhatwadekar-Dutta and Chakrabarty-Dasgu
pta-Dutta-Gupta.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amartya Datta (ISI\, Kolkata)
DTSTART:20201113T120000Z
DTEND:20201113T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/34
DESCRIPTION:Title: G_a-actions on Affine Varieties: Some Applications - Part 2\nby A
martya Datta (ISI\, Kolkata) as part of IIT Bombay Virtual Commutative Alg
ebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryo Takahashi (Nagoya University)
DTSTART:20201127T120000Z
DTEND:20201127T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/35
DESCRIPTION:Title: Getting a module from another and classifying resolving subcategories
\nby Ryo Takahashi (Nagoya University) as part of IIT Bombay Virtual C
ommutative Algebra Seminar\n\n\nAbstract\nLet $R$ be a commutative noether
ian ring. Let $M$ and $N$ be finitely generated $R$-modules. When can we g
et $M$ from $N$ by taking direct summands\, extensions and syzygies? This
question is closely related to classification of resolving subcategories o
f finitely generated $R$-modules. In this talk\, I will explain what I hav
e got so far on this topic.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shreedevi Masuti (IIT Dharwad)
DTSTART:20200807T120000Z
DTEND:20200807T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/36
DESCRIPTION:Title: Normal Hilbert coefficients and blow-up algebras\nby Shreedevi Ma
suti (IIT Dharwad) as part of IIT Bombay Virtual Commutative Algebra Semin
ar\n\n\nAbstract\nThe normal Hilbert coefficients are important numerical
invariants associated with an \nideal in an analytically unramified local
ring. They play an important role in determining \nthe homological proper
ties of the blow-up algebras. This will be an expository talk on the \nnor
mal Hilbert coefficients\, and its relation with blow-up algebras. We will
also discuss \nrecent developments on this topic.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giulio Caviglia (Purdue University)
DTSTART:20201117T130000Z
DTEND:20201117T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/37
DESCRIPTION:Title: The Eisenbud-Green-Harris Conjecture\nby Giulio Caviglia (Purdue
University) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nThe $f$-vector of a simplicial complex is a finite sequence of
integers defined by the number of $i$-dimensional faces of the complex. Al
l possible such vectors are completely characterized thanks to a classical
theorem by Kruskal and Katona. This result\, when rephrased in terms of H
ilbert functions of certain quotients of polynomial rings by monomial idea
ls\, extends the celebrated theorem of Macaulay on lexicographic ideals.\n
The Eisenbud-Green-Harris conjecture is a further generalization of both t
he Kruskal-Katona theorem and the well-known Cayley–Bacharach theorem fo
r plane curves. I will survey the known results on this conjecture includi
ng a recent joint work with Alessandro De Stefani.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marilina Rossi (University of Genoa)
DTSTART:20201201T120000Z
DTEND:20201201T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/38
DESCRIPTION:Title: A constructive approach to one-dimensional Gorenstein k-algebras\
nby Marilina Rossi (University of Genoa) as part of IIT Bombay Virtual Com
mutative Algebra Seminar\n\n\nAbstract\nCodimension three Gorenstein rings
are completely described by Buchsbaum and Eisenbud's structure theorem\,
but despite many attempts the construction of Gorenstein rings is an open
problem in higher codimension. Gorenstein rings are of great interest in
many areas of mathematics and they have appeared as an important component
in a significant number of problems. Our task is to give a procedure for
constructing all $1$-dimensional Gorenstein $k$-algebras. Applications to
the Gorenstein linkage of zero-dimensional schemes and to Gorenstein aff
ine semigroup rings are discussed. The results are based on recent results
obtained jointly with J. Elias.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aldo Conca (University of Genoa)
DTSTART:20201211T120000Z
DTEND:20201211T133000Z
DTSTAMP:20260404T110657Z
UID:VCAS/39
DESCRIPTION:Title: Ideals and algebras associated with subspace arrangements.\nby Al
do Conca (University of Genoa) as part of IIT Bombay Virtual Commutative A
lgebra Seminar\n\n\nAbstract\nI will present some results\, old and new\,
about the algebraic objects that are naturally associated with a finite se
t of subspaces of a given vector space.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajendra Gurjar (IIT Bombay)
DTSTART:20201215T120000Z
DTEND:20201215T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/40
DESCRIPTION:Title: Zariski-Lipman Conjecture for Module of Derivations - Part 1\nby
Rajendra Gurjar (IIT Bombay) as part of IIT Bombay Virtual Commutative Alg
ebra Seminar\n\n\nAbstract\nZariski conjectured that if the module of deri
vations of a local ring $R$ at a point on an algebraic variety defined ove
r a field of chararacteristic $0$ is a free $R$-module then $R$ is regular
. In these two talks we will survey most of the interesting results proved
affirming the conjecture.\n\nResults of Lipman\, Scheja-Storch\, Becker\,
Hochster\, Steenbrink-van Straten\, Flenner\, Kallstrom\, Biswas-Gurjar-K
olte\, and some general results which can be deduced by combining some of
these results will be discussed. An interesting proposed counterexample du
e to Hochster will be introduced. Some unsolved cases in the paper of Bisw
as-Gurjar-Kolte will be mentioned.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajendra Gurjar (IIT Bombay)
DTSTART:20201218T120000Z
DTEND:20201218T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/41
DESCRIPTION:Title: Zariski-Lipman Conjecture for Module of Derivations - Part 2\nby
Rajendra Gurjar (IIT Bombay) as part of IIT Bombay Virtual Commutative Alg
ebra Seminar\n\n\nAbstract\nZariski conjectured that if the module of deri
vations of a local ring $R$ at a point on an algebraic variety defined ove
r a field of chararacteristic $0$ is a free $R$-module then $R$ is regular
. In these two talks we will survey most of the interesting results proved
affirming the conjecture.\n\nResults of Lipman\, Scheja-Storch\, Becker\,
Hochster\, Steenbrink-van Straten\, Flenner\, Kallstrom\, Biswas-Gurjar-K
olte\, and some general results which can be deduced by combining some of
these results will be discussed. An interesting proposed counterexample du
e to Hochster will be introduced. Some unsolved cases in the paper of Bisw
as-Gurjar-Kolte will be mentioned.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hema Srinivasan (University of Missouri\, Columbia\, MO)
DTSTART:20201208T130000Z
DTEND:20201208T143000Z
DTSTAMP:20260404T110657Z
UID:VCAS/42
DESCRIPTION:Title: Semigroup rings\nby Hema Srinivasan (University of Missouri\, Col
umbia\, MO) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nLet $A = \\{a_{ij}\\}$ be an $n \\times m$ matrix of natural nu
mbers $\\mathbb N.$ The $S(A)$ denotes the sub-semigroup of $\\mathbb N^n
$ generated by the columns of $A$. The semigroup ring of $A$ over a field
$k$\, denoted by $k[A]$ is the homomorphic image of $\\phi: k[x_1\, \\ldot
s\, x_m] \\to k[t_1\, \\ldots\, t_n]$ defined by $\\phi (x_j) = \\prod_{i=
1}^nt_i^{a_{ij}}$ and hence $k[A]$ is isomorphic to $k[x_1\, \\ldots\, x_m
]/I_A$. In this talk\, we will discuss various invariants of $k[A]$\, suc
h as depth\, dimension\, Frobenius numbers and homological properties\, su
ch as resolutions\, Betti Numbers\, regularity and Hilbert Series. Recen
t work on gluing and its relation to these invariants will be outlined. We
will compare the situation in numerical semigroups (subgroups of $\\mathb
b N$) to semigroups of higher dimension and which of the many formulas and
structures generalize to higher dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Parangama Sarkar (IIT Palakkad)
DTSTART:20201120T120000Z
DTEND:20201120T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/43
DESCRIPTION:Title: Frobenius Betti numbers of finite length modules\nby Parangama Sa
rkar (IIT Palakkad) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nLet $(R\, m)$ be a Noetherian local ring of dimension $
d > 0$ and $M$ be a finitely generated $R$-module of finite length. Suppos
e char R = $p > 0$ and $d = 1.$ De Stefani\, Huneke and Núñez-Betancourt
explored the question: what vanishing conditions on the Frobenius Betti n
umbers force projective dimension of $M$ to be finite. In this talk we wil
l discuss the question for $d ≥ 1.$ This is joint work with Ian Aberbach
.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tai Huy Ha (University of Tulane)
DTSTART:20201124T130000Z
DTEND:20201124T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/44
DESCRIPTION:Title: The ideal containment problem and vanishing loci of homogeneous polyn
omials\nby Tai Huy Ha (University of Tulane) as part of IIT Bombay Vir
tual Commutative Algebra Seminar\n\n\nAbstract\nWe shall discuss Chudnovsk
y’s and Demailly’s conjectures which provide lower bounds for the answ
er to the following fundamental question: given a set of points in project
ive space and a positive integer m\, what is the least degree of a homogen
eous polynomial vanishing at these points of order at least $m$? Particula
rly\, we shall present the main ideas of the proofs of these conjectures f
or sufficiently many general points.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Aberbach (University of Missouri\, Columbia\, MO)
DTSTART:20201204T130000Z
DTEND:20201204T143000Z
DTSTAMP:20260404T110657Z
UID:VCAS/45
DESCRIPTION:Title: On the equivalence of weak and strong F-regularity\nby Ian Aberba
ch (University of Missouri\, Columbia\, MO) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\nLet $(R\, \\mathfrak m\, k)$ be
a (Noetherian) local ring of positive prime characteristic $p.$ Assume a
lso\, for simplicity\, that $R$ is complete (or\, more generally\, excelle
nt). In such rings we have the notion of tight closure of an ideal\, def
ined by Hochster and Huneke\, using the Frobenius endomorphism. The tight
closure of an ideal sits between the ideal itself and its integral closur
e. When the tight closure of an ideal $I$ is $I$ itself we call $I$ tight
ly closed. For particularly nice rings\, e.g.\, regular rings\, every idea
l is tightly closed. We call such rings weakly $F$-regular. Unfortunatel
y\, tight closure is known not to commute with localization\, and hence th
is property of being weakly $F$-regular is not known to localize. To deal
with this problem\, Hochster and Huneke defined the notion of strongly $F
$-regular (assuming $R$ is $F$-finite)\, which does localize\, and implies
that $R$ is weakly $F$-regular. It is still an open question whether or
not the two notions are equivalent\, although it has been shown in some cl
asses of rings. Not much progress has been made in the last 15-20 years.
I will discuss the problem itself\, the cases that are known\, and also o
utline recent progress made by myself and Thomas Polstra.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bernd Ulrich (Purdue University)
DTSTART:20210226T130000Z
DTEND:20210226T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/48
DESCRIPTION:Title: Generalized multiplicities and integral dependence-II\nby Bernd U
lrich (Purdue University) as part of IIT Bombay Virtual Commutative Algebr
a Seminar\n\n\nAbstract\nThese two talks will give a survey about multipli
city based criteria\nfor the integral dependence of ideals. This subject h
as close connections\nwith equisingularity theory and intersection theory\
, which will be\ndiscussed as well. The first numerical criterion for inte
gral dependence\nwas proved in the 1960s by Rees who treated the case of z
ero-dimensional\nideals using the Hilbert-Samuel multiplicity. Criteria fo
r arbitrary\nideals require generalized notions of multiplicities. We will
discuss\nvarious such notions and talk about how they are used to detect
integral\ndependence. The most recent results are from joint work with Cla
udia\nPolini\, Ngo Viet Trung\, and Javid Validashti.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suprajo Das (Chennai Mathematical Institute)
DTSTART:20210101T120000Z
DTEND:20210101T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/58
DESCRIPTION:Title: An inequality in mixed multiplicities of filtrations\nby Suprajo
Das (Chennai Mathematical Institute) as part of IIT Bombay Virtual Commuta
tive Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Iarrobino (Northeastern University\, Boston\, MA)
DTSTART:20201229T130000Z
DTEND:20201229T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/61
DESCRIPTION:Title: Jordan type and Lefschetz Properties for Artinian algebras\nby An
thony Iarrobino (Northeastern University\, Boston\, MA) as part of IIT Bom
bay Virtual Commutative Algebra Seminar\n\n\nAbstract\nThe Jordan type of
a pair $(A\,x)\,$ where $x$ is in the maximum ideal of a standard graded A
rtinian algebra A\, is the partition P giving the Jordan block decompositi
on of the multiplication map by $x$ on $A.$ When $A$ is Artinian Gorenste
in\, we say that $(A\,x)$ is weak Lefschetz if the number of parts in the
Jordan type $P_x$ is the \nSperner number of $A$ – the highest value of
the Hilbert function H(A). We say that \n$(A\,x)$ is strong Lefschetz i
f $P_x$ is the conjugate of the Hilbert function.\n\n Weak and strong Lef
schetz properties of $A$ for a generic choice of $x$ have been studied\, d
ue to the connection with topology and geometry\, where A is the cohomolog
y ring of a\n\ntopological space or a variety $X.$ We discuss some of the
properties of Jordan type\, and its\n\nuse as an invariant of $A\,$ its be
havior for tensor products and free extensions (defined by \n\nT. Harima a
nd J. Watanabe).\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juergen Herzog (University of Duisberg-Essen)
DTSTART:20210108T120000Z
DTEND:20210108T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/64
DESCRIPTION:Title: Powers of component wise linear ideals\nby Juergen Herzog (Univer
sity of Duisberg-Essen) as part of IIT Bombay Virtual Commutative Algebra
Seminar\n\n\nAbstract\nLet $S=K[x_1\,\\ldots\,x_n]$ be the polynomial ring
over a field and $A$ a standard graded $S$-algebra. In terms of the Gr
oebner basis of the defining ideal $J$ of $A$ we give a condition\, called
the x-condition\, which implies that all graded components $A_k$ of $A$
have linear quotients and with additional assumptions are componentwise li
near. A typical example of such an algebra is the Rees ring $\\mathcal R(I
)$ of a graded ideal or the symmetric algebra $\\text{Sym}(M)$ of a module
$M$. We apply our criterion to study certain symmetric algebras and the p
owers of vertex cover ideals of certain classes of graphs. This is a repor
t on joint work with Takayuki Hibi and Somayeh Moradi.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Viviana Ene (Ovidius University\, Constanta\, Romania)
DTSTART:20210115T120000Z
DTEND:20210115T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/65
DESCRIPTION:Title: Binomial edge ideals\nby Viviana Ene (Ovidius University\, Consta
nta\, Romania) as part of IIT Bombay Virtual Commutative Algebra Seminar\n
\n\nAbstract\nIn this talk we will survey various old and new results on t
he homological and algebraic properties of binomial edge ideals. In the la
st part of the talk\, we will present some new results of a recent joint p
aper with G. Rinaldo and N. Terai on powers of binomial edge ideals with q
uadratic Groebner bases.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Katz (The University of Kansas\, Lawrence\, KS)
DTSTART:20210122T130000Z
DTEND:20210122T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/66
DESCRIPTION:Title: Rees Valuations-I\nby Dan Katz (The University of Kansas\, Lawren
ce\, KS) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAb
stract\nIn these expository talks\, we will discuss the Rees valuations an
d Rees valuation rings associated with an ideal in a Noetherian ring\, a
s well as their applications to asymptotic prime divisors and various mult
iplicities. If time permits\, we will describe the Rees valuations associ
ated with a finitely generated torsion-free module.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Katz (The University of Kansas\, Lawrence\, KS)
DTSTART:20210129T130000Z
DTEND:20210129T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/67
DESCRIPTION:Title: Rees Valuations-II\nby Dan Katz (The University of Kansas\, Lawre
nce\, KS) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nA
bstract\nIn these expository talks\, we will discuss the Rees valuations a
nd Rees valuation rings associated with an ideal in a Noetherian ring\,
as well as their applications to asymptotic prime divisors and various mul
tiplicities. If time permits\, we will describe the Rees valuations assoc
iated with a finitely generated torsion-free module.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijaylaxmi Trivedi (Tata Institute of Fundamental Research\, Mumba
i)
DTSTART:20210205T120000Z
DTEND:20210205T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/68
DESCRIPTION:Title: Hilbert-Kunz density function and its applications\nby Vijaylaxmi
Trivedi (Tata Institute of Fundamental Research\, Mumbai) as part of IIT
Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn this talk\, w
e recall the notion of the Hilbert-Kunz density function for graded rings.
This function was introduced to understand the Hilbert-Kunz multiplicity
which is a difficult characteristic $p$ invariant to compute and to make s
peculation about its properties. It turns out the HK density function is a
lso related to another characteristic $p$-invariant namely $F$-threshold.
Here we describe its properties and give its applications to HK multiplici
ty\, $F$-thershold\, and to a conjecture of Watanabe-Yoshida. The talk is
partly based on joint work with K.I. Watanabe.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Graham Leuschke (Syracuse University\, New York\, NY)
DTSTART:20210212T130000Z
DTEND:20210212T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/69
DESCRIPTION:Title: Matrix Factorizations and Knörrer Periodicity\nby Graham Leuschk
e (Syracuse University\, New York\, NY) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nA matrix factorization of a ring el
ement $f$ is a pair of square matrices so that the product (in either orde
r) is diagonal with $f$ in each diagonal entry. These were introduced by D
avid Eisenbud in 1980. When the ring is regular\, matrix factorizations of
$f$ correspond to maximal Cohen-Macaulay modules over the hypersurface de
fined by $f.$ This talk will give an overview of the theory of matrix fact
orizations\, ending with some recent generalizations to factorizations by
more than two matrices.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bernd Ulrich (Purdue University)
DTSTART:20210219T130000Z
DTEND:20210219T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/70
DESCRIPTION:Title: Generalized multiplicities and integral dependence-I\nby Bernd Ul
rich (Purdue University) as part of IIT Bombay Virtual Commutative Algebra
Seminar\n\n\nAbstract\nThese two talks will give a survey about multiplic
ity based criteria\nfor the integral dependence of ideals. This subject ha
s close connections\nwith equisingularity theory and intersection theory\,
which will be\ndiscussed as well. The first numerical criterion for integ
ral dependence\nwas proved in the 1960s by Rees who treated the case of ze
ro-dimensional\nideals using the Hilbert-Samuel multiplicity. Criteria for
arbitrary\nideals require generalized notions of multiplicities. We will
discuss\nvarious such notions and talk about how they are used to detect i
ntegral\ndependence. The most recent results are from joint work with Clau
dia\nPolini\, Ngo Viet Trung\, and Javid Validashti.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K. Ozeki (Yamaguchi University)
DTSTART:20210305T120000Z
DTEND:20210305T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/71
DESCRIPTION:Title: The reduction number of stretched ideals\nby K. Ozeki (Yamaguchi
University) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nThe homological property of the associated graded ring of an id
eal is an\nimportant problem in commutative algebra and algebraic geometry
.\nIn this paper we explore the structure of the associated graded ring of
\nstretched $\\mathfrak m $-primary ideals in the case where the reduction
number\nattains almost minimal value in a Cohen-Macaulay local ring $(A\,
\\mathfrak m )$.\nAs an application\, we present complete descriptions of
the associated\ngraded ring of stretched $\\mathfrak m $-primary ideals wi
th small reduction\nnumber.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Polstra (University of Virgina)
DTSTART:20210312T130000Z
DTEND:20210312T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/72
DESCRIPTION:Title: Strongly F-regular rings\, maximal Cohen-Macaulay modules\, and the F
-signature\nby Thomas Polstra (University of Virgina) as part of IIT B
ombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nThe singularities
of a local prime characteristic ring are best understood through the beha
vior of the Frobenius endomorphism. A singularity class of central focus i
s the class of strongly $F$-regular rings. Examples of strongly $F$-regula
r rings include normal affine toric rings\, direct summands of regular rin
gs\, and determinantal rings. Every strongly $F$-regular ring enjoys the p
roperty of being a normal Cohen-Macaulay domain. In particular\, the study
of finitely generated maximal Cohen-Macaulay modules over such rings is a
warranted venture. We will demonstrate a surprising uniform behavior enjo
yed by the category of maximal Cohen-Macaulay modules over a strongly $F$-
regular local ring. Consequently\, we can redrive Aberbach and Leuschke's
theorem that the $F$-signature of a strongly $F$-regular ring is positive
in a novel and elementary manner. Time permitting\, we will present applic
ations on the structure of the divisor class group of a local strongly $F$
-regular ring.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Polstra (University of Virgina)
DTSTART:20210319T130000Z
DTEND:20210319T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/73
DESCRIPTION:Title: Prime characteristic singularities and the deformation problem\nb
y Thomas Polstra (University of Virgina) as part of IIT Bombay Virtual Com
mutative Algebra Seminar\n\n\nAbstract\nLet $P$ be a property of local rin
gs (such as regular\, Gorenstein\, or complete). We say that $P$ deforms i
f a local ring $R$ enjoys property $P$ provided there exists a nonzerodivi
sor $x$ such that $R/xR$ is $P$. (For example\, the properties of being re
gular or Gorenstein deform\, but the property of being complete does not d
eform). The deformation problem\, as it pertains to the prime characterist
ic singularity classes of $F$-regular\, $F$-rational\, $F$-pure\, and $F$-
injective singularities\, has a rich history that dates to work of Fedder
in the 1980's and remains an active research area. We will survey the hist
ory of the deformation problem of these four prime characteristic singular
ity classes and discuss a recent solution to the deformation of $F$-purity
problem in rings which are $\\mathbb{Q}$-Gorenstein. This talk is based o
n a collaboration with Austyn Simpson.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason McCullough (Iowa State University)
DTSTART:20210402T130000Z
DTEND:20210402T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/74
DESCRIPTION:Title: Rees-like Algebras\nby Jason McCullough (Iowa State University) a
s part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nGi
ven their importance in constructing counterexamples to the Eisenbud-Goto
Conjecture\, it is reasonable to study the algebra and geometry of Rees-li
ke algebras further. Given a graded ideal I of a polynomial ring S\, its
Rees-like algebra is S[It\, t^2]\, where t is a new variable. Unlike the
Rees algebra\, whose defining equations are difficult to compute in genera
l\, the Rees-like algebra has a concrete minimal generating set in terms o
f the generators and first syzygies of I. Moreover\, the free resolution
of this ideal is well understood. While it is clear that the Rees-like al
gebra of an ideal is never normal and only Cohen-Macaulay if the ideal is
principal\, I will explain that it is often seminormal\, weakly normal\, o
r F-pure. I will also discuss the computation of the singular locus\, how
the singular locus is affected by homogenization\, and the structure of t
he canonical module\, class group\, and Picard group. This talk is joint
work with Paolo Mantero and Lance E. Miller.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason McCullough (Iowa State University)
DTSTART:20210326T130000Z
DTEND:20210326T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/75
DESCRIPTION:Title: The Eisenbud-Goto Conjecture\nby Jason McCullough (Iowa State Uni
versity) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAb
stract\nLet S be a polynomial ring over an algebraically closed field K.
There has been considerable research into effective upper bounds for the C
astelnuovo-Mumford regularity of graded ideals of S. Through the work of
Bertram\, Ein\, Gruson\, Kwak\, Lazarsfeld\, Peskine\, and others\, there
are several good bounds for the defining ideals of smooth projective varie
ties in characteristic zero. However\, for arbitrary ideals\, the best up
per bound is doubly exponential (in terms of the number of variables and d
egrees of generators)\, and this bound is asymptotically close to optimal
due to examples derived from the Mayr-Meyer construction. In 1984\, Eisen
bud and Goto conjectured that the regularity of a nondegenerate prime idea
l P was at most deg(P) – codim(P) + 1\, and proved this when S/P was Coh
en-Macaulay (even if P is not prime). In this talk\, I will explain the c
onstruction of counterexamples to the Eisenbud-Goto Conjecture\, joint wor
k with Irena Peeva\, through the construction of Rees-Like algebras and a
special homogenization. While we show that there is no linear bound on re
gularity in terms of the degree (or multiplicity) of P\, we later showed t
hat some such bound exists. The latter part of this talk is joint work wi
th Giulio Caviglia\, Marc Chardin\, Irena Peeva\, and Matteo Varbaro.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karen Smith (University of Michigan)
DTSTART:20210408T130000Z
DTEND:20210408T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/76
DESCRIPTION:Title: Extremal Singularities in Prime Characteristic\nby Karen Smith (U
niversity of Michigan) as part of IIT Bombay Virtual Commutative Algebra S
eminar\n\n\nAbstract\nWhat is the most singular possible singularity? What
can we say about its geometric and algebraic properties? This seemingly n
aive question has a sensible answer in characteristic p.\nThe "F-pure thre
shold\," which is an analog of the log canonical threshold\, can be used
to "measure" how bad a singularity is. The F-pure threshold is a numerical
invariant of a point on (say) a hypersurface---a positive rational numb
er that is 1 at any smooth point (or more generally\, any F-pure point) bu
t less than one in general\, with "more singular" points having smaller F-
pure thresholds. We explain a recently proved lower bound on the F-pure t
hreshold in terms of the multiplicity of the singularity. We also show tha
t there is a nice class of hypersurfaces---which we call "Extremal hypersu
rfaces"---for which this bound is achieved. These have very nice (extreme!
) geometric properties. For example\, the affine cone over a non Frobenius
split cubic surface of characteristic two is one example of an "extremal
singularity". Geometrically\, these are the only cubic surfaces with the p
roperty that *every* triple of coplanar lines on the surface meets in a si
ngle point (rather than a "triangle" as expected)---a very extreme propert
y indeed.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mitsuhiro Miyazaki (Kyoto University of Education)
DTSTART:20210409T120000Z
DTEND:20210409T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/77
DESCRIPTION:Title: Hibi rings and the Ehrhart rings of chain polytopes - Part 1\nby
Mitsuhiro Miyazaki (Kyoto University of Education) as part of IIT Bombay V
irtual Commutative Algebra Seminar\n\n\nAbstract\nIn 1985\, Stanley submit
ted a paper titled "Two Poset Polytopes"\, which was published in 1986\, i
n which he defined the order and chain polytopes of a finite partially ord
ered set (poset for short).\nOn the other hand\, Hibi presented a notion o
f an algebra with straightening law (ASL for short) on a finite distributi
ve lattice\, which nowadays is called a Hibi ring\, in a conference held i
n Kyoto 1985.\nThis result was published in 1987.\nIt turned out that the
Hibi ring on a distributive lattice D is the Ehrhart ring of the order pol
ytope of the poset consisting of join-irreducible elements of D.\nIn the f
irst talk\, we recall the definition of Ehrhart rings\, order and chain po
lytopes\, and Hibi rings.\nWe recall some basic properties of Ehrhart ring
s and describe the canonical module of them.\nUsing these facts\, we state
some basic facts of Hibi rings\, i.e.\, the Ehrhart rings of the order po
lytopes of posets.\nWe also state some basic facts of the Ehrhart rings of
chain polytopes of posets.\nIn the second talk\, we focus on the structu
re of the canonical modules of the Ehrhart rings of order and chain polyto
pes of a poset.\nWe describe the generators of the canonical modules in te
rms of the combinatorial structure of the poset and characterize the level
property.\nIf time permits\, we describe the radical of the trace of the
canonical module of these rings and describe the non-Gorenstein locus.\nTh
is final part is a joint-work with Janet Page.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mitsuhiro Miyazaki (Kyoto University of Education)
DTSTART:20210416T120000Z
DTEND:20210416T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/78
DESCRIPTION:Title: Hibi rings and the Ehrhart rings of chain polytopes - Part 2\nby
Mitsuhiro Miyazaki (Kyoto University of Education) as part of IIT Bombay V
irtual Commutative Algebra Seminar\n\n\nAbstract\nIn 1985\, Stanley submit
ted a paper titled "Two Poset Polytopes"\, which was published in 1986\, i
n which he defined the order and chain polytopes of a finite partially ord
ered set (poset for short).\nOn the other hand\, Hibi presented a notion o
f an algebra with straightening law (ASL for short) on a finite distributi
ve lattice\, which nowadays called a Hibi ring\, in a conference held in K
yoto 1985.\nThis result was published in 1987.\nIt turned out that the Hib
i ring on a distributive lattice D is the Ehrhart ring of the order polyto
pe of the poset consisting of join-irreducible elements of D.\nIn the firs
t talk\, we recall the definition of Ehrhart rings\, order and chain polyt
opes\, and Hibi rings.\nWe recall some basic properties of Ehrhart rings a
nd describe the canonical module of them.\nUsing these facts\, we state so
me basic facts of Hibi rings\, i.e.\, the Ehrhart rings of the order polyt
opes of posets.\nWe also state some basic facts of the Ehrhart rings of ch
ain polytopes of posets.\nIn the second talk\, we focus on the structure
of the canonical modules of the Ehrhart rings of order and chain polytopes
of a poset.\nWe describe the generators of the canonical modules in terms
of the combinatorial structure of the poset and characterize the level pr
operty.\nIf time permits\, we describe the radical of the trace of the can
onical module of these rings and describe the non-Gorenstein locus.\nThis
final part is a joint-work with Janet Page.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:L. T. Hoa (Institute of Mathematics\, Hanoi\, Vietnam)
DTSTART:20210423T120000Z
DTEND:20210423T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/79
DESCRIPTION:Title: Asymptotic behavior of Integer Programming and the stability of the C
astelnuovo-Mumford regularity\nby L. T. Hoa (Institute of Mathematics\
, Hanoi\, Vietnam) as part of IIT Bombay Virtual Commutative Algebra Semin
ar\n\n\nAbstract\nIn the talk\, we explain a connection between Commutati
ve Algebra and Integer Programming in two parts. The first one is devoted
to the asymptotic behavior of integer programs with a fixed cost linear fu
nctional and the constraint sets consisting of a finite system of linear e
quations or inequalities with integer coefficients depending linearly on $
n$. An integer $N_*$ is determined such that the optima of these integer
programs are a quasi-linear function of $n$ for all $n\\ge N_*$. Using re
sults in the first part\, one can bound in the second part the indices of
stability of the Castelnuovo-Mumford regularities of integral closures of
powers of a monomial ideal and that of symbolic powers of a square-free m
onomial ideal.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Eisenbud (University of California\, Berkeley and MSRI)
DTSTART:20210430T140000Z
DTEND:20210430T150000Z
DTSTAMP:20260404T110657Z
UID:VCAS/80
DESCRIPTION:Title: Layered resolutions and Cohen-Macaulay approximation\nby David Ei
senbud (University of California\, Berkeley and MSRI) as part of IIT Bomba
y Virtual Commutative Algebra Seminar\n\n\nAbstract\nThe representation th
eory of finite-dimensional algebras has an important generalization to\nth
e study of maximal Cohen-Macaulay modules (MCMs) over local Cohen-Macaulay
rings. In the case of a hypersurface ring\, this is the study of the matr
ix factorizations of the equation of the hypersurface\, and these come fro
m minimal free resolutions of the MCMs. I will talk about the "next" case-
--MCMs and their minimal free resolutions over complete intersections. Thi
s is joint work with Irena Peeva.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Indranath Sengupta (IIT Gandhinagar)
DTSTART:20210507T120000Z
DTEND:20210507T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/81
DESCRIPTION:Title: Some Questions on bounds of Betti Numbers of Numerical Semigroup Ring
s\nby Indranath Sengupta (IIT Gandhinagar) as part of IIT Bombay Virtu
al Commutative Algebra Seminar\n\n\nAbstract\nJ. Herzog proved in 1969 tha
t the possible values of the first Betti number (minimal number of generat
ors of the defining ideal) of numerical semigroup rings in embedding dimen
sion 3 are 2 (complete intersection and Gorenstein) and 3 (the almost comp
lete intersection).\n\nIn a conversation about this work\, O. Zariski indi
cated a possible relation between Gorenstein rings and symmetric value sem
igroups. In response to that\, E.Kunz proved (in 1970) that a one-dimensio
nal\, local\, Noetherian\, reduced\, analytically irreducible ring is Gore
nstein if and only if its value semigroup is symmetric. A question that re
mains open to date is whether the Betti numbers (or at least the first Bet
ti number) of every numerical semigroup ring in embedding dimension e\, ar
e bounded above by a function of e.\n\nIn the years 1974 and 1975\, two in
teresting classes of examples were given by T. Moh and H. Bresinsky. Moh
’s example was that of a family of algebroid space curves and Bresinsky
’s example was about a family of numerical semigroups in embedding dimen
sion 4\, with the common feature that there is no upper bound on the Betti
numbers. Therefore\, for embedding dimension 4 and above\, the Betti numb
ers (or at least the first Betti number) are not bounded above by some “
good” function of the embedding dimension e. A question that emerges is
the following: Is there a natural way to generate such numerical semigroup
s in arbitrary embedding dimension? In this talk\, we will discuss some re
cent observations in this direction\, which is a joint work of the author
with his collaborators Joydip Saha and Ranjana Mehta.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Roemer (University of Osnabrueck)
DTSTART:20210514T120000Z
DTEND:20210514T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/82
DESCRIPTION:Title: Cut and related polytopes in commutative algebra\nby Tim Roemer (
University of Osnabrueck) as part of IIT Bombay Virtual Commutative Algebr
a Seminar\n\n\nAbstract\nThe study of cuts in graphs is an interesting top
ic in discrete mathematics and optimization with relations and application
s to many other fields such as algebraic geometry\, algebraic statistics\,
and commutative algebra. Here we focus on cut algebras\, which are toric
algebras\, and each one is defined by all cuts of a given graph\,\nand sim
ilar constructions. We discuss known and new results as well as open quest
ions related to the algebraic properties of such algebras and their defini
ng ideals.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trung Cuong Doan (Institute of Mathematics\, Hanoi)
DTSTART:20210521T120000Z
DTEND:20210521T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/83
DESCRIPTION:Title: Betti numbers and ideal structure of projective subschemes of almost
maximal degree\nby Trung Cuong Doan (Institute of Mathematics\, Hanoi)
as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\n
Similar to Castelnouvo-Mumford regularity\, reduction number is a measure
of the complexity of the structure of an algebra. For a projective subsche
me $X$\, there is a degree upper bound\n$$\\deg(X)\\leq \\binom{e+r}{r}\,$
$\nwhere $e$ is the codimension and $r$ is the reduction number of the hom
ogeneous coordinate ring. In this talk\, I discuss the maximal case $\\deg
(X)=\\binom{e+r}{r}$ and the almost maximal case $\\deg(X)=\\binom{e+r}{r}
-1$. In these cases\, it is possible to explicitly describe certain initia
l ideal of the defining ideal of $X$ and consequently one obtains an expli
cit description of the Betti table. I also discuss how componentwise linea
rity is helpful for computing the Betti tables of projective varieties wit
h an almost maximal degree.\n\nThis is a joint work with Sijong Kwak.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ravi Rao (NMIMS University\, Mumbai)
DTSTART:20210528T130000Z
DTEND:20210528T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/84
DESCRIPTION:Title: Some approaches to a question of Suslin\nby Ravi Rao (NMIMS Unive
rsity\, Mumbai) as part of IIT Bombay Virtual Commutative Algebra Seminar\
n\n\nAbstract\nIn his Helsinki talk in 1978\, A. Suslin asked if a stably
free module of rank d-1 over an affine algebra of dimension d over an alge
braically closed field is free. Here we discuss this question and explain
why it is true when the affine algebra is non-singular\, and when $\\frac{
1}{d!}$ is in the base field.\nThis is joint work with Jean Fasel and Rich
ard Swan.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anand Sawant (Tata Institute of Fundamental Research)
DTSTART:20210604T120000Z
DTEND:20210604T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/85
DESCRIPTION:Title: Naive $\\mathbb A^1$-homotopies on surfaces\nby Anand Sawant (Tat
a Institute of Fundamental Research) as part of IIT Bombay Virtual Commuta
tive Algebra Seminar\n\n\nAbstract\nWe will describe an algebraic criterio
n for two morphisms of the spectrum of a henselian regular local ring into
a smooth projective surface to be naively $\\mathbb A^1$-homotopic. If t
ime permits\, we will explain the significance of this criterion to purely
algebro-geometric questions related to $\\mathbb A^1$-connected component
s. The talk is based on joint work with Chetan Balwe.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Moty Katzman (University of Sheffield)
DTSTART:20210611T120000Z
DTEND:20210611T131500Z
DTSTAMP:20260404T110657Z
UID:VCAS/86
DESCRIPTION:Title: A generalized-fractions approach to computing local cohomology.\n
by Moty Katzman (University of Sheffield) as part of IIT Bombay Virtual Co
mmutative Algebra Seminar\n\n\nAbstract\nThis talk aims to describe a meth
od to compute Lyubeznik Numbers in prime characteristic by applying genera
lized fractions tools to $F$-finite $F$-modules. The main part of the talk
will consist of a very brief introduction to local cohomology\, Lyubeznik
's notion of $F$-finite $F$-modules\, and the Sharp-Zakeri theory of gener
alized fractions. This introduction will be aimed at non-experts.\n\nThe f
inal part of the talk will introduce Lyubeznik numbers and show how these
can be computed using the tools introduced earlier in the talk.\nAll resul
ts are based on a joint project with Rodney Sharp whose results are availa
ble in "Lyubeznik numbers\, $F$-modules and modules of generalized fractio
ns" (arXiv:2006.05438)\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Chardin (Pierre and Marie Curie University\, Jussieu\, France
)
DTSTART:20210618T120000Z
DTEND:20210618T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/87
DESCRIPTION:Title: Multigraded Sylvester forms\, Duality and Elimination\nby Marc Ch
ardin (Pierre and Marie Curie University\, Jussieu\, France) as part of II
T Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nThis talk will
report on joint work with Laurent Busé and Navid Nemati. First\, the cla
ssical situation of the theory of resultants and Sylvester forms in a stan
dard graded algebra will be presented\, as it was developed by Jouanolou i
n a series of monographs. Then we will explain the extension of this theor
y to the multi-graded case (which corresponds to a product of projective s
paces\, in place of a single one). This builds on the previous work of two
Ph.D. students of Jouanolou (Chaichaa and Chkiriba) and an extension of a
duality result from the classical case to this more general setting. We w
ill illustrate these in a very simple case\, by providing a family of form
ulas that extends the work of Dixon.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Schenzel (Institut für Informatik. Martin-Luther-Universit
ät Halle-Wittenberg.)
DTSTART:20210625T120000Z
DTEND:20210625T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/88
DESCRIPTION:Title: News about Koszul and \\v{C}ech complexes: Another view at local coho
mology and completion-I\nby Peter Schenzel (Institut für Informatik.
Martin-Luther-Universität Halle-Wittenberg.) as part of IIT Bombay Virtua
l Commutative Algebra Seminar\n\n\nAbstract\nIn the first part of the talk
\, we present some elementary new facts about \nKoszul and \\v{C}ech compl
exes with respect to a single element. \nWe construct free resolutions of
the \\v{C}ech complex for \na system of elements in a commutative ring. \n
\nThis is used in order to construct quasi-isomorphisms \nbetween the \\v
{C}ech complexes and certain Koszul complexes.\nThe free resolution of the
\\v{Cech} complex is applied in order \nto find relations to the left der
ived functors of the completion \nas a certain Koszul homology. \n\nThis m
aterial provides an elementary introduction to some of the results \nof th
e speakers joint work with A.-M. Simon (see "Completion\, \\v{C}ech \nand
local homology and cohomology. Interactions between them. Springer Monogra
ph\, \n2018") as well as some further developments.\n\nOne focus is the s
tudy of weakly proregular sequences and of proregular \nsequences which pr
ovides a new insight for the local cohomology as well as the \nleft derive
d functors of the completion. Finally we shall present an \napplication to
prisms in the sense of Bhatt and Scholze.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Schenzel (Institut für Informatik. Martin-Luther-Universit
ät Halle-Wittenberg.)
DTSTART:20210702T120000Z
DTEND:20210702T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/89
DESCRIPTION:Title: News about Koszul and \\v{C}ech complexes: Another view at local coho
mology and completion-II\nby Peter Schenzel (Institut für Informatik.
Martin-Luther-Universität Halle-Wittenberg.) as part of IIT Bombay Virtu
al Commutative Algebra Seminar\n\n\nAbstract\nIn the first part of the tal
k\, we present some elementary new facts about \nKoszul and \\v{C}ech comp
lexes with respect to a single element. \nWe construct free resolutions of
the \\v{C}ech complex for \na system of elements in a commutative ring. \
n\nThis is used in order to construct quasi-isomorphisms \nbetween the \\
v{C}ech complexes and certain Koszul complexes.\nThe free resolution of th
e \\v{Cech} complex is applied in order \nto find relations to the left de
rived functors of the completion \nas a certain Koszul homology. \n\nThis
material provides an elementary introduction to some of the results of the
speakers joint work with A.-M. Simon (see "Completion\, \\v{C}ech \nand l
ocal homology and cohomology. Interactions between them. Springer Monograp
h\, \n2018") as well as some further developments.\n\nOne focus is the stu
dy of weakly proregular sequences and of proregular \nsequences which prov
ides a new insight for the local cohomology as well as the \nleft derived
functors of the completion. Finally we shall present an application to pri
sms in the sense of Bhatt and Scholze.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuma Shimomoto (Nihon University\, Tokyo\, Japan)
DTSTART:20210723T120000Z
DTEND:20210723T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/90
DESCRIPTION:Title: Perfectoid spaces-I\nby Kazuma Shimomoto (Nihon University\, Toky
o\, Japan) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\n
Abstract\nIn the first talk\, I begin with a historical review of perfect
oid geometry.\nThen I talk about the definition of perfectoid rings and ti
lting operations.\nSome basic examples are examined. I also show how to us
e perfectoid rings\nby introducing recent results obtained by several peop
le.\nI end with a remark on a classical result on valuation rings.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuma Shimomoto (Nihon University\, Tokyo\, Japan)
DTSTART:20210730T120000Z
DTEND:20210730T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/91
DESCRIPTION:Title: Perfectoid spaces-II\nby Kazuma Shimomoto (Nihon University\, Tok
yo\, Japan) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nIn the second talk\, I start talking about rough ideas of Andr
e's proof\nof the direct summand conjecture. Then I move to basics of almo
st rings and\nformulate the almost purity theorem. I also show some guidel
ines for learning\nperfectoids by showing some fundamental results on perf
ectoid geometry.\nFinally\, I talk about my recent contributions (joint wi
th K. Nakazato and S. Ishiro).\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Montano (New Mexico State University\, Las Cruces\, NM\,
USA)
DTSTART:20210709T130000Z
DTEND:20210709T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/92
DESCRIPTION:Title: Mixed multiplicities of graded families of ideals\nby Jonathan Mo
ntano (New Mexico State University\, Las Cruces\, NM\, USA) as part of IIT
Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nWe show the exi
stence (and define) the mixed multiplicities of arbitrary graded families
of ideals under mild assumptions. In particular\, our methods and results
are valid for the case of arbitrary m-primary graded families. Furthermore
\, we provide a “Volume = Multiplicity formula” for the mixed multipli
cities of graded families of ideals. This is joint work with Yairon Cid-Ru
iz.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yairon Cid-Ruiz (Ghent University\, Krijgslaan\, Belgium)
DTSTART:20210716T120000Z
DTEND:20210716T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/93
DESCRIPTION:Title: Convex bodies and graded families of monomial ideals\nby Yairon C
id-Ruiz (Ghent University\, Krijgslaan\, Belgium) as part of IIT Bombay Vi
rtual Commutative Algebra Seminar\n\n\nAbstract\nWe show that the mixed vo
lumes of arbitrary convex bodies are equal to mixed multiplicities of grad
ed families of monomial ideals and to normalized limits of mixed multiplic
ities of monomial ideals. This result evinces the close relationship betwe
en the theories of mixed volumes from convex geometry and mixed multiplici
ties from commutative algebra. This is joint work with Jonathan Montaño.
\n\nTime permitting\, we will also speak about some joint work with Fateme
h Mohammadi and Leonid Monin regarding "multi-graded algebras and multi-gr
aded linear series".\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Purna Bangere (University of Kansas Lawrence\, KS)
DTSTART:20210806T130000Z
DTEND:20210806T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/94
DESCRIPTION:Title: Syzygies and Gonality\nby Purna Bangere (University of Kansas Law
rence\, KS) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nIn this talk we will deal with recent results on the so-called
property $N_p$ and $M_p.$ These concern the structure of free resolutions
associated with a very ample line bundle on \na projective variety. There
are interesting conjectures and ideas related to the structure of free res
olutions and the intrinsic geometry connected with properties $N_p$ and $
M_p.$ \nA lot of attention has been paid of property $N_p\,$ not so much
for property $M_p.$ In this talk we will describe some new results about p
roperties $M_p$ for an algebraic surface\, \nsome higher dimensional varie
ties\, and even interesting everywhere non reduced schemes called carpets.
\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K.-i. Watanabe (Nihon University)
DTSTART:20210813T120000Z
DTEND:20210813T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/95
DESCRIPTION:Title: Inverse polynomials of symmetric numerical semigroups\nby K.-i. W
atanabe (Nihon University) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\n\nAbstract\nThis is a joint work with Kazufumi Eto (Nippon I
nstitute of Technology).\nThis work was inspired by the talk of M.E. Rossi
(Univ. Genova) at VCAS on Dec. 1\, 2020.\nLet $H \\subset \\mathbb N$ be
a numerical semigroup ring and $k[H]$ be its semigroup ring over any fie
ld $K.$ If $H = ⟨n_1\, \\ldots\,n_e)$\, we express $k[H]$ as $k[H] = k[x
_1\,\\ldots\,x_e]/I_H$ and we want to express $k[H]/(t^h)$ by ”Inverse p
olynomials” of Macaulay.\nWe study the defining ideal of a numerical sem
igroup ring $k[H]$ using the inverse poly- nomial attached to the Artinian
ring $k[H]/(t^h)$ for $h \\in H_+$. I believe this method to express by i
nverse polynomials is very powerful and can be used for many purposes. At
present\, we apply this method for the following cases.\n(1) To give a cri
terion for H to be symmetric or almost symmetric.\n(2) Characterization of
symmetric numerical semigroups of small multiplicity.\n(3) A new proof of
Bresinsky’s Theorem for symmetric semigroups generated by 4\nelements.\
n
LOCATION:https://stable.researchseminars.org/talk/VCAS/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mousumi Mandal (IIT Kharagpur)
DTSTART:20210820T120000Z
DTEND:20210820T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/96
DESCRIPTION:by Mousumi Mandal (IIT Kharagpur) as part of IIT Bombay Virtua
l Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jai Laxmi (Tata Institute of Fundamental Research\, Mumbai)
DTSTART:20210827T120000Z
DTEND:20210827T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/97
DESCRIPTION:Title: Gorenstein ideals of codimension four\nby Jai Laxmi (Tata Institu
te of Fundamental Research\, Mumbai) as part of IIT Bombay Virtual Commuta
tive Algebra Seminar\n\n\nAbstract\nWe explore spinor coordinates on free
resolutions of\ncodimension four Gorenstein ideals. Our analysis of spinor
coordinate\nexamples shows notable differences between Gorenstein ideals
with 4\, 6\,\n7\, and 8 generators compared to those with more generators.
Also\, we\ndiscuss the codimension four Gorenstein ideals resulting from
doubling\ncodimension three perfect ideals. We see\, in particular\, the c
onstruction\nof generic doublings of almost complete intersection perfect
ideals of\ncodimension three.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eloisa Grifo (University of Nebraska\, LIncoln\, NE)
DTSTART:20210903T130000Z
DTEND:20210903T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/98
DESCRIPTION:Title: A survey of Harbourne’s Conjecture\nby Eloisa Grifo (University
of Nebraska\, LIncoln\, NE) as part of IIT Bombay Virtual Commutative Alg
ebra Seminar\n\n\nAbstract\nHarbourne’s conjecture on the containment pr
oblem for symbolic and ordinary powers of ideals is not true in its origin
al form\, but it has sparked a lot of different research avenues. We will
discuss some of the known counterexamples but mostly focus on the differen
t variations of the conjecture that are true or still open.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dumitru Stamate (University of Bucharest\, Romania)
DTSTART:20210910T120000Z
DTEND:20210910T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/99
DESCRIPTION:Title: The trace of the canonical module: algebra and combinatorics\nby
Dumitru Stamate (University of Bucharest\, Romania) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\n\nAbstract\nLet R be a Cohen-Macaul
ay local ring (or positively graded K-algebra) with canonical module $\\om
ega_R.$ The trace of the latter\, $tr(\\omega_R)\,$ is by definition\, the
ideal generated by the images of all R-module homomorphisms from $\\omega
_R$ into R. Since this ideal describes the non-Gorenstein locus of R\, it
can be viewed as a way to measure how far is R from being Gorenstein.\n\nI
n terms of this ideal\, new classes of rings have been introduced\, and th
eir properties are under scrutiny. We discuss some of these approaches\, w
ith a special focus on families of examples coming from combinatorics. \n\
nThis talk is based on joint works with J. Herzog\, T. Hibi\, R. Jafari an
d S. Kumashiro.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winfried Bruns (University of Osnabrueck\, Germany)
DTSTART:20210917T120000Z
DTEND:20210917T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/100
DESCRIPTION:Title: Castelnuovo-Mumford regularity over general base rings\nby Winfr
ied Bruns (University of Osnabrueck\, Germany) as part of IIT Bombay Virtu
al Commutative Algebra Seminar\n\n\nAbstract\nOur talk has two goals. The
first is to give a short introduction to Castelnuovo-Mumford regularity
for standard graded rings over general base rings. The second is to prese
nt a simple and concise proof of a classical result of Cutkosky\, Herzog
and Trung and\, independently\, Kodiyalam asserting that the regularity
of powers of a homogeneous ideal is eventually a linear function of the
exponent in this generality. Finally we show how the flexibility of work
ing over general base rings can be used to give a simple proof for the cha
racterization of "linear powers" in terms of the Rees algebra.\n\nThis is
joint work with Aldo Conca and Matteo Varbaro. See "Castelnuovo-Mumford re
gularity and powers"\, arXiv:2107.14734. An extensive version will be part
of the forthcoming book "Determinants\, Gröber bases and cohomology" wit
h Conca\, Varbaro and Claudiu Raicu.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prashant Sridhar (TIFR\, Mumbai\, India)
DTSTART:20210924T120000Z
DTEND:20210924T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/101
DESCRIPTION:Title: Finding Maximal Cohen-Macaulay modules\nby Prashant Sridhar (TIF
R\, Mumbai\, India) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nIn this talk\, we consider a problem that lies in the c
onfluence of two topics.\nOn one hand\, we have maximal Cohen-Macaulay (MC
M) modules - these are classical objects that have been studied extensivel
y from algebraic and geometric viewpoints. There is a rich theory of MCM m
odules over Cohen-Macaulay (CM) rings and many beautiful connections to th
e singularities of the ring have been discovered. However\, in the absence
of the CM property in the ring\, not as much is known - even the object's
existence is largely unclear.\nOn the other hand\, we have a mixed charac
teristic phenomenon. In 1980\, Paul Roberts showed that the integral closu
re of a regular local ring in an Abelian extension of its quotient field i
s CM\, provided the characteristic of the residue field does not divide th
e degree of the extension. This fails in the "modular case" in mixed chara
cteristic.\nWe will look at some past results in the literature before con
sidering the question of existence of MCMs in the modular case of Roberts'
s theorem.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Fasel (Université de Grenoble I\, Grenoble\, France)
DTSTART:20211001T130000Z
DTEND:20211001T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/102
DESCRIPTION:Title: Cohomotopy groups in algebraic geometry and unimodular rows\nby
Jean Fasel (Université de Grenoble I\, Grenoble\, France) as part of IIT
Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nWe will discuss
the notion of cohomotopy groups in algebraic geometry. Our main example an
d motivation will be the study of unimodular rows\, but we will also discu
ss other examples such as unimodular m x n matrices and Euler class groups
. The general context of this study is motivic homotopy theory\, but we wi
ll avoid technicalities and motivate the constructions using classical top
ology.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krishna Hanumanthu (Chennai Mathematical Institute\, Chennai)
DTSTART:20211008T120000Z
DTEND:20211008T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/103
DESCRIPTION:Title: Seshadri constants\nby Krishna Hanumanthu (Chennai Mathematical
Institute\, Chennai) as part of IIT Bombay Virtual Commutative Algebra Sem
inar\n\n\nAbstract\nSeshadri constants of line bundles on projective varie
ties were defined by J-P. Demailly in 1990\, motivated by an ampleness cri
terion of C. S. Seshadri. They are a measure of local positivity of line
bundles and have interesting connections to different areas in mathematics
. We will discuss some important questions that drive research on Seshadri
constants. We will also discuss their relationship with Waldschmidt const
ants which are invariants attached to homogeneous ideals in polynomial rin
gs.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritvik Ramkumar (UC Berkeley\, CA\, USA)
DTSTART:20211015T140000Z
DTEND:20211015T150000Z
DTSTAMP:20260404T110657Z
UID:VCAS/104
DESCRIPTION:Title: An invitation to the fiber-full scheme\nby Ritvik Ramkumar (UC B
erkeley\, CA\, USA) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nI will introduce the fiber-full scheme which can be see
n as the parameter space that generalizes the Hilbert and Quot schemes by
controlling the entire cohomological data. In particular\, given a sequenc
e of functions $(h_0\,..\,h_n)$ the fiber-full scheme parameterizes subsch
emes $X$ of $P^n$ satisfying dim $H^i(X\,O_X(v)) = h_i(v).$ In this talk I
will sketch the construction of the fiber-full scheme and describe some e
xamples associated with classical Hilbert schemes. I will also discuss var
ious future directions one can pursue. This is joint work with Yairon-Cid
Ruiz.\nChairperson - N. Mohan Kumar\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amalendu Krishna (Indian Institute of Science\, Bangalore)
DTSTART:20211022T120000Z
DTEND:20211022T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/105
DESCRIPTION:Title: Chow groups and Euler class groups of affine varieties\nby Amale
ndu Krishna (Indian Institute of Science\, Bangalore) as part of IIT Bomba
y Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn this talk\, we sha
ll present some results regarding the relation between the Chow groups and
Euler class groups of affine varieties over algebraically closed fields.
We shall show how these results allow one to deduce an old conjecture of M
urthy. If time permits\, we shall discuss some related questions on Chow-W
itt groups of affine varieties over non-algebraically closed.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josh Pollitz (University of Utah)
DTSTART:20211029T130000Z
DTEND:20211029T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/106
DESCRIPTION:Title: Symmetries in Bass and Betti sequences over a complete intersection
ring\nby Josh Pollitz (University of Utah) as part of IIT Bombay Virtu
al Commutative Algebra Seminar\n\n\nAbstract\nDespite homological algebra
over a complete intersection ring being unbounded\, resolutions enjoy poly
nomial growth. That is to say\, for a finitely generated module over a com
plete intersection ring\, its sequence of Bass numbers and its sequence Be
tti numbers are eventually given by quasi-polynomials with period two\; th
e leading terms of the quasi-polynomials are independent of the parity. In
this talk\, I will discuss joint worth with Briggs and McCormick where we
show the leading terms of the two quasi-polynomials agree. The main tool
is a higher order support theory which generalizes the well-studied suppor
t varieties of a complete intersection ring.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Eur (Harvard University)
DTSTART:20211105T140000Z
DTEND:20211105T150000Z
DTSTAMP:20260404T110657Z
UID:VCAS/107
DESCRIPTION:Title: Tautological classes of matroids\nby Christopher Eur (Harvard Un
iversity) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nA
bstract\nAlgebraic geometry has furnished fruitful tools for studying matr
oids\, which are combinatorial abstractions of hyperplane arrangements. We
first survey some recent developments\, pointing out how these developmen
ts remained partially disjoint. We then introduce certain vector bundles (
K-classes) on permutohedral varieties\, which we call "tautological bundle
s (classes)" of matroids\, as a new framework that unifies\, recovers\, an
d extends these recent developments. Our framework leads to new questions
that further probe the boundary between combinatorics and geometry. This i
s joint work with Andrew Berget\, Hunter Spink\, and Dennis Tseng.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seyed Amin Seyed Fakhari (University of Tehran)
DTSTART:20211112T120000Z
DTEND:20211112T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/108
DESCRIPTION:Title: Homological properties of symbolic powers of cover ideals of graphs<
/a>\nby Seyed Amin Seyed Fakhari (University of Tehran) as part of IIT Bom
bay Virtual Commutative Algebra Seminar\n\n\nAbstract\nTo every simple gra
ph\, one associates its edge ideal which is generated by quadratic square
free monomials corresponding to edges of the graph. In this talk\, we stud
y the Alexander dual of edge ideals\, which is called the cover ideal. The
reason for this naming is that the cover ideal is minimally generated by
squarefree monomials corresponding to the minimal vertex covers of the giv
en graph. We review the recent results about the symbolic powers of cover
ideals. In particular\, we characterize all graphs with the property that
every symbolic power of its cover ideal has a linear resolution. Also\, we
determine an upper bound for the regularity of symbolic powers of certain
classes of graphs including bipartite graphs\, unmixed graphs and claw-fr
ee graphs. Moreover\, we study the asymptotic behavior of depth of symboli
c powers of cover ideals.\n\nChairperson: Siamak Yassemi\, IPM\, Tehran\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susan Morey (Texas State University)
DTSTART:20211119T130000Z
DTEND:20211119T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/109
DESCRIPTION:Title: Cellular Resolutions and Powers of Monomial Ideals\nby Susan Mor
ey (Texas State University) as part of IIT Bombay Virtual Commutative Alge
bra Seminar\n\n\nAbstract\nUsing combinatorial structures to obtain resolu
tions of monomial ideals is an idea that traces back to Diana Taylor’s t
hesis\, where a simplex associated to the generators of a monomial ideal w
as used to construct a free resolution of the ideal. This concept has been
expanded over the years\, with various authors determining conditions und
er which simplicial or cellular complexes can be associated to monomial id
eals in ways that produce a free resolution. \nIn a research project ini
tiated at a BIRS workshop “Women in Commutative Algebra” in Fall 2019\
, the\nauthors studied simplicial and cellular structures that produced re
solutions of powers of monomial\nideals. The optimal structure to use depe
nds upon the structure of the monomial ideal. This talk will focus on powe
rs of square-free monomial ideals of projective dimension one. Faridi and
Hersey proved that a monomial ideal has projective dimension one if and on
ly if there is an associated tree (one dimensional acyclic simplicial comp
lex) that supports a free resolution of the ideal. The talk will show how\
, for each power $r >\;1$\, to use the tree associated to a square-free
monomial ideal $I$ of projective dimension one to produce a cellular compl
ex that supports a free resolution of $I^r$. Moreover\, each of these reso
lutions will be minimal resolutions. These cellular resolutions can also b
e viewed as strands of the resolution of the Rees algebra of $I$. This tal
k will contain joint work with Susan Cooper\, Sabine El Khoury\, Sara Fari
di\, Sarah Mayes-Tang\, Liana Sega\, and Sandra Spiroff.\nChairperson - Ta
kayuki Hibi\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nursel Erey (Gebze Technical University)
DTSTART:20211126T120000Z
DTEND:20211126T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/110
DESCRIPTION:Title: Squarefree powers of edge ideals\nby Nursel Erey (Gebze Technica
l University) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\
n\nAbstract\nLet $G$ be a finite simple graph. The edge ideal of $G$\, den
oted by $I(G)$\, is a monomial ideal generated by the monomials that corre
spond to the edges of the graph. In this talk\, we will be interested in r
esolutions of squarefree powers of edge ideals. The $k$th squarefree power
$I(G)^{[k]}$ of $I(G)$ is generated by the squarefree monomials in $I(G)^
k$. We will explore the question of when squarefree powers of edge ideals
are linearly related or have linear resolution. This talk is based on join
t work with Jürgen Herzog\, Takayuki Hibi and Sara Saeedi Madani.\nChairp
erson - Takayuki Hibi\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jerzy Weyman (Jagiellonian University\, Poland)
DTSTART:20211203T130000Z
DTEND:20211203T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/111
DESCRIPTION:Title: Gorenstein ideals of codimension 4\nby Jerzy Weyman (Jagiellonia
n University\, Poland) as part of IIT Bombay Virtual Commutative Algebra S
eminar\n\n\nAbstract\nIn this talk I will describe old and new results on
free resolutions of Gorenstein ideals of codimension 4.\nIn the first part
I will discuss situation in codimension 3 and Kustin-Miller results. Then
I will describe new ideas related to spinor structures and connection to
the exceptional root systems.\nChairperson - Bernd Ulrich\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudiu Raicu (University of Notre Dame)
DTSTART:20211210T130000Z
DTEND:20211210T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/112
DESCRIPTION:Title: Cohomology of line bundles on the incidence correspondence\nby C
laudiu Raicu (University of Notre Dame) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nLet X denote the incidence correspo
ndence (or partial flag variety) parametrizing pairs consisting of a point
in projective space and a hyperplane containing it. I will explain how to
characterize the vanishing and non-vanishing behavior of the cohomology g
roups of line bundles on X over an arbitrary field. For the projective pla
ne\, the results are contained in the thesis of Griffith from the 70s\, wh
ile in characteristic zero the cohomology groups are described in any dime
nsion by the Borel-Weil-Bott theorem. Joint work with Zhao Gao.\nChairpers
on - Bernd Ulrich\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dipankar Ghosh (Indian Institute of Technology\, Hyderabad)
DTSTART:20210107T130000Z
DTEND:20210107T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/113
DESCRIPTION:by Dipankar Ghosh (Indian Institute of Technology\, Hyderabad)
as part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract: TB
A\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naoki Terai (Okayama University)
DTSTART:20211217T120000Z
DTEND:20211217T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/114
DESCRIPTION:Title: Cohen-Macaulay property of weighted edge ideal of very well-covered
graphs\nby Naoki Terai (Okayama University) as part of IIT Bombay Virt
ual Commutative Algebra Seminar\n\n\nAbstract\nWe consider the edge ideals
of edge-weighted very well-covered graphs and discuss their unmixed and C
ohen-Macaulay properties. This is based on a joint work with S.A. Seyed Fa
khari\, K. Shibata and S. Yassemi.\nChairperson - Siamak Yassemi\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Joseph (IIT Bombay)
DTSTART:20210108T120000Z
DTEND:20210108T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/115
DESCRIPTION:by Tony Joseph (IIT Bombay) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dipankar Ghosh (IIT Kharagpur)
DTSTART:20220107T120000Z
DTEND:20220107T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/116
DESCRIPTION:Title: (Non) linear behavior of Castelnuovo-Mumford regularity\nby Dipa
nkar Ghosh (IIT Kharagpur) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\n\nAbstract\nIn the first half\, we shall discuss the importa
nce of Castelnuovo-Mumford regularity and the main motivation to define th
is invariant along with examples. This part is aimed for the graduate stud
ents and should be elementary.\nIn the second half\, we discuss the asympt
otic behavior of Castelnuovo-Mumford regularity of powers of ideals\, and
that of Ext and Tor\, with respect to the homological degree\, over graded
complete intersection rings. We derive from a theorem of Gulliksen\, a li
nearity result for the regularity of Ext modules in high homological degre
es. We show a similar result for Tor\, under the additional hypothesis tha
t high enough Tor modules are supported in dimension at most one\; we then
provide examples showing that the behavior could be pretty hectic when th
e latter condition is not satisfied.\nThis talk is based on our joint pape
rs with Marc Chardin\, Navid Nemati and Tony Puthenpurakal.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Puthenpurakal (IIT Bombay)
DTSTART:20220114T120000Z
DTEND:20220114T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/117
DESCRIPTION:Title: Itoh's conjecture for normal ideals\nby Tony Puthenpurakal (IIT
Bombay) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbs
tract\nLet $(A\,m)$ be a CM local ring and let $I$ be a normal $m$-primar
y ideal with $e_3(I) = 0.$\nThen $G_I(A)$ is CM.\n(note Itoh only conjectu
red it for Gorenstein local rings)\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fatemeh Mohammadi (Ghent University and University of Tromsø)
DTSTART:20220121T120000Z
DTEND:20220121T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/118
DESCRIPTION:Title: Toric degenerations of Grassmannians\nby Fatemeh Mohammadi (Ghen
t University and University of Tromsø) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nToric varieties are popular objects
in algebraic geometry\, as they can be modelled on polytopes and polyhedr
al fans. This is mainly because there is a dictionary between their geomet
ric properties and the combinatorial invariants of their polytopes. This d
ictionary can be extended from toric varieties to arbitrary varieties thro
ugh toric degenerations. \n\nIn this talk\, I will introduce the notion of
toric degenerations which generalizes the fruitful correspondence between
toric varieties and polytopes\, to arbitrary varieties. There are prototy
pic examples of toric degenerations (of Grassmannians) which are related t
o Young tableaux and Gelfand-Cetlin polytopes. I will describe how to obta
in such degenerations using the theory of Gröbner fans and tropical geome
try\, and how their associated polytopes are connected by mutations.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shinya Kumashiro (Oyama College)
DTSTART:20220128T120000Z
DTEND:20220128T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/119
DESCRIPTION:Title: Graded Bourbaki ideals of graded modules and Ideals of reduction num
ber two\nby Shinya Kumashiro (Oyama College) as part of IIT Bombay Vir
tual Commutative Algebra Seminar\n\n\nAbstract\nIn the first part of this
talk\, we explore graded Bourbaki ideals. It is a well-known fact that for
torsionfree modules over Noetherian normal domains\, Bourbaki sequences e
xist. We give criteria in terms of certain attached matrices for a homomor
phism of modules to induce a Bourbaki sequences. In the second part of thi
s talk\, we give an application of graded Bourbaki sequences to Hilbert fu
nctions of m-primary ideals. We give the inequality of the first three Hil
bert coefficients for ideals of reduction number two.\n\nThis talk is base
d on the joint work with J. Herzog and D. I. Stamate.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Van Tuyl (McMaster University)
DTSTART:20220204T130000Z
DTEND:20220204T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/120
DESCRIPTION:Title: Toric ideals of graphs and some of their homological invariants\
nby Adam Van Tuyl (McMaster University) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nThe study of toric ideals of graphs
lies in the intersection of commutative algebra\, algebraic geometry\, an
d combinatorics. Formally\, if $G = (V\,E)$ is a finite simple graph with
edge set $E =\\{e_1\,\\ldots\,e_s\\}$ and vertex set $V = \\{x_1\,\\ldots
\,x_n\\}\,$ then the toric ideal of $G$ is the kernel of the ring homomorp
hism $\\varphi:k[e_1\,\\ldots\,e_s] \\rightarrow k[x_1\,\\ldots\,x_n]$ whe
re $\\varphi(e_i) = x_jx_k$ if the edge $e_i = \\{x_j\,x_k\\}$. Ideally\,
one would like to understand how the homological invariants (e.g. graded
Betti numbers) of $I_G$ are related to the graph $G$. In this talk I will
survey some results connected to this theme\, with an emphasis on the Cas
telnuovo-Mumford regularity of these ideals.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuke Takagi (University of Tokyo)
DTSTART:20220211T120000Z
DTEND:20220211T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/121
DESCRIPTION:Title: Kodaira vanishing for thickenings of globally $F$-regular varieties<
/a>\nby Shunsuke Takagi (University of Tokyo) as part of IIT Bombay Virtua
l Commutative Algebra Seminar\n\n\nAbstract\nBlickle-Bhatt-Lyubeznik-Singh
-Zhang proved that if $X$ is a projective variety over a field $k$ of char
acteristic zero with isolated complete intersection singularities\, then t
he Kodaira vanishing theorem holds for all thickenings of $X$. What if $k$
is of positive characteristic? Kodaira vanishing can fail in positive cha
racteristic\, but it still holds for Frobenius split varieties. In this ta
lk\, I will discuss Kodaira vanishing for thickenings of globally $F$-regu
lar varieties\, a special class of Frobenius split varieties. This talk is
based on joint work with Kenta Sato.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liran Shaul (Charles University\, Prague)
DTSTART:20220218T120000Z
DTEND:20220218T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/122
DESCRIPTION:Title: Special classes of rings in derived commutative algebra\nby Lira
n Shaul (Charles University\, Prague) as part of IIT Bombay Virtual Commut
ative Algebra Seminar\n\n\nAbstract\nThe classes of regular\, Gorenstein a
nd Cohen-Macaulay rings are among the most important classes of rings in c
ommutative algebra and algebraic geometry.\nIn this talk we recall the def
initions and basic properties of these classes\,\nand then explain how to
generalize each of them to derived commutative\nalgebra\, in the context o
f commutative differential graded algebras.\nWe further explain how each o
f these generalizations arise naturally\nin various algebraic geometry con
texts and discuss some applications.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keri Sather-Wagstaff (Clemson University)
DTSTART:20220225T130000Z
DTEND:20220225T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/123
DESCRIPTION:Title: Monomial Ideals Arising from Graph Domination Problems\nby Keri
Sather-Wagstaff (Clemson University) as part of IIT Bombay Virtual Commuta
tive Algebra Seminar\n\n\nAbstract\nGraph domination problems are ubiquito
us in graph theory. In the broadest terms\, they ask how one can ‘observ
e’ an entire graph by designating a certain list of vertices\, following
a proscribed list of rules. An example of this is the vertex covering pro
blem which happens to describe the irredundant irreducible decomposition o
f the edge ideal of a graph. In this talk\, we will survey recent work wit
h various collaborators on other monomial ideal constructions that arise f
rom other graph domination problems\, including one coming from electrical
engineering.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Gubeladze (San Francisco State University)
DTSTART:20220304T140000Z
DTEND:20220304T150000Z
DTSTAMP:20260404T110657Z
UID:VCAS/124
DESCRIPTION:Title: Normal polytopes and ellispoids\nby Joseph Gubeladze (San Franci
sco State University) as part of IIT Bombay Virtual Commutative Algebra Se
minar\n\n\nAbstract\nLattice polytopes are the combinatorial backbone of t
oric varieties. Many important properties of these varieties admit purely
combinatorial description in terms of the underlying polytopes. These incl
ude normality and projective normality. On the other hand\, there are geom
etric properties of polytopes of integer programming/discrete optimization
origin\, which can be used to deduce the aforementioned combinatorial pro
perties: existence of unimodular triangulations or unimodular covers. In t
his talk we present the following recent results: (1) unimodular simplices
in a lattice 3-polytope cover a neighborhood of the boundary if and only
if the polytope is very ample\, (2) the convex hull of lattice points in e
very ellipsoid in R^3 has a unimodular cover\, and (3) for every d at leas
t 5\, there are ellipsoids in R^d\, such that the convex hulls of the latt
ice points in these ellipsoids are not even normal. Part (3) answers a que
stion of Bruns\, Michalek\, and the speaker.\nChaiperson - Siamak Yassemi\
n
LOCATION:https://stable.researchseminars.org/talk/VCAS/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Craig Huneke (University of Virginia)
DTSTART:20220311T130000Z
DTEND:20220311T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/125
DESCRIPTION:Title: Torsion in Commutative Algebra\nby Craig Huneke (University of V
irginia) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAb
stract\nThis talk will be a somewhat historical one\, concerning three pro
blems\ndealing with the idea of torsion. The three problems are those on
symbolic powers\,\nthe Huneke-Wiegand conjecture\, and Berger's conjecture
. Besides talking about\nmy own memories\, we will focus on torsion in te
nsor products.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Smirnov (BCAM-Basque Center for Applied Mathematics)
DTSTART:20220318T120000Z
DTEND:20220318T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/126
DESCRIPTION:Title: Lech's inequality can be sharpened uniformly\nby Ilya Smirnov (B
CAM-Basque Center for Applied Mathematics) as part of IIT Bombay Virtual C
ommutative Algebra Seminar\n\n\nAbstract\nThe classical Lech's inequality
can be viewed as a uniform\, independent of an ideal\, upper bound on the
ratio of the multiplicity and the colength of an m-primary ideal of a loca
l ring. It was also observed by Lech that\, if the dimension is at least t
wo\, it is not sharp for any given ideal. Recently\, we were able to show
more: most of the time\, it is possible to improve Lech's upper bound so t
hat it works for all ideals. I will present the proof of this result and a
ll required background in multiplicity theory.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dharm Veer (Chennai Mathematical Institute)
DTSTART:20220325T120000Z
DTEND:20220325T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/127
DESCRIPTION:Title: On Green-Lazarsfeld property $N_p$ for Hibi rings\nby Dharm Veer
(Chennai Mathematical Institute) as part of IIT Bombay Virtual Commutativ
e Algebra Seminar\n\n\nAbstract\nLet $L$ be a finite distributive lattice.
By Birkhoff's fundamental structure theorem\, $L$ is the ideal lattice of
its subposet $P$ of join-irreducible elements. Write $P=\\{p_1\,\\ldots\,
p_n\\}$ and let $K[t\,z_1\,\\ldots\,z_n]$ be a polynomial ring in $n+1$ va
riables over a field $K.$ The {\\em Hibi ring} associated with $L$ is the
subring of $K[t\,z_1\,\\ldots\,z_n]$ generated by the monomials $u_{\\alp
ha}=t\\prod_{p_i\\in \\alpha}z_i$ where $\\alpha\\in L$. In this talk\, we
show that a Hibi ring satisfies property $N_4$ if and only if it is a pol
ynomial ring or it has a linear resolution. We also discuss a few results
about the property $N_p$ of Hibi rings for $p=2$ and 3. For example\, we s
how that if a Hibi ring satisfies property $N_2$\, then its Segre product
with a polynomial ring in finitely many variables also satisfies property
$N_2$.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumi Tikader (Diamond Harbour Women's University)
DTSTART:20220401T120000Z
DTEND:20220401T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/128
DESCRIPTION:Title: Monic inversion principle of local complete intersection ideal\n
by Soumi Tikader (Diamond Harbour Women's University) as part of IIT Bomba
y Virtual Commutative Algebra Seminar\n\n\nAbstract\nThe renowned Quillen
–Suslin Theorem is closely associated to the Affine Horrocks’ Theorem
on algebraic vector bundles. It says : If $R$ is any commutative\nring a
nd $E$ is a vector bundle on $\\mathbb{A}_{R}^1$ and $E$ extends to a vect
or bundle on $\\mathbb{P}^1\n_R\,$ then $E$ is extended\nfrom $Spec(R).$ T
his is also known as "Monic inversion principle" for projective modules.\n
Here we discuss about \nanalogue of the Monic inversion principle for loc
al complete intersection ideals\nof height $n$ in $R[T]\,$ where $R$ is a
regular domain of dimension $d\,$ which is essentially of\nfinite type ov
er an infinite perfect field of characteristic unequal to $2\,$ and $2n \\
geq d + 3.$ This is a joint work with Mrinal Kanti Das and Md. Ali Zinna.\
n
LOCATION:https://stable.researchseminars.org/talk/VCAS/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Sammartano (Politecnico di Milano)
DTSTART:20220408T120000Z
DTEND:20220408T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/129
DESCRIPTION:Title: Nested Hilbert schemes of the plane\nby Alessio Sammartano (Poli
tecnico di Milano) as part of IIT Bombay Virtual Commutative Algebra Semin
ar\n\n\nAbstract\nThe Hilbert scheme of points Hilb^n(A^2)\, parametrizing
finite subschemes of the plane of degree n\, is a well studied and well b
ehaved parameter space. A classical theorem of Fogarty states that it is a
smooth variety of dimension 2n. By contrast\, the nested Hilbert scheme H
ilb^(n_1\,n_2)(A^2)\, parametrizing nested pairs of subschemes of degrees
n_1 and n_2\, are usually singular\, and very little is known about their
singularities. Using techniques from commutative algebra\, we prove that t
he nested Hilbert scheme Hilb^(n\,2)(A^2) has rational singularities. This
is a joint work with Ritvik Ramkumar.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro De Stefani (Università di Genova)
DTSTART:20220415T120000Z
DTEND:20220415T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/130
DESCRIPTION:Title: A uniform Chevalley theorem for direct summands in mixed characteris
tic\nby Alessandro De Stefani (Università di Genova) as part of IIT B
ombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nLet R be a graded
direct summand of a positively graded polynomial ring over the p-adic int
egers. We exhibit an explicit constant D such that\, for any positive inte
ger n and any homogeneous prime ideal Q of R\, the Dn-th symbolic power of
Q is contained in the n-th power of the homogeneous maximal ideal (p)R +
R_+. The strategy relies on the introduction of a new type of differential
powers\, which do not require the existence of a p-derivation on R. The t
alk will be based on joint work with E. Grifo and J. Jeffries.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuji Yoshino (Okayama University)
DTSTART:20220422T120000Z
DTEND:20220422T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/131
DESCRIPTION:Title: Naive liftings of dg modules\nby Yuji Yoshino (Okayama Universit
y) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract
\nThe naive lifting for dg modules is the new concept introduced by M.Ono\
, S.Nasseh and myself for the purpose of unifying the ideas of lifting and
weak lifting for modules over commutative rings.\nIn this talk I will sho
w how we get the obstruction class of naive liftings\, which in fact coinc
ides with the Atiyah class that has been introduced by Buchweitz-Flenner.\
nThis is a joint work with Saeed Nasseh and Maiko Ono.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eleonore Faber (University of Leeds)
DTSTART:20220429T120000Z
DTEND:20220429T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/132
DESCRIPTION:Title: Cluster structures for the A-infinity singularity\nby Eleonore F
aber (University of Leeds) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\n\nAbstract\nThis talk is about a categorification of the coo
rdinate rings of Grassmannians\nof infinite rank in terms of graded maxima
l Cohen-Macaulay modules over the ring $\\mathbb{C}[x\,y]/(x^k).$ This yie
lds an infinite rank analog of the Grassmannian cluster categories introdu
ced by Jensen\, King\, and Su.\nIn the special case\, $k=2\,$ $\\text{Spec
}(\\mathbb{C}[x\,y]/(x^2))$ is a type $A$-infinity singularity and the ung
raded version of the category of maximal Cohen-Macaulay modules over $\\ma
thbb{C}[x\,y]/(x^2))$ has been studied by Buchweitz\, Greuel\, and Schreye
r in the 1980s. We demonstrate that his category has infinite type $A$ clu
ster combinatorics. In particular\, we show that it has cluster-tilting su
bcategories modeled by certain triangulations of the (completed) infinity-
gon and we can also interpret certain mutations of the category in this mo
del. This is joint work with Jenny August\, Man-Wai Cheung\, Sira Gratz\,
and Sibylle Schroll.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mircea Mustata (University of Michigan)
DTSTART:20220909T130000Z
DTEND:20220909T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/139
DESCRIPTION:Title: An estimate for the F-pure threshold via the roots of the Bernstein-
Sato polynomial\nby Mircea Mustata (University of Michigan) as part of
IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nGiven a smo
oth complex algebraic variety X and a nonzero regular function f\non X\, I
will describe an estimate for the difference between the log canonical th
reshold of f\nand the F-pure threshold of a reduction mod p of f\, in term
s of the roots of the Bernstein-Sato polynomial bf of f. This is based on
some old work with S. Takagi and K.-i. Watanabe on one hand\, and with W.
Zhang on the other hand\, plus one simple observation. Most of the talk wi
ll be devoted to an introduction to the invariants of singularities that\n
feature in the result.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joachim Jelisiejew (University of Warsaw)
DTSTART:20220930T120000Z
DTEND:20220930T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/140
DESCRIPTION:Title: When is a homogeneous ideal a limit of saturated ones?\nby Joach
im Jelisiejew (University of Warsaw) as part of IIT Bombay Virtual Commuta
tive Algebra Seminar\n\n\nAbstract\nLet I be a homogeneous ideal in a poly
nomial ring S. If the Hilbert function of S/I is admissible\, for example
(1\,n\,n\,n\,...) is it natural to ask whether I is a limit of homogeneous
ideals: does there exist a ideal F in S[t] such that F(t = 0) is equal to
I\, while F(t = lambda) is a saturated homogeneous ideal for lambda gener
al. Examples of such limits (for the above Hilbert function) can be constr
ucted e.g. by degenerating I(Gamma)\, where Gamma is a tuple of n general
points on the projective space associated to S. However\, to decide whethe
r a given ideal I is a limit is very much nontrivial. This problem very re
cently became of key interest for applications in the theory of tensors: p
roving that certain ideals are not limits would improve best known lower b
ounds on border ranks of certain important tensors.\nIn the talk I will re
port how surprisingly little is known and present some recent results and
some challenges\, both theoretical and computational. All this is a joint
work with Tomasz Mandziuk.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Parnashree Ghosh (Indian Statistical Institute\, Kolkata\, India)
DTSTART:20221014T120000Z
DTEND:20221014T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/141
DESCRIPTION:Title: On the triviality of a family of linear hyperplanes\nby Parnashr
ee Ghosh (Indian Statistical Institute\, Kolkata\, India) as part of IIT B
ombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nLet k be a field\
, m a positive integer\, $\\mathbb{V}$ an affine subvariety of $\\mathbb{A
}^{m+3}$ defined by a linear relation of the form $x_1^{ r_1}\\cdots x_r^{
r_m} y = F(x_1\,\\ldots \, x_m\, z\, t)\,$ A the coordinate ring of $\\mat
hbb{V}$ and $G = X_1^{ r_1} \\cdots X_r^{r_m} Y − F(X_1\, \\ldots \, X_m
\, Z\, T).$ We exhibit several necessary and sufficient conditions for V t
o be isomorphic $\\mathbb{A}^{m+2}$ and G to be a coordinate in $k[X_1\, \
\ldots \, X_m\, Y\, Z\, T]\,$ under a certain hypothesis on F. Our main re
sult immediately yields a family of higher-dimensional linear hyperplanes
for which the Abhyankar-Sathaye Conjecture holds.\nWe also describe the is
omorphism classes and automorphisms of integral domains of the type A unde
r certain conditions. These results show that for each integer $d\\geq 3\,
$ there is a family of infinitely many pairwise non-isomorphic rings which
are counterexamples to the Zariski Cancellation Problem for dimension d i
n positive characteristic. \nThis is a joint work with Neena Gupta.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louiza Fouli (New Mexico State University\, Las Cruces\, NM\, USA)
DTSTART:20221021T130000Z
DTEND:20221021T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/142
DESCRIPTION:Title: Regular Sequences and the depth function for monomial ideals\nby
Louiza Fouli (New Mexico State University\, Las Cruces\, NM\, USA) as par
t of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn join
t work with T\\`ai Huy H\\`a and Susan Morey we introduced the notion of
initially regular sequences on $R/I$\, where $I$ is any homogeneous ideal
in a polynomial ring $R$. We will discuss this notion\, and show how we ca
n construct certain types of (initially) regular sequences on $R/I$ that g
ive effective bounds on the depth of $R/I$. Moreover\, we will discuss wh
en these sequences remain (initially) regular sequences on $R/I^t$ and giv
e lower bounds on $\\depth R/I^t$ for $t\\ge 2$.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/142/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Xianglong Ni (University of California\, Berkeley\, United StatesU
niversity of California\, Berkeley\, CA\, USA)
DTSTART:20221028T130000Z
DTEND:20221028T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/143
DESCRIPTION:Title: Linkage in codimension three\nby Xianglong Ni (University of Cal
ifornia\, Berkeley\, United StatesUniversity of California\, Berkeley\, CA
\, USA) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbs
tract\nAll perfect ideals of codimension two are in the linkage class of a
complete intersection (licci)\, but in codimension three and beyond this
is no longer the case. I will share some ongoing work\, joint with Lorenzo
Guerrieri and Jerzy Weyman\, which illustrates how the theory of "higher
structure maps" originating from Weyman's generic ring may be used to dist
inguish licci ideals within the broader class of perfect ideals of codimen
sion three.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Erman (University of Wisconsin)
DTSTART:20221007T130000Z
DTEND:20221007T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/144
DESCRIPTION:Title: Matrix factorizations of generic polynomials\nby Daniel Erman (U
niversity of Wisconsin) as part of IIT Bombay Virtual Commutative Algebra
Seminar\n\n\nAbstract\nI’ll discuss the Buchweitz-Greuel-Schreyer Conjec
ture on the minimal size of a matrix factorization\, and my recent proof t
hat the conjecture holds for generic polynomials.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivek Sadhu (IISER Bhopal\, Madhya Pradesh\, India)
DTSTART:20221104T120000Z
DTEND:20221104T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/145
DESCRIPTION:Title: Injectivity of Brauer groups for valuation rings\nby Vivek Sadhu
(IISER Bhopal\, Madhya Pradesh\, India) as part of IIT Bombay Virtual Com
mutative Algebra Seminar\n\n\nAbstract\nIn the non noetherian situation\,
valuation rings often behave like regular rings. We will discuss several s
uch results which are classically known to be true for regular rings\, but
also true for valuation rings. We then focus on Brauer groups. It is well
known that Br(R) injects into Br(K) provided R is a regular domain and K=
qt(R). We observe that the same is true for valuation rings. In fact\, we
will discuss a more general result in the setting of etale cohomology.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ramakrishna Nanduri (IIT Kharagpur\, West Bengal\, India)
DTSTART:20221111T120000Z
DTEND:20221111T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/146
DESCRIPTION:Title: On regularity of (symbolic) Rees algebra and (symbolic) powers of ed
ge & vertex cover ideals of graphs\nby Ramakrishna Nanduri (IIT Kharag
pur\, West Bengal\, India) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\n\nAbstract\nIn this talk\, we discuss about the Castelnuovo-
Mumford regularity (or regularity)\nof Rees algebras and symbolic Rees alg
ebras of certain ideals associated to finite\nsimple graphs and we give va
rious combinatorial upper bounds. Also we study\nupper bounds for symbolic
and ordinary powers of edge and vertex cover ideals of\nsimple graphs.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kohsuke Shibata (Okayama University\, Okayama\, Japan)
DTSTART:20221125T120000Z
DTEND:20221125T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/147
DESCRIPTION:Title: Bounds of the multiplicity of abelian quotient complete intersection
singularities\nby Kohsuke Shibata (Okayama University\, Okayama\, Jap
an) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstrac
t\nWatanabe classified all abelian quotient complete intersection singular
ities. Watanabe defined a special datum in order to classify abelian quoti
ent complete intersection singularities. In this talk\, I investigate the
multiplicities and the log canonical thresholds of abelian quotient comple
te intersection singularities in terms of the special datum. Moreover I gi
ve bounds of the multiplicity of abelian quotient complete intersection si
ngularities.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mina Bigdeli (IPM\, Tehran\, Iran)
DTSTART:20221118T120000Z
DTEND:20221118T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/148
DESCRIPTION:Title: Quadratic monomial ideals with almost linear free resolutions\nb
y Mina Bigdeli (IPM\, Tehran\, Iran) as part of IIT Bombay Virtual Commuta
tive Algebra Seminar\n\n\nAbstract\nThis talk will be about the minimal fr
ee resolution of quadratic monomial ideals. It is well known that a quadra
tic monomial ideal $I$ in the polynomial ring $\\mathbb{K}[x_1\,\\ldots\,
x_n]$\, $\\mathbb{K}$ a field\, has a linear resolution if and only if $I$
is the edge ideal of the complement of a chordal graph\, and this is equi
valent to the linearity of the resolution of all powers of $I$. \n\nIn thi
s talk we will discuss the case that the resolution of a quadratic monomia
l ideal $I$ is linear up to the homological degree $t$ with $t\\geq\\pro
jdim(I)-2$\, where $\\projdim(I)$ denotes the projective dimension of $I$
. As an outcome\, we give a combinatorial classification of such ideals
and also check whether their high powers have a linear resolution.\n\n\nC
hairperson: Siamak Yassemi\, University of Tehran\, Iran\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Utsav Chowdhury (Indian Statical Institute\, Kolkata\, India)
DTSTART:20221216T120000Z
DTEND:20221216T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/150
DESCRIPTION:Title: Characterisation of the affine plane using $\\mathbb{A}^1$-homotopy
theory\nby Utsav Chowdhury (Indian Statical Institute\, Kolkata\, Indi
a) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract
\nCharacterisation of the affine $n$-space is one of the major problem in
affine algebraic geometry. Miyanishi showed an affine complex surface $X$
is isomorphic to $\\mathbb{C}^2$ if $\\mathscr{O}(X)$ is a U.F.D.\, $\\mat
hscr{O}(X)^*= \\mathbb{C}^∗$ and $X$ has a non-trivial $\\mathbb{G}_a$-a
ction [3\, Theorem 1]. Since the orbits of a $\\mathbb{G}_a$-action are af
fine lines\, existence of a non-trivial $\\mathbb{G}_a$-action says that t
here is a non-constant $\\mathbb{A}^1$ in X. Ramanujam showed that a smoot
h complex surface is isomorphic to $\\mathbb{C}^2$ if it is topologically
contractible and it is simply connected at infinity [5]. Topological contr
actibility\, in particular pathconnectedness says that there are non-const
ant intervals in X. On the other hand\, $\\mathbb{A}^1$-homotopy theory ha
s been developed by F.Morel and V.Voevodsky [4] as a connection between al
gebra and topology. An algebrogeometric analogue of topological connectedn
ess is $\\mathbb{A}^1$-connectedness. In this talk\, using ghost homotopy
techniques [2\, Section 3] we will prove that if a surface $X$ is $\\mathb
b{A}^1$-connected\, then there is an open dense subset such that through e
very point there is a non-constant $\\mathbb{A}^1$ in X. As a consequence
using the algebraic characterisation\, we will prove that $\\mathbb{C}^2$
is the only $\\mathbb{A}^1$-contractible smooth complex surface. This answ
ers the conjecture appeared in [1\, Conjecture 5.2.3]. We will also see so
me other useful consequences of this result. This is a joint work with Bim
an Roy. \n\nReferences\n[1] A. Asok\, P. A. Østvær\; A 1 -homotopy theor
y and contractible varieties: a survey\, Homotopy Theory and Arithmetic Ge
ometry – Motivic and Diophantine Aspects. Lecture Notes in Mathematics\,
vol 2292. Springer\, Cham. https://doi.org/10.1007/978-3-030-78977-05. \n
[2] C. Balwe\, A. Hogadi and A. Sawant\; A 1 -connected components of sche
mes. Adv Math\, Volume 282\, 2016. \n[3] M. Miyanishi\; An algebraic chara
cterization of the affine plane. J. Math. Kyoto Univ. 15-1 (1975) 19-184.\
n
LOCATION:https://stable.researchseminars.org/talk/VCAS/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shigeru Kuroda (Tokyo Metropolitan University\, Hachioji\, Japan)
DTSTART:20221223T120000Z
DTEND:20221223T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/151
DESCRIPTION:Title: Z/pZ-actions on the affine space: classification\, invariant ring\,
and plinth ideal\nby Shigeru Kuroda (Tokyo Metropolitan University\, H
achioji\, Japan) as part of IIT Bombay Virtual Commutative Algebra Seminar
\n\n\nAbstract\nLet k be a field of characteristic p>0. In this talk\, we
consider the Z/pZ-actions on the affine n-space over k\, or equivalently t
he order p automorphisms of the polynomial ring k[X] in n variables over k
. For example\, every automorphism induced from a G_a-action is of order p
. Hence\, the famous automorphism of Nagata is of order p. Such an automor
phism is important to study the automorphism group of the k-algebra k[X].\
nWe discuss two topics: (1) classification\, and (2) the relation between
polynomiality of the invariant ring and principality of the plinth ideal.
We also present some conjectures and open problems.\n\ndoi: 10.1007/s00031
-022-09764-2\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vu Quang Thanh
DTSTART:20220106T120000Z
DTEND:20220106T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/152
DESCRIPTION:by Vu Quang Thanh as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Seccia (University of Genoa)
DTSTART:20220113T120000Z
DTEND:20220113T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/153
DESCRIPTION:by Lisa Seccia (University of Genoa) as part of IIT Bombay Vir
tual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A.V. Jayanthan (IIT Madras)
DTSTART:20220120T120000Z
DTEND:20220120T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/154
DESCRIPTION:by A.V. Jayanthan (IIT Madras) as part of IIT Bombay Virtual C
ommutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Caminata (University of Genoa)
DTSTART:20220127T120000Z
DTEND:20220127T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/155
DESCRIPTION:by Alessio Caminata (University of Genoa) as part of IIT Bomba
y Virtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vu Quang Thanh (Hanoi University of Science and Technology\, Hanoi
\, Vietnam)
DTSTART:20230106T120000Z
DTEND:20230106T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/156
DESCRIPTION:Title: Regularity of powers and symbolic powers of squarefree monomial idea
ls\nby Vu Quang Thanh (Hanoi University of Science and Technology\, Ha
noi\, Vietnam) as part of IIT Bombay Virtual Commutative Algebra Seminar\n
\n\nAbstract\nI will discuss the problem of comparing/computing the regula
rity of symbolic powers and regular powers of certain classes of squarefre
e monomial ideals focusing on edge ideals of graphs.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Seccia (University of Genoa\, Genoa\, Italy)
DTSTART:20230113T120000Z
DTEND:20230113T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/157
DESCRIPTION:Title: Weakly- closed graphs and F-purity of binomial edge ideals\nby L
isa Seccia (University of Genoa\, Genoa\, Italy) as part of IIT Bombay Vir
tual Commutative Algebra Seminar\n\n\nAbstract\nHerzog et al. characterize
d closed graphs as the graphs whose binomial edge ideals have a quadratic
Groebner basis. In this talk\, we focus on a generalization of closed grap
hs\, namely weakly-closed graphs (or co-comparability graphs). Building on
known results about Knutson ideals of generic matrices\, we characterize
weakly-closed graphs as the only graphs whose binomial edge ideals are Knu
tson ideals (associated with a certain polynomial f). In doing so\, we re-
prove Matsuda's theorem about the F-purity of binomial edge ideals of weak
ly-closed graphs in prime characteristic and we extend it to generalized b
inomial edge ideals. \nLastly\, we will discuss some open conjectures on t
he F-purity of binomial edge ideals and on the relation between Knutson id
eals and compatible ideals.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A.V. Jayanthan (IIT Madras\, Chennai\, India)
DTSTART:20230120T120000Z
DTEND:20230120T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/158
DESCRIPTION:Title: On the resurgence and asymptotic resurgence of homogeneous ideals\nby A.V. Jayanthan (IIT Madras\, Chennai\, India) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\n\nAbstract\nLet $k$ be a field and
$R = k[x_1\, \\ldots\, x_n]$. We obtain an improved upper bound for asympt
otic resurgence of squarefree monomial ideals in $R$. We study the effect
on the resurgence when sum\, product and intersection of ideals are taken.
We obtain sharp upper and lower bounds for the resurgence and asymptotic
resurgence of cover ideals of finite simple graphs in terms of associated
combinatorial invariants. We also explicitly compute the resurgence and as
ymptotic resurgence of cover ideals of several classes of graphs. We chara
cterize a graph being bipartite in terms of the resurgence and asymptotic
resurgence of edge and cover ideals. We also compute explicitly the resurg
ence and asymptotic resurgence of edge ideals of some classes of graphs.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Caminata (University of Genoa\, Genoa\, Italy)
DTSTART:20230127T120000Z
DTEND:20230127T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/159
DESCRIPTION:Title: Determinantal varieties from point configurations on hypersurfaces\nby Alessio Caminata (University of Genoa\, Genoa\, Italy) as part of I
IT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nPoint configu
rations appear naturally in different contexts\, ranging from the study of
the geometry of data sets to questions in commutative algebra and algebra
ic geometry concerning determinantal varieties and invariant theory. In th
is talk\, we bring these perspectives together: we consider the scheme X_{
r\,d\,n} parametrizing n ordered points in r-dimensional projective space
that lie on a common hypersurface of degree d. We show that this scheme ha
s a determinantal structure and\, if r>1\, we prove that it is irreducible
\, Cohen-Macaulay\, and normal. Moreover\, we give an algebraic and geomet
ric description of the singular locus of X_{r\,d\,n} in terms of Castelnuo
vo-Mumford regularity and d-normality. This yields a complete characteriza
tion of the singular locus of X_{2\,d\,n} and X_{3\,2\,n}. This is joint w
ork with Han-Bom Moon and Luca Schaffler.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Duarte (University of Genoa\, Genoa\, Italy)
DTSTART:20230224T120000Z
DTEND:20230224T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/160
DESCRIPTION:Title: Perturbations of ideals in local rings\nby Luis Duarte (Universi
ty of Genoa\, Genoa\, Italy) as part of IIT Bombay Virtual Commutative Alg
ebra Seminar\n\n\nAbstract\nLet I be an ideal of a Noetherian local ring R
. We study how properties of the ideal change under small perturbations\,
that is\, when I is replaced by an ideal J which is the same as I modulo a
large \npower of the maximal ideal. In particular\, assuming that R/J has
the same Hilbert function as R/I\, we show that the Betti numbers of R/J
coincide with those of R/I. We also compare the local cohomology \nmodules
of R/J with those of R/I.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Blickle (Johannes Gutenberg University Mainz\, Rhineland Pa
latinate\, Germany)
DTSTART:20230217T120000Z
DTEND:20230217T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/161
DESCRIPTION:by Manuel Blickle (Johannes Gutenberg University Mainz\, Rhine
land Palatinate\, Germany) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Seceleanu (University of Nebraska-Lincoln\, Lincoln\, NE
\, USA)
DTSTART:20230331T130000Z
DTEND:20230331T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/162
DESCRIPTION:Title: Principal symmetric ideals\nby Alexandra Seceleanu (University o
f Nebraska-Lincoln\, Lincoln\, NE\, USA) as part of IIT Bombay Virtual Com
mutative Algebra Seminar\n\n\nAbstract\nConsider a homogeneous polynomial
f in variables x_1\,...\,x_n. The set of polynomials obtained from f by pe
rmuting the variables in all possible ways generates an ideal\, which we c
all a principal symmetric ideal. What can we say about the Betti numbers o
f a principal symmetric ideal? I will give a general answer in this talk.
This is a joint work with Megumi Harada and Liana Sega.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Tucker (University of Illinois Chicago\, IL\, USA)
DTSTART:20210203T130000Z
DTEND:20210203T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/163
DESCRIPTION:by Kevin Tucker (University of Illinois Chicago\, IL\, USA) as
part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vaibhav Pandey (Purdue University\, West Lafayette\, IN\, USA)
DTSTART:20230303T130000Z
DTEND:20230303T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/164
DESCRIPTION:Title: Linkage and F-regularity of generic determinantal rings\nby Vaib
hav Pandey (Purdue University\, West Lafayette\, IN\, USA) as part of IIT
Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nWe prove that th
e generic link of a generic determinantal ring of maximal minors is strong
ly F-regular\, hence it has rational singularities. In the process\, we st
rengthen a result of Chardin and Ulrich. They showed that the generic resi
dual intersections of a complete intersection ring with rational singulari
ties again have rational singularities. We show that they are\, in fact\,
strongly F-regular. \n\nIn the mid 1990s\, Hochster and Huneke showed that
generic determinantal rings are strongly F-regular\; however\, their proo
f is quite involved. The techniques that we discuss will allow us to give
a new and simple proof of the strong F-regularity of generic determinantal
rings defined by maximal minors. Time permitting\, we will also share a n
ew proof of the strong F-regularity of determinantal rings defined by mino
rs of any size. This is joint work with Yevgeniya Tarasova.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Meng (Cheng Meng\, Purdue University\, West Lafayette\, IN\,
USA)
DTSTART:20220310T130000Z
DTEND:20220310T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/165
DESCRIPTION:by Cheng Meng (Cheng Meng\, Purdue University\, West Lafayette
\, IN\, USA) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n
Abstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA)
DTSTART:20220324T130000Z
DTEND:20220324T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/166
DESCRIPTION:by Takumi Murayama (Purdue University\, West Lafayette\, IN\,
USA) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract
: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Meng (Purdue University\, West Lafayette\, IN\, USA)
DTSTART:20220310T130000Z
DTEND:20220310T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/167
DESCRIPTION:by Cheng Meng (Purdue University\, West Lafayette\, IN\, USA)
as part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract: TBA
\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/167/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA)
DTSTART:20220324T130000Z
DTEND:20220324T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/168
DESCRIPTION:by Takumi Murayama (Purdue University\, West Lafayette\, IN\,
USA) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract
: TBA\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Meng (Purdue University\, West Lafayette\, IN\, USA)
DTSTART:20230310T120000Z
DTEND:20230310T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/169
DESCRIPTION:Title: Multiplicities in flat local extensions\nby Cheng Meng (Purdue U
niversity\, West Lafayette\, IN\, USA) as part of IIT Bombay Virtual Commu
tative Algebra Seminar\n\n\nAbstract\nWe introduce the notion of strongly
Lech-independent ideals as a generalization of Lech-independent ideals def
ined by Lech and Hanes\, and use this notion to derive inequalities on mul
tiplicities of ideals. In particular we prove that if (R\,m) and (S\,n) ar
e Noetherian local rings of the same dimension\, S is a flat local extensi
on of R\,and up to completion S is standard graded over a field and I=mS i
s homogeneous\, then the multiplicity of R is no greater than that of S.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA)
DTSTART:20230324T120000Z
DTEND:20230324T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/170
DESCRIPTION:Title: Uniform bounds on symbolic powers in regular rings via closure theor
y\nby Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA)
as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nT
he containment problem asks: For a fixed ideal I\, which symbolic powers o
f I are contained in an ordinary power of I? We present a closure-theoreti
c proof of the theorem which says that for ideals I in regular rings R\, t
here is a uniform containment of symbolic powers of I in ordinary powers o
f I.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mitsuyasu Hashimoto (Osaka Metropolitan University\, Sumiyoshi-ku\
, Osaka\, Japan)
DTSTART:20221230T120000Z
DTEND:20221230T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/171
DESCRIPTION:Title: Asymptotic behaviors of the Frobenius pushforwards of the ring of in
variants\nby Mitsuyasu Hashimoto (Osaka Metropolitan University\, Sumi
yoshi-ku\, Osaka\, Japan) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\n\nAbstract\nLet k be an algebraically closed field of char
acteristic p > 0\, n a positive integer\, and V = k^d. Let G be a
finite subgroup of GL(V) without pseudoreflections. Let S = Sym V be t
he symmetric algebra of V\, and A = S^G be the ring of invariants. The
functor (?)^G gives an equivalence between the category {}^*Ref(G\,S)
\, the category of Q-graded S-finite S-reflexive (G\,S)-modules and the
category {}^*Ref(A)\, the category of Q-graded A-finite A-reflexive A-m
odules. As the ring of invariants of the Frobenius pushforward ({}^e S)^
G is the Frobenius pushforward {}^eA\, the study of the (G\,S)-module
{}^e S for various e yields good information on {}^eA. Using this pri
nciple\, we get some results on the properties of A coming from the asym
ptotic behaviors of {}^eA. In this talk\, we talk about the following:\n
\nthe generalized F-signature of A (with Y. Nakajima and with P. Symonds)
.\nExamples of G and V such that A is F-rational\, but not F-regular.
\nExamples of G and V such that (the completion of) A is not of fini
te F-representation type (work in progress with A. Singh).\nGeneralizing
finite groups to finite group schemes G\, we have that s(A)>0 if and onl
y if G is linearly reductive\, and if this is the case\, s(A)=1/|G|\, w
here |G| is the dimension of the coordinate ring k[G] of G\, provided t
he action of G on Spec S is ‘small’ (with F. Kobayashi).\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajib Sarkar (TIFR\, Mumbai\, India)
DTSTART:20221209T120000Z
DTEND:20221209T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/172
DESCRIPTION:Title: Level and pseudo-Gorenstein binomial edge ideals\nby Rajib Sarka
r (TIFR\, Mumbai\, India) as part of IIT Bombay Virtual Commutative Algebr
a Seminar\n\n\nAbstract\nGorenstein binomial edge ideals have been complet
ely characterized and they are the paths only. There are two interesting g
eneralizations of Gorenstein rings: level rings and pseudo-Gorenstein ring
s. In the first half\, we will talk about the behavior of the levelness an
d pseudo-Gorensteinness on the decomposable graphs and cone graphs. \nIn t
he next half\, we will discuss the characterization of Cohen-Macaulay bino
mial edge ideals of bipartite graphs and then their levelness and pseudo-G
orensteinness.\nThis talk is based on the joint work with Giancarlo Rinald
o.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christine Berkesh (University of Minnesota\, Minneapolis\, MN\, US
A)
DTSTART:20230317T120000Z
DTEND:20230317T130000Z
DTSTAMP:20260404T110657Z
UID:VCAS/173
DESCRIPTION:Title: Differential operators\, retracts\, and toric face rings\nby Chr
istine Berkesh (University of Minnesota\, Minneapolis\, MN\, USA) as part
of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nToric fac
e rings\, introduced by Stanley\, are simultaneous generalizations of Stan
ley–Reisner rings and affine semigroup rings\, among others. We use the
combinatorics of the fan underlying these rings to inductively compute the
ir rings of differential operators. Along the way\, we discover a new diff
erential characterization of the Gorenstein property for affine semigroup
rings. Our approach applies to a more general class of rings\, which we ca
ll algebras realized by retracts. This is joint work with C-Y. Chan\, P. K
lein\, L. Matusevich\, J. Page\, and J. Vassilev.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/173/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Saugata Basu (Purdue University\, West Lafayette\, IN\, USA)
DTSTART:20230210T130000Z
DTEND:20230210T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/174
DESCRIPTION:Title: Homology of symmetric semi-algebraic sets\nby Saugata Basu (Purd
ue University\, West Lafayette\, IN\, USA) as part of IIT Bombay Virtual C
ommutative Algebra Seminar\n\n\nAbstract\nStudying the homology groups of
semi-algebraic subsets of $\\mathbb{R}^n$ and obtaining upper bounds\non t
he Betti numbers has been a classical topic in real algebraic geometry beg
inning with the\nwork of Petrovskii and Oleinik\, Thom\, and Milnor. In th
is talk I will consider semi-algebraic\nsubsets of $\\mathbb{R}^n$ which a
re defined by symmetric polynomials and are thus stable under the\nstandar
d action of the symmetric group $\\mathfrak{S}_n$ on $\\mathbb{R}^n$.\nThe
homology groups (with rational coefficients) of such sets\nthus acquire e
xtra structure as $\\mathfrak{S}_n$-modules leading to\npossible refinemen
ts on the classical bounds. I will also mention some \nconnections with a
homological stability conjecture.\n \nJoint work (separately) with Daniel
Perrucci and Cordian Riener.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Tucker (University of Illinois Chicago\, IL\, USA)
DTSTART:20230203T130000Z
DTEND:20230203T140000Z
DTSTAMP:20260404T110657Z
UID:VCAS/175
DESCRIPTION:Title: The Theory of F-rational Signature\nby Kevin Tucker (University
of Illinois Chicago\, IL\, USA) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nThe celebrated results of Smith\, Hara\, an
d Mehta-Srinivas connect rational singularities in characteristic zero aft
er reduction to characteristic p > 0 with F-rational singularities. In rec
ent years\, a number of invariants defined via Frobenius in positive chara
cteristics have been introduced as quantitative measures of F-rationality.
These include the F-rational signature (Hochster-Yao)\, relative F-ration
al signature (Smirnov-Tucker)\, and dual F-signature (Sannai). In this tal
k\, I will discuss new results in joint work with Smirnov relating each of
these invariants. In particular\, we show that the relative F-rational si
gnature and dual F-signature coincide\, while also verifying that the dual
F-signature limit converges.\n
LOCATION:https://stable.researchseminars.org/talk/VCAS/175/
END:VEVENT
END:VCALENDAR