BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Melvin Hochster (University of MIchigan)
DTSTART;VALUE=DATE-TIME:20200721T130000Z
DTEND;VALUE=DATE-TIME:20200721T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/1
DESCRIPTION:Title: Tig
ht Closure\, lim Cohen-Maculay sequences\, content of local cohomology\, a
nd related open questions - Part 1\nby Melvin Hochster (University of
MIchigan) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nA
bstract\nThe talks will give multiple characterizations of tight closure\,
discuss some of its applications\, indicate connections with the existen
ce of big and small Cohen-Macaulay algebras and modules\, as well as varia
nt notions\, and also explain connections with the theory of content. Th
ere will be some discussion of the many open questions in the area\, inclu
ding the very long standing problem of proving that Serre intersection mul
tiplicities have the behavior one expects.\n
LOCATION:https://researchseminars.org/talk/VCAS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hai Long Dao (The University of Kansas)
DTSTART;VALUE=DATE-TIME:20200724T130000Z
DTEND;VALUE=DATE-TIME:20200724T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/2
DESCRIPTION:Title: Ref
lexive modules over curve singularities\nby Hai Long Dao (The Universi
ty of Kansas) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\
n\nAbstract\nA finitely generated module $M$ over a commutative ring $R$ i
s called reflexive if the natural map from $M$ to $M^{**} = Hom(Hom(M\,R)\
, R)$ is an isomorphism. In understanding reflexive modules\, the case of
dimension one is crucial. If $R$ is Gorenstein\, then any maximal Cohen-Ma
caulay module is reflexive\, but in general it is quite hard to understand
reflexive modules even over well-studied one-dimensional singularities. I
n this work\, joint with Sarasij Maitra and Prashanth Sridhar\, we will ad
dress this problem and give some partial answers.\n
LOCATION:https://researchseminars.org/talk/VCAS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Melvin Hochster (University of Michigan)
DTSTART;VALUE=DATE-TIME:20200728T130000Z
DTEND;VALUE=DATE-TIME:20200728T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/3
DESCRIPTION:Title: Tig
ht Closure\, lim Cohen-Maculay sequences\, content of local cohomology\, a
nd related open questions - Part 2\nby Melvin Hochster (University of
Michigan) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbs
tract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Linquan Ma (Purdue University)
DTSTART;VALUE=DATE-TIME:20200804T130000Z
DTEND;VALUE=DATE-TIME:20200804T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/4
DESCRIPTION:Title: The
deformation problem for $F$-injective singularities\nby Linquan Ma (P
urdue University) as part of IIT Bombay Virtual Commutative Algebra Semina
r\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudia Polini (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20201103T130000Z
DTEND;VALUE=DATE-TIME:20201103T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/5
DESCRIPTION:Title: The
core of ideals\nby Claudia Polini (University of Notre Dame) as part
of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nLet I be
an ideal in a Noetherian commutative ring. Among all the closures\nof I\,
the integral closure plays a central role. A reduction of I\nis a sub idea
l with the same integral closure.\nWe can think of reductions as simplific
ations of the given ideal\,\nwhich carry most of the information about I i
tself but\, in general\,\nwith fewer generators. Minimal reductions\, redu
ctions\nminimal with respect to inclusion\, are loosely speaking the\ncoun
terpart of the integral closure. However\,\nunlike the integral closure\,
minimal reductions are not unique.\nFor this reason\, we consider their i
ntersection\, called the core of\nI. The core is related to adjoint and\n
multiplier ideals. Motivation for studying\nthis object comes from the Bri
ancon-Skoda theorem. Furthermore\,\na better understanding of the core cou
ld lead\nto solving Kawamata's conjecture on the non-vanishing of\nsection
s of a certain line bundle. In this talk\, I will discuss the\nimportance
of the core\, its ubiquity in algebra and geometry\,\nand some effective f
ormulas for its computation.\n
LOCATION:https://researchseminars.org/talk/VCAS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudia Polini (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20201106T130000Z
DTEND;VALUE=DATE-TIME:20201106T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/6
DESCRIPTION:Title: The
core of monomial ideals\nby Claudia Polini (University of Notre Dame)
as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\n
Let $I$ be a monomial ideal. Even though there may not exist any\nproper r
eduction of $I$ which is monomial (or even homogeneous)\, the\nintersectio
n of all reductions\, the core\, is again a monomial ideal.\nThe integral
closure and the adjoint of a monomial ideal are again\nmonomial ideals and
can be described in terms of the Newton\npolyhedron of $I$. Such a descri
ption cannot exist for the core\,\nsince the Newton polyhedron only recove
rs the integral closure of\nthe ideal\, whereas the core may change when p
assing from $I$ to\nits integral closure. When attempting to derive any ki
nd of combinatorial\ndescription for the core of a monomial ideal from the
known colon\nformulas\, one faces the problem that the colon formula invo
lves\nnon-monomial ideals\, unless $I$ has a reduction $J$ generated by a\
nmonomial regular sequence. Instead\, in joint work with Ulrich and\nVitul
li we exploit the existence of such non-monomial reductions to\ndevise an
interpretation of the core in terms of monomial\noperations. This algorit
hm provides a new interpretation of the\ncore as the largest monomial idea
l contained in a general locally\nminimal reduction of $I$. In recent join
t 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 monomial
ideals generated in one-degree.\n
LOCATION:https://researchseminars.org/talk/VCAS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Jeffries (University of Nebraska)
DTSTART;VALUE=DATE-TIME:20201023T130000Z
DTEND;VALUE=DATE-TIME:20201023T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/7
DESCRIPTION:Title: Fai
thfulness of top local cohomology modules in domains\nby Jack Jeffries
(University of Nebraska) as part of IIT Bombay Virtual Commutative Algebr
a Seminar\n\n\nAbstract\nInspired by a question of Lynch\, we consider the
following question: under what conditions is the highest nonvanishing loc
al cohomology module of a domain $R$ with support in an ideal $I$ faithful
as an R-module? We will review some of what is known about this question\
, and provide an affirmative answer in positive characteristic when the co
homological dimension is equal to the number of generators of the ideal. T
his is based on joint work with Mel Hochster.\n
LOCATION:https://researchseminars.org/talk/VCAS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ken-ichi Yoshida (Nihon University\, Japan)
DTSTART;VALUE=DATE-TIME:20201222T120000Z
DTEND;VALUE=DATE-TIME:20201222T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/8
DESCRIPTION:Title: Low
er bound on Hilbert-Kunz multiplicities and some related results.\nby
Ken-ichi Yoshida (Nihon University\, Japan) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\nIn my talk\, we introduce some
results of lower bounds on Hilbert-Kunz multiplicities\nfor non-regular lo
cal rings. In the later half\, we will discuss the upper bound\non F-signa
ture.\n
LOCATION:https://researchseminars.org/talk/VCAS/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro De Stefani\, (University of Genoa)
DTSTART;VALUE=DATE-TIME:20200811T120000Z
DTEND;VALUE=DATE-TIME:20200811T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/9
DESCRIPTION:Title: Def
ormation and stability of F-injective singularities\nby Alessandro De
Stefani\, (University of Genoa) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nPicking up from the talk given by Linquan M
a\, I will discuss some more cases where deformation of F-injectivity is k
nown to hold\, and I will discuss the related notion of m-adic stability.
The talk will be based on joint projects with Linquan Ma (deformation) and
Ilya Smirnov (stability).\n
LOCATION:https://researchseminars.org/talk/VCAS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Núñez Betancourt (CIMAT\, Mexico)
DTSTART;VALUE=DATE-TIME:20200814T130000Z
DTEND;VALUE=DATE-TIME:20200814T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/10
DESCRIPTION:Title: Sp
littings and symbolic powers of Ideals\nby Luis Núñez Betancourt (CI
MAT\, Mexico) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\
n\nAbstract\nSplittings of Frobenius have been employed to study the singu
larities\nand cohomology of rings. In this talk we will employ ideas inspi
red by\nthis technique to obtain results of symbolic powers of monomial an
d\ndeterminantal ideals. This is joint work with Jonathan Montaño.\n
LOCATION:https://researchseminars.org/talk/VCAS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pham Hung Quy (FPT University\, Hanoi -)
DTSTART;VALUE=DATE-TIME:20200818T120000Z
DTEND;VALUE=DATE-TIME:20200818T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/11
DESCRIPTION:Title: Fr
obenius closure of parameter ideals\nby Pham Hung Quy (FPT University\
, Hanoi -) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\n
Abstract\nWe discuss recent results about Frobenius closure of parameter i
deals and $F$-singularities as well as the Frobenius test exponent of para
meter ideals.\n
LOCATION:https://researchseminars.org/talk/VCAS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arindam Banerjee (RKM Vivekananda Institute\, Belur)
DTSTART;VALUE=DATE-TIME:20200821T120000Z
DTEND;VALUE=DATE-TIME:20200821T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/12
DESCRIPTION:Title: Ly
ubeznik numbers\nby Arindam Banerjee (RKM Vivekananda Institute\, Belu
r) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract
\nLyubeznik numbers are certain Bass numbers of local cohomology modules a
ssociated to local rings containing a field. This numerical invariants are
known to have many interesting homological\, geometric and topological pr
operties and have been an active area of research. In this talk we plan to
give a brief overview of these.\n
LOCATION:https://researchseminars.org/talk/VCAS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:N. V. Trung (Hanoi Institute of Mathematics)
DTSTART;VALUE=DATE-TIME:20201029T120000Z
DTEND;VALUE=DATE-TIME:20201029T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/13
DESCRIPTION:Title: Mu
ltiplicity sequence and integral dependence\nby N. V. Trung (Hanoi Ins
titute of Mathematics) as part of IIT Bombay Virtual Commutative Algebra S
eminar\n\n\nAbstract\nThe first numerical criterion for integral dependenc
e was proved by Rees in 1961 which states that two m-primary ideals $I \\s
ubset J$ in an equidimensional and universally catenary local ring $(R\, m
)$ have the same integral closure if and only if they have the same Hilber
t-Samuel multiplicity. This result plays an important role in Teissier's w
ork on the equisingularity of families of hypersurfaces with isolated sing
ularities. For hypersurfaces with non-isolated singularities\, one needs
a similar numerical criterion for integral dependence of non-$m$-primary i
deals. Since the Hilbert-Samuel multiplicity is no longer defined for non-
$m$-primary ideals\, one has to use other notions of multiplicities that c
an be used to check for integral dependence. A possibility is the multipli
city sequence which was introduced by Achilles and Manaresi in 1997 and ha
s its origin in the intersection numbers of the Stuckrad-Vogel algorithm.
It was conjectured that two arbitrary ideals $I \\subset J$ in an equidime
nsional and universally catenary local ring have the same integral closure
if and only if they have the same multiplicity sequence. This talk will p
resent a recent solution of this conjecture by Polini\, Trung\, Ulrich and
Validashti.\n
LOCATION:https://researchseminars.org/talk/VCAS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Neena Gupta (Indian Statistical Institute\, Kolkata)
DTSTART;VALUE=DATE-TIME:20200731T120000Z
DTEND;VALUE=DATE-TIME:20200731T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/14
DESCRIPTION:Title: On
the triviality of the affine threefold $x^my = F(x\, z\, t)$ - Part 2
\nby Neena Gupta (Indian Statistical Institute\, Kolkata) as part of IIT B
ombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn this talk we w
ill 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 co
ordinate ring of such a threefold to be a polynomial ring. For instance\,
we will see that this problem of four variables reduces to the equivalent
but simpler two-variable question as to whether F(0\, z\, t) defines an e
mbedded line in the affine plane. As one immediate consequence\, one read
ily 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 t
he obvious fact that $z^2+t^3$ is not a coordinate. The theory on the abov
e threefolds connects several central problems on Affine Algebraic Geometr
y. It links the study of these threefolds with the famous Abhyankar-Moh
“Epimorphism Theorem” in characteristic zero and the Segre-Nagata line
s in positive characteristic. We will also see a simplified proof of the
triviality of most of the Asanuma threefolds (to be defined in the talk) a
nd an affirmative solution to a special case of the Abhyankar-Sathaye Conj
ecture. Using the theory\, we will also give a recipe for constructing inf
initely many counterexample to the Zariski Cancellation Problem (ZCP) in p
ositive characteristic. This will give a simplified proof of the speaker's
earlier result on the negative solution for the ZCP.\n
LOCATION:https://researchseminars.org/talk/VCAS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivek Mukundan (Indian Institute of Technoogy Delhi)
DTSTART;VALUE=DATE-TIME:20200825T120000Z
DTEND;VALUE=DATE-TIME:20200825T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/15
DESCRIPTION:Title: Re
duction to characteristic p - Part 1\nby Vivek Mukundan (Indian Instit
ute of Technoogy Delhi) as part of IIT Bombay Virtual Commutative Algebra
Seminar\n\n\nAbstract\nThis is an expository talk introducing the methods
of reducing to characteristic $p$. The main tools and general notions ne
cessary to reduce a problem to characteristic p will be discussed in this
talk. It is based on chapter 2 of the excellent resource "Tight Ccousres i
n Characterisitic zero" by Hochster and Huneke. We will be restricting our
selves to the case of affine algebras since it is more accessible.\n
LOCATION:https://researchseminars.org/talk/VCAS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivek Mukundan (Indian Institute of Technoogy Delhi)
DTSTART;VALUE=DATE-TIME:20200828T120000Z
DTEND;VALUE=DATE-TIME:20200828T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/16
DESCRIPTION:Title: Re
duction to characteristic p- Part 2\nby Vivek Mukundan (Indian Institu
te of Technoogy Delhi) as part of IIT Bombay Virtual Commutative Algebra S
eminar\n\n\nAbstract\nThis talk presents problems solved by using the meth
od of reduction to characteristic $p.$ Mainly\, we present two nice proble
ms which have been solved using the reduction to characteristic p methods.
We also present a recent result in the field of symbolic power which uses
the reduction to characteristic $p.$\n
LOCATION:https://researchseminars.org/talk/VCAS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Varbaro (University of Genoa)
DTSTART;VALUE=DATE-TIME:20200901T120000Z
DTEND;VALUE=DATE-TIME:20200901T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/17
DESCRIPTION:Title: F-
splittings of the polynomial ring and compatibly split homogeneous ideals<
/a>\nby Matteo Varbaro (University of Genoa) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\nA polynomial ring R in n varia
bles over a field K of positive characteristic is F-split. It has many F-s
plittings. When K is a perfect field every F-splitting is given by a polyn
omial g in R with the monomial u^{p-1} in its support (where u is the prod
uct of all the variables) occurring with coefficient 1\, plus a further co
ndition\, which is not needed if g is homogeneous (w.r.t. any positive gra
ding). 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 th
is 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 compatibly spl
it 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 so
me examples\, focusing especially on determinantal ideals.\n
LOCATION:https://researchseminars.org/talk/VCAS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mandira Mondal (Chennai Mathematical Institute)
DTSTART;VALUE=DATE-TIME:20200904T120000Z
DTEND;VALUE=DATE-TIME:20200904T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/18
DESCRIPTION:Title: De
nsity functions for the coefficients of the Hilbert-Kunz function of polyt
opal monoid algebra\nby Mandira Mondal (Chennai Mathematical Institute
) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\
nWe shall discuss Hilbert-Kunz density function of a\nNoetherian standard
graded ring over a perfect field of characteristic $p>0$. We will \nalso
talk about the second coeffcient of the Hilbert-Kunz function and the poss
ibility of existence\nof a $\\beta$-density function for this coefficient.
\n\nWatanabe and Eto have shown that Hilbert-Kunz multiplicity of affine
monoid rings with respect to a monomial ideal of finite colength can be ex
pressed as relative\nvolume of certain nice set arising from the convex ge
ometry associated to the ring. In this talk\, we shall discuss similar exp
ression for the density functions of polytopal monoid algebra with respect
to the homogeneous maximal ideal in terms of the associated convex geomet
ric structure. This is a joint work with Prof. V. Trivedi. We shall also d
iscuss the existence of $\\beta$-density function for monomial prime ideal
s of hight one of these rings in this context.\n
LOCATION:https://researchseminars.org/talk/VCAS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irena Swanson (Purdue University)
DTSTART;VALUE=DATE-TIME:20200908T130000Z
DTEND;VALUE=DATE-TIME:20200908T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/19
DESCRIPTION:Title: Pr
imary decomposition and powers of ideals\nby Irena Swanson (Purdue Uni
versity) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAb
stract\nThis talk is about associated primes of powers of an ideal in Noet
herian\ncommutative rings. Brodmann proved that the set of associated pri
mes\nstabilizes for large powers. In general\, the number of associated p
rimes can\ngo up or down as the exponent increases. This talk is about se
quences\n$\\{a_n\\}$ for which there exists an ideal $I$ in a Noetherian c
ommutative\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://researchseminars.org/talk/VCAS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Puthenpurakal (IIT Bombay)
DTSTART;VALUE=DATE-TIME:20200911T120000Z
DTEND;VALUE=DATE-TIME:20200911T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/20
DESCRIPTION:Title: Ho
mological algebra over complete intersections\nby Tony Puthenpurakal (
IIT Bombay) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nWe discuss Eisenbud operators over a complete intersection. As
an application we prove that if A is a strict complete intersection of pos
itive dimension and if M is\na maximal CM A-module with bounded betti numb
ers then the Hilbert function of M is non-decreasing\n
LOCATION:https://researchseminars.org/talk/VCAS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Briggs (University of Utah)
DTSTART;VALUE=DATE-TIME:20200915T133000Z
DTEND;VALUE=DATE-TIME:20200915T143000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/21
DESCRIPTION:Title: On
a conjecture of Vasconcelos - Part 1\nby Ben Briggs (University of Ut
ah) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstrac
t\nThese two talks are about the following theorem: If I is an ideal of fi
nite projective dimension in a ring $R\,$ and the conormal module $I/I^2$
has finite projective dimension over R/I\, then I is locally generated by
a regular sequence. This was conjectured by Vasconcelos\, after he and (se
parately) Ferrand established the case that the conormal module is project
ive.\n\nThe key tool is the homotopy Lie algebra\, an object sitting at th
e centre of a bridge between commutative algebra and rational homotopy the
ory. In the first part I will explain what the homotopy Lie algebra is\, a
nd how it can be constructed by differential graded algebra techniques\, f
ollowing the work of Avramov. In the second part I will bring all of the i
ngredients together and\, hopefully\, present the proof of Vasconcelos' co
njecture.\n
LOCATION:https://researchseminars.org/talk/VCAS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ben Briggs (University of Utah)
DTSTART;VALUE=DATE-TIME:20200918T133000Z
DTEND;VALUE=DATE-TIME:20200918T143000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/22
DESCRIPTION:Title: On
a conjecture of Vasconcelos - Part 2\nby Ben Briggs (University of Ut
ah) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract:
TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuke Takagi (University of Tokyo)
DTSTART;VALUE=DATE-TIME:20200922T120000Z
DTEND;VALUE=DATE-TIME:20200922T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/23
DESCRIPTION:Title: $F
$-singularities and singularities in birational geometry - Part 1\nby
Shunsuke Takagi (University of Tokyo) as part of IIT Bombay Virtual Commut
ative Algebra Seminar\n\n\nAbstract\n$F$-singularities are singularities i
n positive characteristic defined using the Frobenius map and there are fo
ur basic classes of $F$-singularities: $F$-regular\, $F$-pure\, $F$-ration
al and $F$-injective singularities. They conjecturally correspond via redu
ction modulo $p$ to singularities appearing in complex birational geometry
. In the first talk\, I will survey basic properties of $F$-singularities
. In the second talk\, I will explain what is known and what is not known
about the correspondence of $F$-singularities and singularities in biratio
nal geometry. If the time permits\, I will also discuss its geometric appl
ications.\n
LOCATION:https://researchseminars.org/talk/VCAS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuke Takagi (University of Tokyo)
DTSTART;VALUE=DATE-TIME:20200925T120000Z
DTEND;VALUE=DATE-TIME:20200925T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K. N. Raghavan (Institute of Mathematical Sciences\, Chennai)
DTSTART;VALUE=DATE-TIME:20200929T120000Z
DTEND;VALUE=DATE-TIME:20200929T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/25
DESCRIPTION:Title: Mu
ltiplicities of points on Schubert varieties in the Grassmannian-I\nby
K. N. Raghavan (Institute of Mathematical Sciences\, Chennai) as part of
IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K. N. Raghavan (Institute of Mathematical Sciences\, Chennai)
DTSTART;VALUE=DATE-TIME:20201002T120000Z
DTEND;VALUE=DATE-TIME:20201002T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/26
DESCRIPTION:Title: Mu
ltiplicities of points on Schubert varieties in the Grassmannian-II\nb
y K. N. Raghavan (Institute of Mathematical Sciences\, Chennai) as part of
IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nGiven an ar
bitrary point on a Schubert (sub)variety in a Grassmannian\, how to comp
ute the Hilbert function (and\, in particular\, the multiplicity) of the l
ocal ring at that point? A solution to this problem based on "standard
monomial theory" was conjectured by Kreiman-Lakshmibai circa 2000 and the
conjecture was proved about a year or two later by them and independently
also by Kodiyalam and the speaker. The two talks will be an exposition o
f this material aimed at non-experts in the sense that we will not presume
familiarity with Grassmannians (let alone flag varieties) or Schubert var
ieties. \n\nThere are two steps to the solution. The first translate
s the problem from geometry to algebra and in turn to combinatorics. The
second is a solution of the resulting combinatorial problem\, which invo
lves establishing a bijection between two combinatorially defined sets.
The two talks will roughly deal with these two steps respectively.\n\nThr
ee aspects of the combinatorial formulation of the problem (and its soluti
on) are noteworthy: (A) it shows that the natural determinantal generat
ors 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 mu
ltiplicity as counting certain non-intersecting lattice paths\; and (C) a
s was observed by Kreiman some years later\, the combinatorial bijection
is a kind of Robinson-Schensted-Knuth correspondence\, which he calls th
e "bounded RSK".\n\nDeterminantal varieties arise as tangent cones at poin
ts on Schubert varieties (in the Grassmannian)\, and thus one recovers mul
tiplicity formulas for these obtained earlier by Abhyankar and Herzog-Trun
g. (The multiplicity part of the Kreiman-Lakshmibai conjecture was also p
roved by Krattenthaler\, but by very different methods.)\n\nWhat about Sc
hubert varieties in other (full or partial) flag varieties (G/Q with Q bei
ng a parabolic subgroup of a reductive algebraic group G)? The problem r
emains 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 by vario
us authors using various methods that solve the problem in various special
cases. Time permitting\, we will give some indication of these result
s\, without however any attempt at comprehensiveness.\n
LOCATION:https://researchseminars.org/talk/VCAS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mrinal Das (ISI\, Kolkata)
DTSTART;VALUE=DATE-TIME:20201006T120000Z
DTEND;VALUE=DATE-TIME:20201006T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/27
DESCRIPTION:Title: So
me open problems in projective modules and complete intersections\nby
Mrinal Das (ISI\, Kolkata) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\n\nAbstract\nConsider a surjective $k$-algebra morphism\, whe
re k is a field\, from a polynomial ring of \n$n$ variables to a polynomi
al ring of $m$ variables over $k.$ Is the kernel generated by \n$n - m$ el
ements? Our discussion will primarily be around this question and its vari
ants.\n
LOCATION:https://researchseminars.org/talk/VCAS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sarang Sane (IIT Madras)
DTSTART;VALUE=DATE-TIME:20201009T120000Z
DTEND;VALUE=DATE-TIME:20201009T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/28
DESCRIPTION:Title: $K
_0$ and ideals\nby Sarang Sane (IIT Madras) as part of IIT Bombay Virt
ual Commutative Algebra Seminar\n\n\nAbstract\nWe begin by discussing $K_0
$ and defining $K_1$ for a ring $R$ and the exact sequence connecting them
on localization with respect to a multiplicative set $S$. More generally\
, 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 properties of th
e ideal $I$ with the intermediate term in the sequence.\n
LOCATION:https://researchseminars.org/talk/VCAS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kamran Divaani Aazar (IPM\, Tehran)
DTSTART;VALUE=DATE-TIME:20201013T120000Z
DTEND;VALUE=DATE-TIME:20201013T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/29
DESCRIPTION:Title: A
survey on the finiteness properties of local cohomology modules - Part 1\nby Kamran Divaani Aazar (IPM\, Tehran) as part of IIT Bombay Virtual C
ommutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kamran Divaani Aazar (IPM\, tehran)
DTSTART;VALUE=DATE-TIME:20201016T120000Z
DTEND;VALUE=DATE-TIME:20201016T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/30
DESCRIPTION:Title: A
survey on the finiteness properties of local cohomology modules - Part 2\nby Kamran Divaani Aazar (IPM\, tehran) as part of IIT Bombay Virtual C
ommutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Satya Mandal (The University of Kansas)
DTSTART;VALUE=DATE-TIME:20201027T130000Z
DTEND;VALUE=DATE-TIME:20201027T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/31
DESCRIPTION:Title: Qu
illen $K$-Theory: A reclamation in Commutative Algebra - Part 1\nby Sa
tya Mandal (The University of Kansas) as part of IIT Bombay Virtual Commut
ative Algebra Seminar\n\n\nAbstract\nIn these two talks I take a pedagogic
approach to Quillen $K$-theory. What it takes to teach (and learn) Quille
n $K$-theory? I am at the tail end of completing a book on this\, which wo
uld eventually be available through some outlet. This is based on a course
I taught. Current version has nearly 400 pages\, in eleven chapters. I fi
nish with Swan’s paper on quadrics. I tried to do it in a reader friendl
y way\, and tried to avoid expressions like “left to the readers”. I w
ould give an overview and a road map. To justify the title\, let me rem
ind you that $K$-theory used to be part of Commutative algebra. In this en
deavor\, I consolidate the background needed\, in about 100 pages\, for a
commutative algebraist to pick up the book and give a course\, or learn. T
here is a huge research potential in this direction. This is because\, wit
h it\, topologists have done what they are good at. However\, these higher
$K$-groups have not been descried in a tangible manner. That would be the
job of commutative algebraist\, and would require such expertise.\n
LOCATION:https://researchseminars.org/talk/VCAS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Satya Mandal (The University of Kansas)
DTSTART;VALUE=DATE-TIME:20201030T130000Z
DTEND;VALUE=DATE-TIME:20201030T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/32
DESCRIPTION:Title: Qu
illen $K$-Theory: A reclamation in Commutative Algebra - Part 2\nby Sa
tya Mandal (The University of Kansas) as part of IIT Bombay Virtual Commut
ative Algebra Seminar\n\n\nAbstract\nIn these two talks I take a pedagogic
approach to Quillen $K$-theory. What it takes to teach (and learn) Quille
n $K$-theory? I am at the tail end of completing a book on this\, which wo
uld eventually be available through some outlet. This is based on a course
I taught. Current version has nearly 400 pages\, in eleven chapters. I fi
nish with Swan’s paper on quadrics. I tried to do it in a reader friendl
y way\, and tried to avoid expressions like “left to the readers”. I w
ould give an overview and a road map. To justify the title\, let me rem
ind you that $K$-theory used to be part of Commutative algebra. In this en
deavor\, I consolidate the background needed\, in about 100 pages\, for a
commutative algebraist to pick up the book and give a course\, or learn. T
here is a huge research potential in this direction. This is because\, wit
h it\, topologists have done what they are good at. However\, these higher
$K$-groups have not been descried in a tangible manner. That would be the
job of commutative algebraist\, and would require such expertise.\n
LOCATION:https://researchseminars.org/talk/VCAS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amartya Datta (ISI\, Kolkata)
DTSTART;VALUE=DATE-TIME:20201110T120000Z
DTEND;VALUE=DATE-TIME:20201110T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/33
DESCRIPTION:Title: G_
a-actions on Affine Varieties: Some Applications - Part 1\nby Amartya
Datta (ISI\, Kolkata) as part of IIT Bombay Virtual Commutative Algebra Se
minar\n\n\nAbstract\nOne of the hardest problems that come up in affine al
gebraic geometry is to decide whether a certain d-dimensional factorial a
ffine domain is ``trivial''\, i.e.\, isomorphic to the polynomial ring in
d variables. There are instances when the ring of invariants of a suitabl
y chosen G_a-action has been able to distinguish between two rings (i.e.\,
to prove they are non-isomorphic)\, when all other known invariants faile
d to make the distinction. It was using one such invariant that Makar-Lim
anov proved the non-triviality of the Russell-Koras threefold\, leading to
the solution of the Linearization Problem\; and again\, it was using an
invariant of G_a-actions that Neena Gupta proved the nontriviality of a l
arge class of Asanuma threefolds leading to her solution of the Zariski Ca
ncellation Problem in positive characteristic.\n\nG_a actions are also inv
olved in the algebraic characterisation of the affine plane by M. Miyanish
i and the algebraic characterisation of the affine 3-space.by Nikhilesh D
asgupta and Neena Gupta. Miyanishi's characterisation had led to the solut
ion of Zariski's Cancellation Problem for the affine plane. Using G_a-act
ions\, a simple algebraic proof for this cancellation theorem was obtaine
d three decades later by Makar-Limanov.\n\nIn this talk (in two parts)\, w
e will discuss the concept of G_a-actions along with the above application
s\, and the closely related theme of Invariant Theory. The concept of G_a-
action can be reformulated in the convenient ring-theoretic language of ``
locally nilpotent derivation'' (in characteristic zero) and ``exponential
map'' (in arbitrary characteristic). The ring of invariants of a G_a- acti
on corresponds to the kernel of the corresponding locally nilpotent deriva
tion (in characteristic zero) and the ring of invariants of an exponential
map. We will recall these concepts. We will also mention a theorem on G_
a actions on affine spaces (or polynomial rings) due to C.S. Seshadri.
\n\nWe will also discuss the close alignment of the kernel of a locally n
ilpotent derivation on a polynomial ring over a field of characteristic ze
ro with Hilbert's fourteenth problem. While Hilbert Basis Theorem had its
genesis in a problem on Invariant Theory\, Hilbert's fourteenth problem
seeks a further generalisation: Zariski generalises it still further. The
connection with locally nilpotent derivations has helped construct some l
ow-dimensional counterexamples to Hilbert's problem. We will also mention
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-Lokhande and
Dasgupta-Gupta. Finally\, we will state a few technical results on the ri
ng of invariants of a G_a action on the polynomial ring over a Noetherian
normal domain\, obtained by Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dut
ta-Gupta.\n
LOCATION:https://researchseminars.org/talk/VCAS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amartya Datta (ISI\, Kolkata)
DTSTART;VALUE=DATE-TIME:20201113T120000Z
DTEND;VALUE=DATE-TIME:20201113T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/34
DESCRIPTION:Title: G_
a-actions on Affine Varieties: Some Applications - Part 2\nby Amartya
Datta (ISI\, Kolkata) as part of IIT Bombay Virtual Commutative Algebra Se
minar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ryo Takahashi (Nagoya University)
DTSTART;VALUE=DATE-TIME:20201127T120000Z
DTEND;VALUE=DATE-TIME:20201127T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/35
DESCRIPTION:Title: Ge
tting a module from another and classifying resolving subcategories\nb
y Ryo Takahashi (Nagoya University) as part of IIT Bombay Virtual Commutat
ive Algebra Seminar\n\n\nAbstract\nLet $R$ be a commutative noetherian rin
g. Let $M$ and $N$ be finitely generated $R$-modules. When can we get $M$
from $N$ by taking direct summands\, extensions and syzygies? This questio
n is closely related to classification of resolving subcategories of finit
ely generated $R$-modules. In this talk\, I will explain what I have got s
o far on this topic.\n
LOCATION:https://researchseminars.org/talk/VCAS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shreedevi Masuti (IIT Dharwad)
DTSTART;VALUE=DATE-TIME:20200807T120000Z
DTEND;VALUE=DATE-TIME:20200807T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/36
DESCRIPTION:Title: No
rmal Hilbert coefficients and blow-up algebras\nby Shreedevi Masuti (I
IT Dharwad) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nThe normal Hilbert coefficients are important numerical invaria
nts associated with an \nideal in an analytically unramified local ring.
They play an important role in determining \nthe homological properties of
the blow-up algebras. This will be an expository talk on the \nnormal Hil
bert coefficients\, and its relation with blow-up algebras. We will also d
iscuss \nrecent developments on this topic.\n
LOCATION:https://researchseminars.org/talk/VCAS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giulio Caviglia (Purdue University)
DTSTART;VALUE=DATE-TIME:20201117T130000Z
DTEND;VALUE=DATE-TIME:20201117T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/37
DESCRIPTION:Title: Th
e Eisenbud-Green-Harris Conjecture\nby Giulio Caviglia (Purdue Univers
ity) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstra
ct\nThe $f$-vector of a simplicial complex is a finite sequence of integer
s defined by the number of $i$-dimensional faces of the complex. All possi
ble such vectors are completely characterized thanks to a classical theore
m by Kruskal and Katona. This result\, when rephrased in terms of Hilbert
functions of certain quotients of polynomial rings by monomial ideals\, ex
tends the celebrated theorem of Macaulay on lexicographic ideals.\nThe Eis
enbud-Green-Harris conjecture is a further generalization of both the Krus
kal-Katona theorem and the well-known Cayley–Bacharach theorem for plane
curves. I will survey the known results on this conjecture including a re
cent joint work with Alessandro De Stefani.\n
LOCATION:https://researchseminars.org/talk/VCAS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marilina Rossi (University of Genoa)
DTSTART;VALUE=DATE-TIME:20201201T120000Z
DTEND;VALUE=DATE-TIME:20201201T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/38
DESCRIPTION:Title: A
constructive approach to one-dimensional Gorenstein k-algebras\nby Mar
ilina Rossi (University of Genoa) as part of IIT Bombay Virtual Commutativ
e Algebra Seminar\n\n\nAbstract\nCodimension three Gorenstein rings are co
mpletely described by Buchsbaum and Eisenbud's structure theorem\, but de
spite many attempts the construction of Gorenstein rings is an open proble
m in higher codimension. Gorenstein rings are of great interest in many ar
eas of mathematics and they have appeared as an important component in a s
ignificant number of problems. Our task is to give a procedure for constru
cting all $1$-dimensional Gorenstein $k$-algebras. Applications to the Go
renstein linkage of zero-dimensional schemes and to Gorenstein affine sem
igroup rings are discussed. The results are based on recent results obtain
ed jointly with J. Elias.\n
LOCATION:https://researchseminars.org/talk/VCAS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aldo Conca (University of Genoa)
DTSTART;VALUE=DATE-TIME:20201211T120000Z
DTEND;VALUE=DATE-TIME:20201211T133000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/39
DESCRIPTION:Title: Id
eals and algebras associated with subspace arrangements.\nby Aldo Conc
a (University of Genoa) as part of IIT Bombay Virtual Commutative Algebra
Seminar\n\n\nAbstract\nI will present some results\, old and new\, about t
he algebraic objects that are naturally associated with a finite set of su
bspaces of a given vector space.\n
LOCATION:https://researchseminars.org/talk/VCAS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajendra Gurjar (IIT Bombay)
DTSTART;VALUE=DATE-TIME:20201215T120000Z
DTEND;VALUE=DATE-TIME:20201215T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/40
DESCRIPTION:Title: Za
riski-Lipman Conjecture for Module of Derivations - Part 1\nby Rajendr
a Gurjar (IIT Bombay) as part of IIT Bombay Virtual Commutative Algebra Se
minar\n\n\nAbstract\nZariski conjectured that if the module of derivations
of a local ring $R$ at a point on an algebraic variety defined over a fie
ld of chararacteristic $0$ is a free $R$-module then $R$ is regular. In th
ese two talks we will survey most of the interesting results proved affirm
ing the conjecture.\n\nResults of Lipman\, Scheja-Storch\, Becker\, Hochst
er\, Steenbrink-van Straten\, Flenner\, Kallstrom\, Biswas-Gurjar-Kolte\,
and some general results which can be deduced by combining some of these r
esults will be discussed. An interesting proposed counterexample due to Ho
chster will be introduced. Some unsolved cases in the paper of Biswas-Gurj
ar-Kolte will be mentioned.\n
LOCATION:https://researchseminars.org/talk/VCAS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajendra Gurjar (IIT Bombay)
DTSTART;VALUE=DATE-TIME:20201218T120000Z
DTEND;VALUE=DATE-TIME:20201218T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/41
DESCRIPTION:Title: Za
riski-Lipman Conjecture for Module of Derivations - Part 2\nby Rajendr
a Gurjar (IIT Bombay) as part of IIT Bombay Virtual Commutative Algebra Se
minar\n\n\nAbstract\nZariski conjectured that if the module of derivations
of a local ring $R$ at a point on an algebraic variety defined over a fie
ld of chararacteristic $0$ is a free $R$-module then $R$ is regular. In th
ese two talks we will survey most of the interesting results proved affirm
ing the conjecture.\n\nResults of Lipman\, Scheja-Storch\, Becker\, Hochst
er\, Steenbrink-van Straten\, Flenner\, Kallstrom\, Biswas-Gurjar-Kolte\,
and some general results which can be deduced by combining some of these r
esults will be discussed. An interesting proposed counterexample due to Ho
chster will be introduced. Some unsolved cases in the paper of Biswas-Gurj
ar-Kolte will be mentioned.\n
LOCATION:https://researchseminars.org/talk/VCAS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hema Srinivasan (University of Missouri\, Columbia\, MO)
DTSTART;VALUE=DATE-TIME:20201208T130000Z
DTEND;VALUE=DATE-TIME:20201208T143000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/42
DESCRIPTION:Title: Se
migroup rings\nby Hema Srinivasan (University of Missouri\, Columbia\,
MO) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstra
ct\nLet $A = \\{a_{ij}\\}$ be an $n \\times m$ matrix of natural numbers $
\\mathbb N.$ The $S(A)$ denotes the sub-semigroup of $\\mathbb N^n$ gener
ated by the columns of $A$. The semigroup ring of $A$ over a field $k$\, d
enoted by $k[A]$ is the homomorphic image of $\\phi: k[x_1\, \\ldots\, x_m
] \\to k[t_1\, \\ldots\, t_n]$ 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 $k[A]$\, such as de
pth\, dimension\, Frobenius numbers and homological properties\, such as r
esolutions\, Betti Numbers\, regularity and Hilbert Series. Recent work
on gluing and its relation to these invariants will be outlined. We will c
ompare the situation in numerical semigroups (subgroups of $\\mathbb N$) t
o semigroups of higher dimension and which of the many formulas and struct
ures generalize to higher dimensions.\n
LOCATION:https://researchseminars.org/talk/VCAS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Parangama Sarkar (IIT Palakkad)
DTSTART;VALUE=DATE-TIME:20201120T120000Z
DTEND;VALUE=DATE-TIME:20201120T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/43
DESCRIPTION:Title: Fr
obenius Betti numbers of finite length modules\nby Parangama Sarkar (I
IT Palakkad) as part of IIT Bombay Virtual Commutative Algebra Seminar\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. Suppose char
R = $p > 0$ and $d = 1.$ De Stefani\, Huneke and Núñez-Betancourt explor
ed the question: what vanishing conditions on the Frobenius Betti numbers
force projective dimension of $M$ to be finite. In this talk we will discu
ss the question for $d ≥ 1.$ This is joint work with Ian Aberbach.\n
LOCATION:https://researchseminars.org/talk/VCAS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tai Huy Ha (University of Tulane)
DTSTART;VALUE=DATE-TIME:20201124T130000Z
DTEND;VALUE=DATE-TIME:20201124T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/44
DESCRIPTION:Title: Th
e ideal containment problem and vanishing loci of homogeneous polynomials<
/a>\nby Tai Huy Ha (University of Tulane) as part of IIT Bombay Virtual Co
mmutative Algebra Seminar\n\n\nAbstract\nWe shall discuss Chudnovsky’s a
nd Demailly’s conjectures which provide lower bounds for the answer to t
he following fundamental question: given a set of points in projective spa
ce and a positive integer m\, what is the least degree of a homogeneous po
lynomial vanishing at these points of order at least $m$? Particularly\, w
e shall present the main ideas of the proofs of these conjectures for suff
iciently many general points.\n
LOCATION:https://researchseminars.org/talk/VCAS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ian Aberbach (University of Missouri\, Columbia\, MO)
DTSTART;VALUE=DATE-TIME:20201204T130000Z
DTEND;VALUE=DATE-TIME:20201204T143000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/45
DESCRIPTION:Title: On
the equivalence of weak and strong F-regularity\nby Ian Aberbach (Uni
versity of Missouri\, Columbia\, MO) as part of IIT Bombay Virtual Commuta
tive Algebra Seminar\n\n\nAbstract\nLet $(R\, \\mathfrak m\, k)$ be a (Noe
therian) local ring of positive prime characteristic $p.$ Assume also\, f
or simplicity\, that $R$ is complete (or\, more generally\, excellent).
In such rings we have the notion of tight closure of an ideal\, defined by
Hochster and Huneke\, using the Frobenius endomorphism. The tight closur
e of an ideal sits between the ideal itself and its integral closure. Whe
n the tight closure of an ideal $I$ is $I$ itself we call $I$ tightly clos
ed. For particularly nice rings\, e.g.\, regular rings\, every ideal is ti
ghtly closed. We call such rings weakly $F$-regular. Unfortunately\, tig
ht closure is known not to commute with localization\, and hence this prop
erty of being weakly $F$-regular is not known to localize. To deal with t
his problem\, Hochster and Huneke defined the notion of strongly $F$-regul
ar (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 classes o
f 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 outline
recent progress made by myself and Thomas Polstra.\n
LOCATION:https://researchseminars.org/talk/VCAS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bernd Ulrich (Purdue University)
DTSTART;VALUE=DATE-TIME:20210226T130000Z
DTEND;VALUE=DATE-TIME:20210226T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/48
DESCRIPTION:Title: Ge
neralized multiplicities and integral dependence-II\nby Bernd Ulrich (
Purdue University) as part of IIT Bombay Virtual Commutative Algebra Semin
ar\n\n\nAbstract\nThese two talks will give a survey about multiplicity ba
sed criteria\nfor the integral dependence of ideals. This subject has clos
e connections\nwith equisingularity theory and intersection theory\, which
will be\ndiscussed as well. The first numerical criterion for integral de
pendence\nwas proved in the 1960s by Rees who treated the case of zero-dim
ensional\nideals using the Hilbert-Samuel multiplicity. Criteria for arbit
rary\nideals require generalized notions of multiplicities. We will discus
s\nvarious such notions and talk about how they are used to detect integra
l\ndependence. The most recent results are from joint work with Claudia\nP
olini\, Ngo Viet Trung\, and Javid Validashti.\n
LOCATION:https://researchseminars.org/talk/VCAS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Suprajo Das (Chennai Mathematical Institute)
DTSTART;VALUE=DATE-TIME:20210101T120000Z
DTEND;VALUE=DATE-TIME:20210101T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/58
DESCRIPTION:Title: An
inequality in mixed multiplicities of filtrations\nby Suprajo Das (Ch
ennai Mathematical Institute) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anthony Iarrobino (Northeastern University\, Boston\, MA)
DTSTART;VALUE=DATE-TIME:20201229T130000Z
DTEND;VALUE=DATE-TIME:20201229T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/61
DESCRIPTION:Title: Jo
rdan type and Lefschetz Properties for Artinian algebras\nby Anthony I
arrobino (Northeastern University\, Boston\, MA) as part of IIT Bombay Vir
tual 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 Artinian
algebra A\, is the partition P giving the Jordan block decomposition of t
he multiplication map by $x$ on $A.$ When $A$ is Artinian Gorenstein\, 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 Hi
lbert function H(A). We say that \n$(A\,x)$ is strong Lefschetz if $P_x$
is the conjugate of the Hilbert function.\n\n Weak and strong Lefschetz
properties of $A$ for a generic choice of $x$ have been studied\, due to t
he connection with topology and geometry\, where A is the cohomology ring
of a\n\ntopological space or a variety $X.$ We discuss some of the propert
ies of Jordan type\, and its\n\nuse as an invariant of $A\,$ its behavior
for tensor products and free extensions (defined by \n\nT. Harima and J. W
atanabe).\n
LOCATION:https://researchseminars.org/talk/VCAS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juergen Herzog (University of Duisberg-Essen)
DTSTART;VALUE=DATE-TIME:20210108T120000Z
DTEND;VALUE=DATE-TIME:20210108T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/64
DESCRIPTION:Title: Po
wers of component wise linear ideals\nby Juergen Herzog (University 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 Groebner
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 li
near quotients and with additional assumptions are componentwise linear. 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 $M$. W
e apply our criterion to study certain symmetric algebras and the powers o
f vertex cover ideals of certain classes of graphs. This is a report on jo
int work with Takayuki Hibi and Somayeh Moradi.\n
LOCATION:https://researchseminars.org/talk/VCAS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Viviana Ene (Ovidius University\, Constanta\, Romania)
DTSTART;VALUE=DATE-TIME:20210115T120000Z
DTEND;VALUE=DATE-TIME:20210115T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/65
DESCRIPTION:Title: Bi
nomial edge ideals\nby Viviana Ene (Ovidius University\, Constanta\, R
omania) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbs
tract\nIn this talk we will survey various old and new results on the homo
logical and algebraic properties of binomial edge ideals. In the last part
of the talk\, we will present some new results of a recent joint paper wi
th G. Rinaldo and N. Terai on powers of binomial edge ideals with quadrati
c Groebner bases.\n
LOCATION:https://researchseminars.org/talk/VCAS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Katz (The University of Kansas\, Lawrence\, KS)
DTSTART;VALUE=DATE-TIME:20210122T130000Z
DTEND;VALUE=DATE-TIME:20210122T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/66
DESCRIPTION:Title: Re
es Valuations-I\nby Dan Katz (The University of Kansas\, Lawrence\, KS
) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\
nIn these expository talks\, we will discuss the Rees valuations and Rees
valuation rings associated with an ideal in a Noetherian ring\, as well
as their applications to asymptotic prime divisors and various multiplicit
ies. If time permits\, we will describe the Rees valuations associated wi
th a finitely generated torsion-free module.\n
LOCATION:https://researchseminars.org/talk/VCAS/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Katz (The University of Kansas\, Lawrence\, KS)
DTSTART;VALUE=DATE-TIME:20210129T130000Z
DTEND;VALUE=DATE-TIME:20210129T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/67
DESCRIPTION:Title: Re
es Valuations-II\nby Dan Katz (The University of Kansas\, Lawrence\, K
S) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract
\nIn these expository talks\, we will discuss the Rees valuations and Ree
s valuation rings associated with an ideal in a Noetherian ring\, as well
as their applications to asymptotic prime divisors and various multiplici
ties. If time permits\, we will describe the Rees valuations associated w
ith a finitely generated torsion-free module.\n
LOCATION:https://researchseminars.org/talk/VCAS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vijaylaxmi Trivedi (Tata Institute of Fundamental Research\, Mumba
i)
DTSTART;VALUE=DATE-TIME:20210205T120000Z
DTEND;VALUE=DATE-TIME:20210205T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/68
DESCRIPTION:Title: Hi
lbert-Kunz density function and its applications\nby Vijaylaxmi Trived
i (Tata Institute of Fundamental Research\, Mumbai) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn this talk\, we recal
l the notion of the Hilbert-Kunz density function for graded rings. This f
unction was introduced to understand the Hilbert-Kunz multiplicity which i
s a difficult characteristic $p$ invariant to compute and to make speculat
ion about its properties. It turns out the HK density function is also rel
ated to another characteristic $p$-invariant namely $F$-threshold. Here we
describe its properties and give its applications to HK multiplicity\, $F
$-thershold\, and to a conjecture of Watanabe-Yoshida. The talk is partly
based on joint work with K.I. Watanabe.\n
LOCATION:https://researchseminars.org/talk/VCAS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Graham Leuschke (Syracuse University\, New York\, NY)
DTSTART;VALUE=DATE-TIME:20210212T130000Z
DTEND;VALUE=DATE-TIME:20210212T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/69
DESCRIPTION:Title: Ma
trix Factorizations and Knörrer Periodicity\nby Graham Leuschke (Syra
cuse University\, New York\, NY) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nA matrix factorization of a ring element $
f$ is a pair of square matrices so that the product (in either order) is d
iagonal with $f$ in each diagonal entry. These were introduced by David Ei
senbud in 1980. When the ring is regular\, matrix factorizations of $f$ co
rrespond to maximal Cohen-Macaulay modules over the hypersurface defined b
y $f.$ This talk will give an overview of the theory of matrix factorizati
ons\, ending with some recent generalizations to factorizations by more th
an two matrices.\n
LOCATION:https://researchseminars.org/talk/VCAS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bernd Ulrich (Purdue University)
DTSTART;VALUE=DATE-TIME:20210219T130000Z
DTEND;VALUE=DATE-TIME:20210219T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/70
DESCRIPTION:Title: Ge
neralized multiplicities and integral dependence-I\nby Bernd Ulrich (P
urdue University) as part of IIT Bombay Virtual Commutative Algebra Semina
r\n\n\nAbstract\nThese two talks will give a survey about multiplicity bas
ed criteria\nfor the integral dependence of ideals. This subject has close
connections\nwith equisingularity theory and intersection theory\, which
will be\ndiscussed as well. The first numerical criterion for integral dep
endence\nwas proved in the 1960s by Rees who treated the case of zero-dime
nsional\nideals using the Hilbert-Samuel multiplicity. Criteria for arbitr
ary\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 Claudia\nPo
lini\, Ngo Viet Trung\, and Javid Validashti.\n
LOCATION:https://researchseminars.org/talk/VCAS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K. Ozeki (Yamaguchi University)
DTSTART;VALUE=DATE-TIME:20210305T120000Z
DTEND;VALUE=DATE-TIME:20210305T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/71
DESCRIPTION:Title: Th
e reduction number of stretched ideals\nby K. Ozeki (Yamaguchi Univers
ity) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstra
ct\nThe homological property of the associated graded ring of an ideal is
an\nimportant problem in commutative algebra and algebraic geometry.\nIn t
his paper we explore the structure of the associated graded ring of\nstret
ched $\\mathfrak m $-primary ideals in the case where the reduction number
\nattains almost minimal value in a Cohen-Macaulay local ring $(A\,\\mathf
rak m )$.\nAs an application\, we present complete descriptions of the ass
ociated\ngraded ring of stretched $\\mathfrak m $-primary ideals with smal
l reduction\nnumber.\n
LOCATION:https://researchseminars.org/talk/VCAS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Polstra (University of Virgina)
DTSTART;VALUE=DATE-TIME:20210312T130000Z
DTEND;VALUE=DATE-TIME:20210312T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/72
DESCRIPTION:Title: St
rongly F-regular rings\, maximal Cohen-Macaulay modules\, and the F-signat
ure\nby Thomas Polstra (University of Virgina) as part of IIT Bombay V
irtual Commutative Algebra Seminar\n\n\nAbstract\nThe singularities of a l
ocal prime characteristic ring are best understood through the behavior of
the Frobenius endomorphism. A singularity class of central focus is the c
lass of strongly $F$-regular rings. Examples of strongly $F$-regular rings
include normal affine toric rings\, direct summands of regular rings\, an
d determinantal rings. Every strongly $F$-regular ring enjoys the property
of being a normal Cohen-Macaulay domain. In particular\, the study of fin
itely generated maximal Cohen-Macaulay modules over such rings is a warran
ted venture. We will demonstrate a surprising uniform behavior enjoyed 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 no
vel and elementary manner. Time permitting\, we will present applications
on the structure of the divisor class group of a local strongly $F$-regula
r ring.\n
LOCATION:https://researchseminars.org/talk/VCAS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Polstra (University of Virgina)
DTSTART;VALUE=DATE-TIME:20210319T130000Z
DTEND;VALUE=DATE-TIME:20210319T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/73
DESCRIPTION:Title: Pr
ime characteristic singularities and the deformation problem\nby Thoma
s Polstra (University of Virgina) as part of IIT Bombay Virtual Commutativ
e Algebra Seminar\n\n\nAbstract\nLet $P$ be a property of local rings (suc
h as regular\, Gorenstein\, or complete). We say that $P$ deforms if a loc
al ring $R$ enjoys property $P$ provided there exists a nonzerodivisor $x$
such that $R/xR$ is $P$. (For example\, the properties of being regular o
r Gorenstein deform\, but the property of being complete does not deform).
The deformation problem\, as it pertains to the prime characteristic sing
ularity classes of $F$-regular\, $F$-rational\, $F$-pure\, and $F$-injecti
ve 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 history of
the deformation problem of these four prime characteristic singularity cla
sses and discuss a recent solution to the deformation of $F$-purity proble
m in rings which are $\\mathbb{Q}$-Gorenstein. This talk is based on a col
laboration with Austyn Simpson.\n
LOCATION:https://researchseminars.org/talk/VCAS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason McCullough (Iowa State University)
DTSTART;VALUE=DATE-TIME:20210402T130000Z
DTEND;VALUE=DATE-TIME:20210402T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/74
DESCRIPTION:Title: Re
es-like Algebras\nby Jason McCullough (Iowa State University) as part
of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nGiven the
ir importance in constructing counterexamples to the Eisenbud-Goto Conject
ure\, it is reasonable to study the algebra and geometry of Rees-like alge
bras further. Given a graded ideal I of a polynomial ring S\, its Rees-li
ke algebra is S[It\, t^2]\, where t is a new variable. Unlike the Rees al
gebra\, whose defining equations are difficult to compute in general\, the
Rees-like algebra has a concrete minimal generating set in terms of the g
enerators and first syzygies of I. Moreover\, the free resolution of this
ideal is well understood. While it is clear that the Rees-like algebra o
f an ideal is never normal and only Cohen-Macaulay if the ideal is princip
al\, I will explain that it is often seminormal\, weakly normal\, or F-pur
e. I will also discuss the computation of the singular locus\, how the si
ngular locus is affected by homogenization\, and the structure of the cano
nical module\, class group\, and Picard group. This talk is joint work wi
th Paolo Mantero and Lance E. Miller.\n
LOCATION:https://researchseminars.org/talk/VCAS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason McCullough (Iowa State University)
DTSTART;VALUE=DATE-TIME:20210326T130000Z
DTEND;VALUE=DATE-TIME:20210326T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/75
DESCRIPTION:Title: Th
e Eisenbud-Goto Conjecture\nby Jason McCullough (Iowa State University
) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\
nLet S be a polynomial ring over an algebraically closed field K. There h
as been considerable research into effective upper bounds for the Castelnu
ovo-Mumford regularity of graded ideals of S. Through the work of Bertram
\, Ein\, Gruson\, Kwak\, Lazarsfeld\, Peskine\, and others\, there are sev
eral good bounds for the defining ideals of smooth projective varieties in
characteristic zero. However\, for arbitrary ideals\, the best upper bou
nd is doubly exponential (in terms of the number of variables and degrees
of generators)\, and this bound is asymptotically close to optimal due to
examples derived from the Mayr-Meyer construction. In 1984\, Eisenbud and
Goto conjectured that the regularity of a nondegenerate prime ideal P was
at most deg(P) – codim(P) + 1\, and proved this when S/P was Cohen-Maca
ulay (even if P is not prime). In this talk\, I will explain the construc
tion of counterexamples to the Eisenbud-Goto Conjecture\, joint work with
Irena Peeva\, through the construction of Rees-Like algebras and a special
homogenization. While we show that there is no linear bound on regularit
y in terms of the degree (or multiplicity) of P\, we later showed that som
e such bound exists. The latter part of this talk is joint work with Giul
io Caviglia\, Marc Chardin\, Irena Peeva\, and Matteo Varbaro.\n
LOCATION:https://researchseminars.org/talk/VCAS/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karen Smith (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210408T130000Z
DTEND;VALUE=DATE-TIME:20210408T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/76
DESCRIPTION:Title: Ex
tremal Singularities in Prime Characteristic\nby Karen Smith (Universi
ty of Michigan) as part of IIT Bombay Virtual Commutative Algebra Seminar\
n\n\nAbstract\nWhat is the most singular possible singularity? What can we
say about its geometric and algebraic properties? This seemingly naive qu
estion has a sensible answer in characteristic p.\nThe "F-pure threshold\,
" which is an analog of the log canonical threshold\, can be used to "mea
sure" how bad a singularity is. The F-pure threshold is a numerical invari
ant of a point on (say) a hypersurface---a positive rational number that
is 1 at any smooth point (or more generally\, any F-pure point) but less
than one in general\, with "more singular" points having smaller F-pure th
resholds. We explain a recently proved lower bound on the F-pure threshol
d in terms of the multiplicity of the singularity. We also show that there
is a nice class of hypersurfaces---which we call "Extremal hypersurfaces"
---for which this bound is achieved. These have very nice (extreme!) geome
tric properties. For example\, the affine cone over a non Frobenius split
cubic surface of characteristic two is one example of an "extremal singula
rity". Geometrically\, these are the only cubic surfaces with the property
that *every* triple of coplanar lines on the surface meets in a single po
int (rather than a "triangle" as expected)---a very extreme property indee
d.\n
LOCATION:https://researchseminars.org/talk/VCAS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mitsuhiro Miyazaki (Kyoto University of Education)
DTSTART;VALUE=DATE-TIME:20210409T120000Z
DTEND;VALUE=DATE-TIME:20210409T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/77
DESCRIPTION:Title: Hi
bi rings and the Ehrhart rings of chain polytopes - Part 1\nby Mitsuhi
ro Miyazaki (Kyoto University of Education) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\nIn 1985\, Stanley submitted a p
aper titled "Two Poset Polytopes"\, which was published in 1986\, in which
he defined the order and chain polytopes of a finite partially ordered se
t (poset for short).\nOn the other hand\, Hibi presented a notion of an al
gebra with straightening law (ASL for short) on a finite distributive latt
ice\, which nowadays is called a Hibi ring\, in a conference held in Kyoto
1985.\nThis result was published in 1987.\nIt turned out that the Hibi ri
ng on a distributive lattice D is the Ehrhart ring of the order polytope o
f the poset consisting of join-irreducible elements of D.\nIn the first ta
lk\, we recall the definition of Ehrhart rings\, order and chain polytopes
\, and Hibi rings.\nWe recall some basic properties of Ehrhart rings and d
escribe the canonical module of them.\nUsing these facts\, we state some b
asic facts of Hibi rings\, i.e.\, the Ehrhart rings of the order polytopes
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 structure of t
he 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 proper
ty.\nIf time permits\, we describe the radical of the trace of the canonic
al module of these rings and describe the non-Gorenstein locus.\nThis fina
l part is a joint-work with Janet Page.\n
LOCATION:https://researchseminars.org/talk/VCAS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mitsuhiro Miyazaki (Kyoto University of Education)
DTSTART;VALUE=DATE-TIME:20210416T120000Z
DTEND;VALUE=DATE-TIME:20210416T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/78
DESCRIPTION:Title: Hi
bi rings and the Ehrhart rings of chain polytopes - Part 2\nby Mitsuhi
ro Miyazaki (Kyoto University of Education) as part of IIT Bombay Virtual
Commutative Algebra Seminar\n\n\nAbstract\nIn 1985\, Stanley submitted a p
aper titled "Two Poset Polytopes"\, which was published in 1986\, in which
he defined the order and chain polytopes of a finite partially ordered se
t (poset for short).\nOn the other hand\, Hibi presented a notion of an al
gebra with straightening law (ASL for short) on a finite distributive latt
ice\, which nowadays called a Hibi ring\, in a conference held in Kyoto 19
85.\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 polytope of t
he poset consisting of join-irreducible elements of D.\nIn the first talk\
, we recall the definition of Ehrhart rings\, order and chain polytopes\,
and Hibi rings.\nWe recall some basic properties of Ehrhart rings and desc
ribe the canonical module of them.\nUsing these facts\, we state some basi
c facts of Hibi rings\, i.e.\, the Ehrhart rings of the order polytopes of
posets.\nWe also state some basic facts of the Ehrhart rings of chain pol
ytopes 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 p
oset.\nWe describe the generators of the canonical modules in terms 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.\nThis final p
art is a joint-work with Janet Page.\n
LOCATION:https://researchseminars.org/talk/VCAS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:L. T. Hoa (Institute of Mathematics\, Hanoi\, Vietnam)
DTSTART;VALUE=DATE-TIME:20210423T120000Z
DTEND;VALUE=DATE-TIME:20210423T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/79
DESCRIPTION:Title: As
ymptotic behavior of Integer Programming and the stability of the Castelnu
ovo-Mumford regularity\nby L. T. Hoa (Institute of Mathematics\, Hanoi
\, Vietnam) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\
nAbstract\nIn the talk\, we explain a connection between Commutative Alge
bra and Integer Programming in two parts. The first one is devoted to the
asymptotic behavior of integer programs with a fixed cost linear functiona
l and the constraint sets consisting of a finite system of linear equation
s or inequalities with integer coefficients depending linearly on $n$. An
integer $N_*$ is determined such that the optima of these integer program
s are a quasi-linear function of $n$ for all $n\\ge N_*$. Using results i
n the first part\, one can bound in the second part the indices of stabili
ty of the Castelnuovo-Mumford regularities of integral closures of powers
of a monomial ideal and that of symbolic powers of a square-free monomial
ideal.\n
LOCATION:https://researchseminars.org/talk/VCAS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Eisenbud (University of California\, Berkeley and MSRI)
DTSTART;VALUE=DATE-TIME:20210430T140000Z
DTEND;VALUE=DATE-TIME:20210430T150000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/80
DESCRIPTION:Title: La
yered resolutions and Cohen-Macaulay approximation\nby David Eisenbud
(University of California\, Berkeley and MSRI) as part of IIT Bombay Virtu
al Commutative Algebra Seminar\n\n\nAbstract\nThe representation theory of
finite-dimensional algebras has an important generalization to\nthe 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 matrix fact
orizations of the equation of the hypersurface\, and these come from minim
al free resolutions of the MCMs. I will talk about the "next" case---MCMs
and their minimal free resolutions over complete intersections. This is jo
int work with Irena Peeva.\n
LOCATION:https://researchseminars.org/talk/VCAS/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Indranath Sengupta (IIT Gandhinagar)
DTSTART;VALUE=DATE-TIME:20210507T120000Z
DTEND;VALUE=DATE-TIME:20210507T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/81
DESCRIPTION:Title: So
me Questions on bounds of Betti Numbers of Numerical Semigroup Rings\n
by Indranath Sengupta (IIT Gandhinagar) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nJ. Herzog proved in 1969 that the p
ossible values of the first Betti number (minimal number of generators of
the defining ideal) of numerical semigroup rings in embedding dimension 3
are 2 (complete intersection and Gorenstein) and 3 (the almost complete in
tersection).\n\nIn a conversation about this work\, O. Zariski indicated a
possible relation between Gorenstein rings and symmetric value semigroups
. In response to that\, E.Kunz proved (in 1970) that a one-dimensional\, l
ocal\, Noetherian\, reduced\, analytically irreducible ring is Gorenstein
if and only if its value semigroup is symmetric. A question that remains o
pen to date is whether the Betti numbers (or at least the first Betti numb
er) of every numerical semigroup ring in embedding dimension e\, are bound
ed above by a function of e.\n\nIn the years 1974 and 1975\, two interesti
ng classes of examples were given by T. Moh and H. Bresinsky. Moh’s exam
ple was that of a family of algebroid space curves and Bresinsky’s examp
le was about a family of numerical semigroups in embedding dimension 4\, w
ith the common feature that there is no upper bound on the Betti numbers.
Therefore\, for embedding dimension 4 and above\, the Betti numbers (or at
least the first Betti number) are not bounded above by some “good” fu
nction of the embedding dimension e. A question that emerges is the follow
ing: Is there a natural way to generate such numerical semigroups in arbit
rary embedding dimension? In this talk\, we will discuss some recent obser
vations in this direction\, which is a joint work of the author with his c
ollaborators Joydip Saha and Ranjana Mehta.\n
LOCATION:https://researchseminars.org/talk/VCAS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Roemer (University of Osnabrueck)
DTSTART;VALUE=DATE-TIME:20210514T120000Z
DTEND;VALUE=DATE-TIME:20210514T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/82
DESCRIPTION:Title: Cu
t and related polytopes in commutative algebra\nby Tim Roemer (Univers
ity of Osnabrueck) as part of IIT Bombay Virtual Commutative Algebra Semin
ar\n\n\nAbstract\nThe study of cuts in graphs is an interesting topic in d
iscrete mathematics and optimization with relations and applications to ma
ny other fields such as algebraic geometry\, algebraic statistics\, and co
mmutative algebra. Here we focus on cut algebras\, which are toric algebra
s\, and each one is defined by all cuts of a given graph\,\nand similar co
nstructions. We discuss known and new results as well as open questions re
lated to the algebraic properties of such algebras and their defining idea
ls.\n
LOCATION:https://researchseminars.org/talk/VCAS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Trung Cuong Doan (Institute of Mathematics\, Hanoi)
DTSTART;VALUE=DATE-TIME:20210521T120000Z
DTEND;VALUE=DATE-TIME:20210521T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/83
DESCRIPTION:Title: Be
tti numbers and ideal structure of projective subschemes of almost maximal
degree\nby Trung Cuong Doan (Institute of Mathematics\, Hanoi) as par
t of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nSimilar
to Castelnouvo-Mumford regularity\, reduction number is a measure of the
complexity of the structure of an algebra. For a projective subscheme $X$\
, there is a degree upper bound\n$$\\deg(X)\\leq \\binom{e+r}{r}\,$$\nwher
e $e$ is the codimension and $r$ is the reduction number of the homogeneou
s coordinate ring. In this talk\, I discuss the maximal case $\\deg(X)=\\b
inom{e+r}{r}$ and the almost maximal case $\\deg(X)=\\binom{e+r}{r}-1$. In
these cases\, it is possible to explicitly describe certain initial ideal
of the defining ideal of $X$ and consequently one obtains an explicit des
cription of the Betti table. I also discuss how componentwise linearity is
helpful for computing the Betti tables of projective varieties with an al
most maximal degree.\n\nThis is a joint work with Sijong Kwak.\n
LOCATION:https://researchseminars.org/talk/VCAS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ravi Rao (NMIMS University\, Mumbai)
DTSTART;VALUE=DATE-TIME:20210528T130000Z
DTEND;VALUE=DATE-TIME:20210528T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/84
DESCRIPTION:Title: So
me approaches to a question of Suslin\nby Ravi Rao (NMIMS University\,
Mumbai) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAb
stract\nIn his Helsinki talk in 1978\, A. Suslin asked if a stably free mo
dule of rank d-1 over an affine algebra of dimension d over an algebraical
ly 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 Richard Swa
n.\n
LOCATION:https://researchseminars.org/talk/VCAS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anand Sawant (Tata Institute of Fundamental Research)
DTSTART;VALUE=DATE-TIME:20210604T120000Z
DTEND;VALUE=DATE-TIME:20210604T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/85
DESCRIPTION:Title: Na
ive $\\mathbb A^1$-homotopies on surfaces\nby Anand Sawant (Tata Insti
tute of Fundamental Research) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\n\nAbstract\nWe will describe an algebraic criterion for t
wo morphisms of the spectrum of a henselian regular local ring into a smoo
th projective surface to be naively $\\mathbb A^1$-homotopic. If time per
mits\, we will explain the significance of this criterion to purely algebr
o-geometric questions related to $\\mathbb A^1$-connected components. The
talk is based on joint work with Chetan Balwe.\n
LOCATION:https://researchseminars.org/talk/VCAS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Moty Katzman (University of Sheffield)
DTSTART;VALUE=DATE-TIME:20210611T120000Z
DTEND;VALUE=DATE-TIME:20210611T131500Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/86
DESCRIPTION:Title: A
generalized-fractions approach to computing local cohomology.\nby Moty
Katzman (University of Sheffield) as part of IIT Bombay Virtual Commutati
ve Algebra Seminar\n\n\nAbstract\nThis talk aims to describe a method to c
ompute Lyubeznik Numbers in prime characteristic by applying generalized f
ractions tools to $F$-finite $F$-modules. The main part of the talk will c
onsist of a very brief introduction to local cohomology\, Lyubeznik's noti
on of $F$-finite $F$-modules\, and the Sharp-Zakeri theory of generalized
fractions. This introduction will be aimed at non-experts.\n\nThe final pa
rt of the talk will introduce Lyubeznik numbers and show how these can be
computed using the tools introduced earlier in the talk.\nAll results are
based on a joint project with Rodney Sharp whose results are available in
"Lyubeznik numbers\, $F$-modules and modules of generalized fractions" (ar
Xiv:2006.05438)\n
LOCATION:https://researchseminars.org/talk/VCAS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marc Chardin (Pierre and Marie Curie University\, Jussieu\, France
)
DTSTART;VALUE=DATE-TIME:20210618T120000Z
DTEND;VALUE=DATE-TIME:20210618T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/87
DESCRIPTION:Title: Mu
ltigraded Sylvester forms\, Duality and Elimination\nby Marc Chardin (
Pierre and Marie Curie University\, Jussieu\, France) as part of IIT Bomba
y Virtual Commutative Algebra Seminar\n\n\nAbstract\nThis talk will report
on joint work with Laurent Busé and Navid Nemati. First\, the classical
situation of the theory of resultants and Sylvester forms in a standard gr
aded algebra will be presented\, as it was developed by Jouanolou in a ser
ies of monographs. Then we will explain the extension of this theory to th
e multi-graded case (which corresponds to a product of projective spaces\,
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 dualit
y result from the classical case to this more general setting. We will ill
ustrate these in a very simple case\, by providing a family of formulas th
at extends the work of Dixon.\n
LOCATION:https://researchseminars.org/talk/VCAS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Schenzel (Institut für Informatik. Martin-Luther-Universit
ät Halle-Wittenberg.)
DTSTART;VALUE=DATE-TIME:20210625T120000Z
DTEND;VALUE=DATE-TIME:20210625T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/88
DESCRIPTION:Title: Ne
ws about Koszul and \\v{C}ech complexes: Another view at local cohomology
and completion-I\nby Peter Schenzel (Institut für Informatik. Martin-
Luther-Universität Halle-Wittenberg.) as part of IIT Bombay Virtual Commu
tative Algebra Seminar\n\n\nAbstract\nIn the first part of the talk\, we p
resent some elementary new facts about \nKoszul and \\v{C}ech complexes wi
th 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{Ce
ch} complex is applied in order \nto find relations to the left derived fu
nctors of the completion \nas a certain Koszul homology. \n\nThis material
provides an elementary introduction to some of the results \nof the speak
ers joint work with A.-M. Simon (see "Completion\, \\v{C}ech \nand local h
omology and cohomology. Interactions between them. Springer Monograph\, \n
2018") as well as some further developments.\n\nOne focus is the study of
weakly proregular sequences and of proregular \nsequences which provides
a new insight for the local cohomology as well as the \nleft derived funct
ors of the completion. Finally we shall present an \napplication to prisms
in the sense of Bhatt and Scholze.\n
LOCATION:https://researchseminars.org/talk/VCAS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Peter Schenzel (Institut für Informatik. Martin-Luther-Universit
ät Halle-Wittenberg.)
DTSTART;VALUE=DATE-TIME:20210702T120000Z
DTEND;VALUE=DATE-TIME:20210702T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/89
DESCRIPTION:Title: Ne
ws about Koszul and \\v{C}ech complexes: Another view at local cohomology
and completion-II\nby Peter Schenzel (Institut für Informatik. Martin
-Luther-Universität Halle-Wittenberg.) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nIn the first part of the talk\, we
present some elementary new facts about \nKoszul and \\v{C}ech complexes w
ith 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{C
ech} complex is applied in order \nto find relations to the left derived f
unctors of the completion \nas a certain Koszul homology. \n\nThis materia
l provides an elementary introduction to some of the results of the speake
rs joint work with A.-M. Simon (see "Completion\, \\v{C}ech \nand local ho
mology and cohomology. Interactions between them. Springer Monograph\, \n2
018") as well as some further developments.\n\nOne focus is the study of w
eakly proregular sequences and of proregular \nsequences which provides a
new insight for the local cohomology as well as the \nleft derived functor
s of the completion. Finally we shall present an application to prisms in
the sense of Bhatt and Scholze.\n
LOCATION:https://researchseminars.org/talk/VCAS/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuma Shimomoto (Nihon University\, Tokyo\, Japan)
DTSTART;VALUE=DATE-TIME:20210723T120000Z
DTEND;VALUE=DATE-TIME:20210723T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/90
DESCRIPTION:Title: Pe
rfectoid spaces-I\nby Kazuma Shimomoto (Nihon University\, Tokyo\, Jap
an) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstrac
t\nIn the first talk\, I begin with a historical review of perfectoid geo
metry.\nThen I talk about the definition of perfectoid rings and tilting o
perations.\nSome basic examples are examined. I also show how to use perfe
ctoid rings\nby introducing recent results obtained by several people.\nI
end with a remark on a classical result on valuation rings.\n
LOCATION:https://researchseminars.org/talk/VCAS/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kazuma Shimomoto (Nihon University\, Tokyo\, Japan)
DTSTART;VALUE=DATE-TIME:20210730T120000Z
DTEND;VALUE=DATE-TIME:20210730T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/91
DESCRIPTION:Title: Pe
rfectoid spaces-II\nby Kazuma Shimomoto (Nihon University\, Tokyo\, Ja
pan) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstra
ct\nIn the second talk\, I start talking about rough ideas of Andre's pro
of\nof the direct summand conjecture. Then I move to basics of almost ring
s and\nformulate the almost purity theorem. I also show some guidelines fo
r learning\nperfectoids by showing some fundamental results on perfectoid
geometry.\nFinally\, I talk about my recent contributions (joint with K. N
akazato and S. Ishiro).\n
LOCATION:https://researchseminars.org/talk/VCAS/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Montano (New Mexico State University\, Las Cruces\, NM\,
USA)
DTSTART;VALUE=DATE-TIME:20210709T130000Z
DTEND;VALUE=DATE-TIME:20210709T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/92
DESCRIPTION:Title: Mi
xed multiplicities of graded families of ideals\nby Jonathan Montano (
New Mexico State University\, Las Cruces\, NM\, USA) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\n\nAbstract\nWe show the existence
(and define) the mixed multiplicities of arbitrary graded families of idea
ls under mild assumptions. In particular\, our methods and results are val
id for the case of arbitrary m-primary graded families. Furthermore\, we p
rovide a “Volume = Multiplicity formula” for the mixed multiplicities
of graded families of ideals. This is joint work with Yairon Cid-Ruiz.\n
LOCATION:https://researchseminars.org/talk/VCAS/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yairon Cid-Ruiz (Ghent University\, Krijgslaan\, Belgium)
DTSTART;VALUE=DATE-TIME:20210716T120000Z
DTEND;VALUE=DATE-TIME:20210716T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/93
DESCRIPTION:Title: Co
nvex bodies and graded families of monomial ideals\nby Yairon Cid-Ruiz
(Ghent University\, Krijgslaan\, Belgium) as part of IIT Bombay Virtual C
ommutative Algebra Seminar\n\n\nAbstract\nWe show that the mixed volumes o
f arbitrary convex bodies are equal to mixed multiplicities of graded fami
lies of monomial ideals and to normalized limits of mixed multiplicities o
f monomial ideals. This result evinces the close relationship between the
theories of mixed volumes from convex geometry and mixed multiplicities fr
om commutative algebra. This is joint work with Jonathan Montaño. \n\nTim
e permitting\, we will also speak about some joint work with Fatemeh Moham
madi and Leonid Monin regarding "multi-graded algebras and multi-graded li
near series".\n
LOCATION:https://researchseminars.org/talk/VCAS/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Purna Bangere (University of Kansas Lawrence\, KS)
DTSTART;VALUE=DATE-TIME:20210806T130000Z
DTEND;VALUE=DATE-TIME:20210806T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/94
DESCRIPTION:Title: Sy
zygies and Gonality\nby Purna Bangere (University of Kansas Lawrence\,
KS) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstra
ct\nIn this talk we will deal with recent results on the so-called propert
y $N_p$ and $M_p.$ These concern the structure of free resolutions associa
ted with a very ample line bundle on \na projective variety. There are int
eresting conjectures and ideas related to the structure of free resolution
s 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 pro
perty $M_p.$ In this talk we will describe some new results about properti
es $M_p$ for an algebraic surface\, \nsome higher dimensional varieties\,
and even interesting everywhere non reduced schemes called carpets.\n
LOCATION:https://researchseminars.org/talk/VCAS/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:K.-i. Watanabe (Nihon University)
DTSTART;VALUE=DATE-TIME:20210813T120000Z
DTEND;VALUE=DATE-TIME:20210813T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/95
DESCRIPTION:Title: In
verse polynomials of symmetric numerical semigroups\nby K.-i. Watanabe
(Nihon University) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nThis is a joint work with Kazufumi Eto (Nippon Institut
e 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 nume
rical semigroup ring and $k[H]$ be its semigroup ring over any field $K.$
If $H = ⟨n_1\, \\ldots\,n_e)$\, we express $k[H]$ as $k[H] = k[x_1\,\\l
dots\,x_e]/I_H$ and we want to express $k[H]/(t^h)$ by ”Inverse polynomi
als” of Macaulay.\nWe study the defining ideal of a numerical semigroup
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 inverse
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 criterion
for H to be symmetric or almost symmetric.\n(2) Characterization of symmet
ric numerical semigroups of small multiplicity.\n(3) A new proof of Bresin
sky’s Theorem for symmetric semigroups generated by 4\nelements.\n
LOCATION:https://researchseminars.org/talk/VCAS/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mousumi Mandal (IIT Kharagpur)
DTSTART;VALUE=DATE-TIME:20210820T120000Z
DTEND;VALUE=DATE-TIME:20210820T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jai Laxmi (Tata Institute of Fundamental Research\, Mumbai)
DTSTART;VALUE=DATE-TIME:20210827T120000Z
DTEND;VALUE=DATE-TIME:20210827T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/97
DESCRIPTION:Title: Go
renstein ideals of codimension four\nby Jai Laxmi (Tata Institute of F
undamental Research\, Mumbai) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\n\nAbstract\nWe explore spinor coordinates on free resolut
ions of\ncodimension four Gorenstein ideals. Our analysis of spinor coordi
nate\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 doublin
g\ncodimension three perfect ideals. We see\, in particular\, the construc
tion\nof generic doublings of almost complete intersection perfect ideals
of\ncodimension three.\n
LOCATION:https://researchseminars.org/talk/VCAS/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eloisa Grifo (University of Nebraska\, LIncoln\, NE)
DTSTART;VALUE=DATE-TIME:20210903T130000Z
DTEND;VALUE=DATE-TIME:20210903T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/98
DESCRIPTION:Title: A
survey of Harbourne’s Conjecture\nby Eloisa Grifo (University of Neb
raska\, LIncoln\, NE) as part of IIT Bombay Virtual Commutative Algebra Se
minar\n\n\nAbstract\nHarbourne’s conjecture on the containment problem f
or symbolic and ordinary powers of ideals is not true in its original form
\, but it has sparked a lot of different research avenues. We will discuss
some of the known counterexamples but mostly focus on the different varia
tions of the conjecture that are true or still open.\n
LOCATION:https://researchseminars.org/talk/VCAS/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dumitru Stamate (University of Bucharest\, Romania)
DTSTART;VALUE=DATE-TIME:20210910T120000Z
DTEND;VALUE=DATE-TIME:20210910T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/99
DESCRIPTION:Title: Th
e 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-Macaulay loca
l ring (or positively graded K-algebra) with canonical module $\\omega_R.$
The trace of the latter\, $tr(\\omega_R)\,$ is by definition\, the ideal
generated by the images of all R-module homomorphisms from $\\omega_R$ int
o 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\nIn terms
of this ideal\, new classes of rings have been introduced\, and their pro
perties are under scrutiny. We discuss some of these approaches\, with a s
pecial focus on families of examples coming from combinatorics. \n\nThis t
alk is based on joint works with J. Herzog\, T. Hibi\, R. Jafari and S. Ku
mashiro.\n
LOCATION:https://researchseminars.org/talk/VCAS/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Winfried Bruns (University of Osnabrueck\, Germany)
DTSTART;VALUE=DATE-TIME:20210917T120000Z
DTEND;VALUE=DATE-TIME:20210917T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/100
DESCRIPTION:Title: C
astelnuovo-Mumford regularity over general base rings\nby Winfried Bru
ns (University of Osnabrueck\, Germany) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nOur talk has two goals. The first
is to give a short introduction to Castelnuovo-Mumford regularity for sta
ndard graded rings over general base rings. The second is to present a si
mple and concise proof of a classical result of Cutkosky\, Herzog and Tru
ng and\, independently\, Kodiyalam asserting that the regularity of pow
ers of a homogeneous ideal is eventually a linear function of the exponen
t in this generality. Finally we show how the flexibility of working ove
r general base rings can be used to give a simple proof for the characteri
zation of "linear powers" in terms of the Rees algebra.\n\nThis is joint w
ork with Aldo Conca and Matteo Varbaro. See "Castelnuovo-Mumford regularit
y and powers"\, arXiv:2107.14734. An extensive version will be part of the
forthcoming book "Determinants\, Gröber bases and cohomology" with Conca
\, Varbaro and Claudiu Raicu.\n
LOCATION:https://researchseminars.org/talk/VCAS/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Prashant Sridhar (TIFR\, Mumbai\, India)
DTSTART;VALUE=DATE-TIME:20210924T120000Z
DTEND;VALUE=DATE-TIME:20210924T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/101
DESCRIPTION:Title: F
inding Maximal Cohen-Macaulay modules\nby Prashant Sridhar (TIFR\, Mum
bai\, India) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n
\nAbstract\nIn this talk\, we consider a problem that lies in the confluen
ce of two topics.\nOn one hand\, we have maximal Cohen-Macaulay (MCM) modu
les - these are classical objects that have been studied extensively from
algebraic and geometric viewpoints. There is a rich theory of MCM modules
over Cohen-Macaulay (CM) rings and many beautiful connections to the singu
larities 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 existe
nce is largely unclear.\nOn the other hand\, we have a mixed characteristi
c phenomenon. In 1980\, Paul Roberts showed that the integral closure of a
regular local ring in an Abelian extension of its quotient field is CM\,
provided the characteristic of the residue field does not divide the degre
e of the extension. This fails in the "modular case" in mixed characterist
ic.\nWe will look at some past results in the literature before considerin
g the question of existence of MCMs in the modular case of Roberts's theor
em.\n
LOCATION:https://researchseminars.org/talk/VCAS/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Fasel (Université de Grenoble I\, Grenoble\, France)
DTSTART;VALUE=DATE-TIME:20211001T130000Z
DTEND;VALUE=DATE-TIME:20211001T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/102
DESCRIPTION:Title: C
ohomotopy groups in algebraic geometry and unimodular rows\nby Jean Fa
sel (Université de Grenoble I\, Grenoble\, France) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\n\nAbstract\nWe will discuss the not
ion of cohomotopy groups in algebraic geometry. Our main example and motiv
ation will be the study of unimodular rows\, but we will also discuss othe
r examples such as unimodular m x n matrices and Euler class groups. The g
eneral context of this study is motivic homotopy theory\, but we will avoi
d technicalities and motivate the constructions using classical topology.\
n
LOCATION:https://researchseminars.org/talk/VCAS/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Krishna Hanumanthu (Chennai Mathematical Institute\, Chennai)
DTSTART;VALUE=DATE-TIME:20211008T120000Z
DTEND;VALUE=DATE-TIME:20211008T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/103
DESCRIPTION:Title: S
eshadri constants\nby Krishna Hanumanthu (Chennai Mathematical Institu
te\, Chennai) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\
n\nAbstract\nSeshadri constants of line bundles on projective varieties we
re defined by J-P. Demailly in 1990\, motivated by an ampleness criterion
of C. S. Seshadri. They are a measure of local positivity of line bundles
and have interesting connections to different areas in mathematics. We wi
ll discuss some important questions that drive research on Seshadri consta
nts. We will also discuss their relationship with Waldschmidt constants wh
ich are invariants attached to homogeneous ideals in polynomial rings.\n
LOCATION:https://researchseminars.org/talk/VCAS/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ritvik Ramkumar (UC Berkeley\, CA\, USA)
DTSTART;VALUE=DATE-TIME:20211015T140000Z
DTEND;VALUE=DATE-TIME:20211015T150000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/104
DESCRIPTION:Title: A
n invitation to the fiber-full scheme\nby Ritvik Ramkumar (UC Berkeley
\, CA\, USA) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n
\nAbstract\nI will introduce the fiber-full scheme which can be seen as th
e parameter space that generalizes the Hilbert and Quot schemes by control
ling the entire cohomological data. In particular\, given a sequence of fu
nctions $(h_0\,..\,h_n)$ the fiber-full scheme parameterizes subschemes $X
$ of $P^n$ satisfying dim $H^i(X\,O_X(v)) = h_i(v).$ In this talk I will s
ketch the construction of the fiber-full scheme and describe some examples
associated with classical Hilbert schemes. I will also discuss various fu
ture directions one can pursue. This is joint work with Yairon-Cid Ruiz.\n
Chairperson - N. Mohan Kumar\n
LOCATION:https://researchseminars.org/talk/VCAS/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amalendu Krishna (Indian Institute of Science\, Bangalore)
DTSTART;VALUE=DATE-TIME:20211022T120000Z
DTEND;VALUE=DATE-TIME:20211022T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/105
DESCRIPTION:Title: C
how groups and Euler class groups of affine varieties\nby Amalendu Kri
shna (Indian Institute of Science\, Bangalore) as part of IIT Bombay Virtu
al Commutative Algebra Seminar\n\n\nAbstract\nIn this talk\, we shall pres
ent some results regarding the relation between the Chow groups and Euler
class groups of affine varieties over algebraically closed fields. We shal
l show how these results allow one to deduce an old conjecture of Murthy.
If time permits\, we shall discuss some related questions on Chow-Witt gro
ups of affine varieties over non-algebraically closed.\n
LOCATION:https://researchseminars.org/talk/VCAS/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josh Pollitz (University of Utah)
DTSTART;VALUE=DATE-TIME:20211029T130000Z
DTEND;VALUE=DATE-TIME:20211029T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/106
DESCRIPTION:Title: S
ymmetries in Bass and Betti sequences over a complete intersection ring\nby Josh Pollitz (University of Utah) as part of IIT Bombay Virtual Comm
utative Algebra Seminar\n\n\nAbstract\nDespite homological algebra over a
complete intersection ring being unbounded\, resolutions enjoy polynomial
growth. That is to say\, for a finitely generated module over a complete i
ntersection ring\, its sequence of Bass numbers and its sequence Betti num
bers are eventually given by quasi-polynomials with period two\; the leadi
ng terms of the quasi-polynomials are independent of the parity. In this t
alk\, I will discuss joint worth with Briggs and McCormick where we show t
he leading terms of the two quasi-polynomials agree. The main tool is a hi
gher order support theory which generalizes the well-studied support varie
ties of a complete intersection ring.\n
LOCATION:https://researchseminars.org/talk/VCAS/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christopher Eur (Harvard University)
DTSTART;VALUE=DATE-TIME:20211105T140000Z
DTEND;VALUE=DATE-TIME:20211105T150000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/107
DESCRIPTION:Title: T
autological classes of matroids\nby Christopher Eur (Harvard Universit
y) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract
\nAlgebraic geometry has furnished fruitful tools for studying matroids\,
which are combinatorial abstractions of hyperplane arrangements. We first
survey some recent developments\, pointing out how these developments rema
ined partially disjoint. We then introduce certain vector bundles (K-class
es) on permutohedral varieties\, which we call "tautological bundles (clas
ses)" of matroids\, as a new framework that unifies\, recovers\, and exten
ds these recent developments. Our framework leads to new questions that fu
rther probe the boundary between combinatorics and geometry. This is joint
work with Andrew Berget\, Hunter Spink\, and Dennis Tseng.\n
LOCATION:https://researchseminars.org/talk/VCAS/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seyed Amin Seyed Fakhari (University of Tehran)
DTSTART;VALUE=DATE-TIME:20211112T120000Z
DTEND;VALUE=DATE-TIME:20211112T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/108
DESCRIPTION:Title: H
omological properties of symbolic powers of cover ideals of graphs\nby
Seyed Amin Seyed Fakhari (University of Tehran) as part of IIT Bombay Vir
tual Commutative Algebra Seminar\n\n\nAbstract\nTo every simple graph\, on
e associates its edge ideal which is generated by quadratic squarefree mo
nomials corresponding to edges of the graph. In this talk\, we study the A
lexander dual of edge ideals\, which is called the cover ideal. The reason
for this naming is that the cover ideal is minimally generated by squaref
ree monomials corresponding to the minimal vertex covers of the given grap
h. We review the recent results about the symbolic powers of cover ideals.
In particular\, we characterize all graphs with the property that every s
ymbolic power of its cover ideal has a linear resolution. Also\, we determ
ine an upper bound for the regularity of symbolic powers of certain classe
s of graphs including bipartite graphs\, unmixed graphs and claw-free grap
hs. Moreover\, we study the asymptotic behavior of depth of symbolic power
s of cover ideals.\n\nChairperson: Siamak Yassemi\, IPM\, Tehran\n
LOCATION:https://researchseminars.org/talk/VCAS/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Susan Morey (Texas State University)
DTSTART;VALUE=DATE-TIME:20211119T130000Z
DTEND;VALUE=DATE-TIME:20211119T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/109
DESCRIPTION:Title: C
ellular Resolutions and Powers of Monomial Ideals\nby Susan Morey (Tex
as State University) as part of IIT Bombay Virtual Commutative Algebra Sem
inar\n\n\nAbstract\nUsing combinatorial structures to obtain resolutions o
f monomial ideals is an idea that traces back to Diana Taylor’s thesis\,
where a simplex associated to the generators of a monomial ideal was used
to construct a free resolution of the ideal. This concept has been expand
ed over the years\, with various authors determining conditions under whic
h simplicial or cellular complexes can be associated to monomial ideals in
ways that produce a free resolution. \nIn a research project initiated
at a BIRS workshop “Women in Commutative Algebra” in Fall 2019\, the\n
authors studied simplicial and cellular structures that produced resolutio
ns of powers of monomial\nideals. The optimal structure to use depends upo
n the structure of the monomial ideal. This talk will focus on powers of s
quare-free monomial ideals of projective dimension one. Faridi and Hersey
proved that a monomial ideal has projective dimension one if and only if t
here is an associated tree (one dimensional acyclic simplicial complex) th
at supports a free resolution of the ideal. The talk will show how\, for e
ach power $r >\;1$\, to use the tree associated to a square-free monomia
l ideal $I$ of projective dimension one to produce a cellular complex that
supports a free resolution of $I^r$. Moreover\, each of these resolutions
will be minimal resolutions. These cellular resolutions can also be viewe
d as strands of the resolution of the Rees algebra of $I$. This talk will
contain joint work with Susan Cooper\, Sabine El Khoury\, Sara Faridi\, Sa
rah Mayes-Tang\, Liana Sega\, and Sandra Spiroff.\nChairperson - Takayuki
Hibi\n
LOCATION:https://researchseminars.org/talk/VCAS/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nursel Erey (Gebze Technical University)
DTSTART;VALUE=DATE-TIME:20211126T120000Z
DTEND;VALUE=DATE-TIME:20211126T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/110
DESCRIPTION:Title: S
quarefree powers of edge ideals\nby Nursel Erey (Gebze Technical Unive
rsity) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbst
ract\nLet $G$ be a finite simple graph. The edge ideal of $G$\, denoted by
$I(G)$\, is a monomial ideal generated by the monomials that correspond t
o the edges of the graph. In this talk\, we will be interested in resoluti
ons 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 lin
early related or have linear resolution. This talk is based on joint work
with Jürgen Herzog\, Takayuki Hibi and Sara Saeedi Madani.\nChairperson -
Takayuki Hibi\n
LOCATION:https://researchseminars.org/talk/VCAS/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jerzy Weyman (Jagiellonian University\, Poland)
DTSTART;VALUE=DATE-TIME:20211203T130000Z
DTEND;VALUE=DATE-TIME:20211203T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/111
DESCRIPTION:Title: G
orenstein ideals of codimension 4\nby Jerzy Weyman (Jagiellonian Unive
rsity\, Poland) as part of IIT Bombay Virtual Commutative Algebra Seminar\
n\n\nAbstract\nIn this talk I will describe old and new results on free re
solutions 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 exc
eptional root systems.\nChairperson - Bernd Ulrich\n
LOCATION:https://researchseminars.org/talk/VCAS/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudiu Raicu (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20211210T130000Z
DTEND;VALUE=DATE-TIME:20211210T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/112
DESCRIPTION:Title: C
ohomology of line bundles on the incidence correspondence\nby Claudiu
Raicu (University of Notre Dame) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nLet X denote the incidence correspondence
(or partial flag variety) parametrizing pairs consisting of a point in pro
jective space and a hyperplane containing it. I will explain how to charac
terize the vanishing and non-vanishing behavior of the cohomology groups o
f line bundles on X over an arbitrary field. For the projective plane\, th
e results are contained in the thesis of Griffith from the 70s\, while in
characteristic zero the cohomology groups are described in any dimension b
y the Borel-Weil-Bott theorem. Joint work with Zhao Gao.\nChairperson - Be
rnd Ulrich\n
LOCATION:https://researchseminars.org/talk/VCAS/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dipankar Ghosh (Indian Institute of Technology\, Hyderabad)
DTSTART;VALUE=DATE-TIME:20210107T130000Z
DTEND;VALUE=DATE-TIME:20210107T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/113/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Naoki Terai (Okayama University)
DTSTART;VALUE=DATE-TIME:20211217T120000Z
DTEND;VALUE=DATE-TIME:20211217T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/114
DESCRIPTION:Title: C
ohen-Macaulay property of weighted edge ideal of very well-covered graphs<
/a>\nby Naoki Terai (Okayama University) as part of IIT Bombay Virtual Com
mutative Algebra Seminar\n\n\nAbstract\nWe consider the edge ideals of edg
e-weighted very well-covered graphs and discuss their unmixed and Cohen-Ma
caulay properties. This is based on a joint work with S.A. Seyed Fakhari\,
K. Shibata and S. Yassemi.\nChairperson - Siamak Yassemi\n
LOCATION:https://researchseminars.org/talk/VCAS/114/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Joseph (IIT Bombay)
DTSTART;VALUE=DATE-TIME:20210108T120000Z
DTEND;VALUE=DATE-TIME:20210108T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/115/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dipankar Ghosh (IIT Kharagpur)
DTSTART;VALUE=DATE-TIME:20220107T120000Z
DTEND;VALUE=DATE-TIME:20220107T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/116
DESCRIPTION:Title: (
Non) linear behavior of Castelnuovo-Mumford regularity\nby Dipankar Gh
osh (IIT Kharagpur) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nIn the first half\, we shall discuss the importance of
Castelnuovo-Mumford regularity and the main motivation to define this inva
riant along with examples. This part is aimed for the graduate students an
d should be elementary.\nIn the second half\, we discuss the asymptotic be
havior of Castelnuovo-Mumford regularity of powers of ideals\, and that of
Ext and Tor\, with respect to the homological degree\, over graded comple
te intersection rings. We derive from a theorem of Gulliksen\, a linearity
result for the regularity of Ext modules in high homological degrees. We
show a similar result for Tor\, under the additional hypothesis that high
enough Tor modules are supported in dimension at most one\; we then provid
e examples showing that the behavior could be pretty hectic when the latte
r condition is not satisfied.\nThis talk is based on our joint papers with
Marc Chardin\, Navid Nemati and Tony Puthenpurakal.\n
LOCATION:https://researchseminars.org/talk/VCAS/116/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tony Puthenpurakal (IIT Bombay)
DTSTART;VALUE=DATE-TIME:20220114T120000Z
DTEND;VALUE=DATE-TIME:20220114T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/117
DESCRIPTION:Title: I
toh's conjecture for normal ideals\nby Tony Puthenpurakal (IIT Bombay)
as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\n
Let $(A\,m)$ be a CM local ring and let $I$ be a normal $m$-primary ideal
with $e_3(I) = 0.$\nThen $G_I(A)$ is CM.\n(note Itoh only conjectured it
for Gorenstein local rings)\n
LOCATION:https://researchseminars.org/talk/VCAS/117/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fatemeh Mohammadi (Ghent University and University of Tromsø)
DTSTART;VALUE=DATE-TIME:20220121T120000Z
DTEND;VALUE=DATE-TIME:20220121T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/118
DESCRIPTION:Title: T
oric degenerations of Grassmannians\nby Fatemeh Mohammadi (Ghent Unive
rsity and University of Tromsø) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nToric varieties are popular objects in alg
ebraic geometry\, as they can be modelled on polytopes and polyhedral fans
. This is mainly because there is a dictionary between their geometric pro
perties and the combinatorial invariants of their polytopes. This dictiona
ry can be extended from toric varieties to arbitrary varieties through tor
ic 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 prototypic exa
mples of toric degenerations (of Grassmannians) which are related to Young
tableaux and Gelfand-Cetlin polytopes. I will describe how to obtain such
degenerations using the theory of Gröbner fans and tropical geometry\, a
nd how their associated polytopes are connected by mutations.\n
LOCATION:https://researchseminars.org/talk/VCAS/118/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shinya Kumashiro (Oyama College)
DTSTART;VALUE=DATE-TIME:20220128T120000Z
DTEND;VALUE=DATE-TIME:20220128T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/119
DESCRIPTION:Title: G
raded Bourbaki ideals of graded modules and Ideals of reduction number two
\nby Shinya Kumashiro (Oyama College) as part of IIT Bombay Virtual Co
mmutative 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 torsio
nfree modules over Noetherian normal domains\, Bourbaki sequences exist. W
e give criteria in terms of certain attached matrices for a homomorphism o
f modules to induce a Bourbaki sequences. In the second part of this talk\
, we give an application of graded Bourbaki sequences to Hilbert functions
of m-primary ideals. We give the inequality of the first three Hilbert co
efficients for ideals of reduction number two.\n\nThis talk is based on th
e joint work with J. Herzog and D. I. Stamate.\n
LOCATION:https://researchseminars.org/talk/VCAS/119/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Van Tuyl (McMaster University)
DTSTART;VALUE=DATE-TIME:20220204T130000Z
DTEND;VALUE=DATE-TIME:20220204T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/120
DESCRIPTION:Title: T
oric ideals of graphs and some of their homological invariants\nby Ada
m Van Tuyl (McMaster University) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nThe study of toric ideals of graphs lies i
n the intersection of commutative algebra\, algebraic geometry\, and combi
natorics. Formally\, if $G = (V\,E)$ is a finite simple graph with edge s
et $E =\\{e_1\,\\ldots\,e_s\\}$ and vertex set $V = \\{x_1\,\\ldots\,x_n\\
}\,$ then the toric ideal of $G$ is the kernel of the ring homomorphism $\
\varphi:k[e_1\,\\ldots\,e_s] \\rightarrow k[x_1\,\\ldots\,x_n]$ where $\\v
arphi(e_i) = x_jx_k$ if the edge $e_i = \\{x_j\,x_k\\}$. Ideally\, one wo
uld like to understand how the homological invariants (e.g. graded Betti n
umbers) 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 Castelnuov
o-Mumford regularity of these ideals.\n
LOCATION:https://researchseminars.org/talk/VCAS/120/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuke Takagi (University of Tokyo)
DTSTART;VALUE=DATE-TIME:20220211T120000Z
DTEND;VALUE=DATE-TIME:20220211T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/121
DESCRIPTION:Title: K
odaira vanishing for thickenings of globally $F$-regular varieties\nby
Shunsuke Takagi (University of Tokyo) as part of IIT Bombay Virtual Commu
tative Algebra Seminar\n\n\nAbstract\nBlickle-Bhatt-Lyubeznik-Singh-Zhang
proved that if $X$ is a projective variety over a field $k$ of characteris
tic zero with isolated complete intersection singularities\, then the Koda
ira vanishing theorem holds for all thickenings of $X$. What if $k$ is of
positive characteristic? Kodaira vanishing can fail in positive characteri
stic\, but it still holds for Frobenius split varieties. In this talk\, I
will discuss Kodaira vanishing for thickenings of globally $F$-regular var
ieties\, a special class of Frobenius split varieties. This talk is based
on joint work with Kenta Sato.\n
LOCATION:https://researchseminars.org/talk/VCAS/121/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liran Shaul (Charles University\, Prague)
DTSTART;VALUE=DATE-TIME:20220218T120000Z
DTEND;VALUE=DATE-TIME:20220218T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/122
DESCRIPTION:Title: S
pecial classes of rings in derived commutative algebra\nby Liran Shaul
(Charles University\, Prague) as part of IIT Bombay Virtual Commutative A
lgebra Seminar\n\n\nAbstract\nThe classes of regular\, Gorenstein and Cohe
n-Macaulay rings are among the most important classes of rings in commutat
ive algebra and algebraic geometry.\nIn this talk we recall the definition
s and basic properties of these classes\,\nand then explain how to general
ize each of them to derived commutative\nalgebra\, in the context of commu
tative differential graded algebras.\nWe further explain how each of these
generalizations arise naturally\nin various algebraic geometry contexts a
nd discuss some applications.\n
LOCATION:https://researchseminars.org/talk/VCAS/122/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Keri Sather-Wagstaff (Clemson University)
DTSTART;VALUE=DATE-TIME:20220225T130000Z
DTEND;VALUE=DATE-TIME:20220225T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/123
DESCRIPTION:Title: M
onomial Ideals Arising from Graph Domination Problems\nby Keri Sather-
Wagstaff (Clemson University) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\n\nAbstract\nGraph domination problems are ubiquitous in g
raph theory. In the broadest terms\, they ask how one can ‘observe’ an
entire graph by designating a certain list of vertices\, following a pros
cribed list of rules. An example of this is the vertex covering problem wh
ich happens to describe the irredundant irreducible decomposition of the e
dge ideal of a graph. In this talk\, we will survey recent work with vario
us collaborators on other monomial ideal constructions that arise from oth
er graph domination problems\, including one coming from electrical engine
ering.\n
LOCATION:https://researchseminars.org/talk/VCAS/123/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Gubeladze (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20220304T140000Z
DTEND;VALUE=DATE-TIME:20220304T150000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/124
DESCRIPTION:Title: N
ormal polytopes and ellispoids\nby Joseph Gubeladze (San Francisco Sta
te University) as part of IIT Bombay Virtual Commutative Algebra Seminar\n
\n\nAbstract\nLattice polytopes are the combinatorial backbone of toric va
rieties. Many important properties of these varieties admit purely combina
torial description in terms of the underlying polytopes. These include nor
mality and projective normality. On the other hand\, there are geometric p
roperties of polytopes of integer programming/discrete optimization origin
\, which can be used to deduce the aforementioned combinatorial properties
: existence of unimodular triangulations or unimodular covers. In this tal
k we present the following recent results: (1) unimodular simplices in a l
attice 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 every el
lipsoid in R^3 has a unimodular cover\, and (3) for every d at least 5\, t
here are ellipsoids in R^d\, such that the convex hulls of the lattice poi
nts in these ellipsoids are not even normal. Part (3) answers a question o
f Bruns\, Michalek\, and the speaker.\nChaiperson - Siamak Yassemi\n
LOCATION:https://researchseminars.org/talk/VCAS/124/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Craig Huneke (University of Virginia)
DTSTART;VALUE=DATE-TIME:20220311T130000Z
DTEND;VALUE=DATE-TIME:20220311T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/125
DESCRIPTION:Title: T
orsion in Commutative Algebra\nby Craig Huneke (University of Virginia
) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\
nThis talk will be a somewhat historical one\, concerning three problems\n
dealing with the idea of torsion. The three problems are those on symboli
c powers\,\nthe Huneke-Wiegand conjecture\, and Berger's conjecture. Besi
des talking about\nmy own memories\, we will focus on torsion in tensor pr
oducts.\n
LOCATION:https://researchseminars.org/talk/VCAS/125/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Smirnov (BCAM-Basque Center for Applied Mathematics)
DTSTART;VALUE=DATE-TIME:20220318T120000Z
DTEND;VALUE=DATE-TIME:20220318T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/126
DESCRIPTION:Title: L
ech's inequality can be sharpened uniformly\nby Ilya Smirnov (BCAM-Bas
que Center for Applied Mathematics) as part of IIT Bombay Virtual Commutat
ive 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 o
f the multiplicity and the colength of an m-primary ideal of a local ring.
It was also observed by Lech that\, if the dimension is at least two\, it
is not sharp for any given ideal. Recently\, we were able to show more: m
ost of the time\, it is possible to improve Lech's upper bound so that it
works for all ideals. I will present the proof of this result and all requ
ired background in multiplicity theory.\n
LOCATION:https://researchseminars.org/talk/VCAS/126/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dharm Veer (Chennai Mathematical Institute)
DTSTART;VALUE=DATE-TIME:20220325T120000Z
DTEND;VALUE=DATE-TIME:20220325T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/127
DESCRIPTION:Title: O
n Green-Lazarsfeld property $N_p$ for Hibi rings\nby Dharm Veer (Chenn
ai Mathematical Institute) as part of IIT Bombay Virtual Commutative Algeb
ra Seminar\n\n\nAbstract\nLet $L$ be a finite distributive lattice. By Bir
khoff's fundamental structure theorem\, $L$ is the ideal lattice of its su
bposet $P$ 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 $n+1$ variables
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 $u_{\\alpha}=t\\
prod_{p_i\\in \\alpha}z_i$ where $\\alpha\\in L$. In this talk\, we show t
hat a Hibi ring satisfies property $N_4$ if and only if it is a polynomial
ring or it has a linear resolution. We also discuss a few results about t
he property $N_p$ of Hibi rings for $p=2$ and 3. For example\, we show tha
t 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://researchseminars.org/talk/VCAS/127/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Soumi Tikader (Diamond Harbour Women's University)
DTSTART;VALUE=DATE-TIME:20220401T120000Z
DTEND;VALUE=DATE-TIME:20220401T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/128
DESCRIPTION:Title: M
onic inversion principle of local complete intersection ideal\nby Soum
i Tikader (Diamond Harbour Women's University) as part of IIT Bombay Virtu
al Commutative Algebra Seminar\n\n\nAbstract\nThe renowned Quillen–Susli
n Theorem is closely associated to the Affine Horrocks’ Theorem on alge
braic vector bundles. It says : If $R$ is any commutative\nring and $E$ i
s a vector bundle on $\\mathbb{A}_{R}^1$ and $E$ extends to a vector bundl
e on $\\mathbb{P}^1\n_R\,$ then $E$ is extended\nfrom $Spec(R).$ This is a
lso known as "Monic inversion principle" for projective modules.\nHere we
discuss about \nanalogue of the Monic inversion principle for local compl
ete intersection ideals\nof height $n$ in $R[T]\,$ where $R$ is a regular
domain of dimension $d\,$ which is essentially of\nfinite type over an in
finite 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://researchseminars.org/talk/VCAS/128/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Sammartano (Politecnico di Milano)
DTSTART;VALUE=DATE-TIME:20220408T120000Z
DTEND;VALUE=DATE-TIME:20220408T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/129
DESCRIPTION:Title: N
ested Hilbert schemes of the plane\nby Alessio Sammartano (Politecnico
di Milano) as part of IIT Bombay Virtual Commutative Algebra Seminar\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 behaved
parameter space. A classical theorem of Fogarty states that it is a smooth
variety of dimension 2n. By contrast\, the nested Hilbert scheme Hilb^(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 singula
rities. Using techniques from commutative algebra\, we prove that the nest
ed Hilbert scheme Hilb^(n\,2)(A^2) has rational singularities. This is a j
oint work with Ritvik Ramkumar.\n
LOCATION:https://researchseminars.org/talk/VCAS/129/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro De Stefani (Università di Genova)
DTSTART;VALUE=DATE-TIME:20220415T120000Z
DTEND;VALUE=DATE-TIME:20220415T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/130
DESCRIPTION:Title: A
uniform Chevalley theorem for direct summands in mixed characteristic
\nby Alessandro De Stefani (Università di Genova) as part of IIT Bombay V
irtual Commutative Algebra Seminar\n\n\nAbstract\nLet R be a graded direct
summand of a positively graded polynomial ring over the p-adic integers.
We exhibit an explicit constant D such that\, for any positive integer n a
nd any homogeneous prime ideal Q of R\, the Dn-th symbolic power of Q is c
ontained in the n-th power of the homogeneous maximal ideal (p)R + R_+. Th
e 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 talk wil
l be based on joint work with E. Grifo and J. Jeffries.\n
LOCATION:https://researchseminars.org/talk/VCAS/130/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuji Yoshino (Okayama University)
DTSTART;VALUE=DATE-TIME:20220422T120000Z
DTEND;VALUE=DATE-TIME:20220422T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/131
DESCRIPTION:Title: N
aive liftings of dg modules\nby Yuji Yoshino (Okayama University) as p
art of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nThe n
aive lifting for dg modules is the new concept introduced by M.Ono\, S.Nas
seh and myself for the purpose of unifying the ideas of lifting and weak l
ifting for modules over commutative rings.\nIn this talk I will show how w
e get the obstruction class of naive liftings\, which in fact coincides wi
th the Atiyah class that has been introduced by Buchweitz-Flenner.\nThis i
s a joint work with Saeed Nasseh and Maiko Ono.\n
LOCATION:https://researchseminars.org/talk/VCAS/131/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eleonore Faber (University of Leeds)
DTSTART;VALUE=DATE-TIME:20220429T120000Z
DTEND;VALUE=DATE-TIME:20220429T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/132
DESCRIPTION:Title: C
luster structures for the A-infinity singularity\nby Eleonore Faber (U
niversity of Leeds) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nThis talk is about a categorification of the coordinate
rings of Grassmannians\nof infinite rank in terms of graded maximal Cohen
-Macaulay modules over the ring $\\mathbb{C}[x\,y]/(x^k).$ This yields an
infinite rank analog of the Grassmannian cluster categories introduced by
Jensen\, King\, and Su.\nIn the special case\, $k=2\,$ $\\text{Spec}(\\mat
hbb{C}[x\,y]/(x^2))$ is a type $A$-infinity singularity and the ungraded v
ersion of the category of maximal Cohen-Macaulay modules over $\\mathbb{C}
[x\,y]/(x^2))$ has been studied by Buchweitz\, Greuel\, and Schreyer in th
e 1980s. We demonstrate that his category has infinite type $A$ cluster co
mbinatorics. In particular\, we show that it has cluster-tilting subcatego
ries modeled by certain triangulations of the (completed) infinity-gon and
we can also interpret certain mutations of the category in this model. Th
is is joint work with Jenny August\, Man-Wai Cheung\, Sira Gratz\, and Sib
ylle Schroll.\n
LOCATION:https://researchseminars.org/talk/VCAS/132/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mircea Mustata (University of Michigan)
DTSTART;VALUE=DATE-TIME:20220909T130000Z
DTEND;VALUE=DATE-TIME:20220909T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/139
DESCRIPTION:Title: A
n estimate for the F-pure threshold via the roots of the Bernstein-Sato po
lynomial\nby Mircea Mustata (University of Michigan) as part of IIT Bo
mbay Virtual Commutative Algebra Seminar\n\n\nAbstract\nGiven a smooth com
plex algebraic variety X and a nonzero regular function f\non X\, I will d
escribe an estimate for the difference between the log canonical threshold
of f\nand the F-pure threshold of a reduction mod p of f\, in terms of th
e roots of the Bernstein-Sato polynomial bf of f. This is based on some ol
d work with S. Takagi and K.-i. Watanabe on one hand\, and with W. Zhang o
n the other hand\, plus one simple observation. Most of the talk will be d
evoted to an introduction to the invariants of singularities that\nfeature
in the result.\n
LOCATION:https://researchseminars.org/talk/VCAS/139/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joachim Jelisiejew (University of Warsaw)
DTSTART;VALUE=DATE-TIME:20220930T120000Z
DTEND;VALUE=DATE-TIME:20220930T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/140
DESCRIPTION:Title: W
hen is a homogeneous ideal a limit of saturated ones?\nby Joachim Jeli
siejew (University of Warsaw) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\n\nAbstract\nLet I be a homogeneous ideal in a polynomial
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\, wh
ile F(t = lambda) is a saturated homogeneous ideal for lambda general. Exa
mples of such limits (for the above Hilbert function) can be constructed 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 whether a giv
en ideal I is a limit is very much nontrivial. This problem very recently
became of key interest for applications in the theory of tensors: proving
that certain ideals are not limits would improve best known lower bounds o
n border ranks of certain important tensors.\nIn the talk I will report ho
w surprisingly little is known and present some recent results and some ch
allenges\, both theoretical and computational. All this is a joint work w
ith Tomasz Mandziuk.\n
LOCATION:https://researchseminars.org/talk/VCAS/140/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Parnashree Ghosh (Indian Statistical Institute\, Kolkata\, India)
DTSTART;VALUE=DATE-TIME:20221014T120000Z
DTEND;VALUE=DATE-TIME:20221014T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/141
DESCRIPTION:Title: O
n the triviality of a family of linear hyperplanes\nby Parnashree Ghos
h (Indian Statistical Institute\, Kolkata\, India) as part of IIT Bombay V
irtual Commutative Algebra Seminar\n\n\nAbstract\nLet k be a field\, m a p
ositive 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 $\\mathbb{V}$
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 to be is
omorphic $\\mathbb{A}^{m+2}$ and G to be a coordinate in $k[X_1\, \\ldots
\, X_m\, Y\, Z\, T]\,$ under a certain hypothesis on F. Our main result im
mediately yields a family of higher-dimensional linear hyperplanes for whi
ch the Abhyankar-Sathaye Conjecture holds.\nWe also describe the isomorphi
sm classes and automorphisms of integral domains of the type A under certa
in conditions. These results show that for each integer $d\\geq 3\,$ there
is a family of infinitely many pairwise non-isomorphic rings which are co
unterexamples to the Zariski Cancellation Problem for dimension d in posit
ive characteristic. \nThis is a joint work with Neena Gupta.\n
LOCATION:https://researchseminars.org/talk/VCAS/141/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louiza Fouli (New Mexico State University\, Las Cruces\, NM\, USA)
DTSTART;VALUE=DATE-TIME:20221021T130000Z
DTEND;VALUE=DATE-TIME:20221021T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/142
DESCRIPTION:Title: R
egular Sequences and the depth function for monomial ideals\nby Louiza
Fouli (New Mexico State University\, Las Cruces\, NM\, USA) as part of II
T Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nIn joint work
with T\\`ai Huy H\\`a and Susan Morey we introduced the notion of initial
ly regular sequences on $R/I$\, where $I$ is any homogeneous ideal in a po
lynomial ring $R$. We will discuss this notion\, and show how we can const
ruct certain types of (initially) regular sequences on $R/I$ that give eff
ective bounds on the depth of $R/I$. Moreover\, we will discuss when thes
e sequences remain (initially) regular sequences on $R/I^t$ and give lower
bounds on $\\depth R/I^t$ for $t\\ge 2$.\n
LOCATION:https://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;VALUE=DATE-TIME:20221028T130000Z
DTEND;VALUE=DATE-TIME:20221028T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/143
DESCRIPTION:Title: L
inkage in codimension three\nby Xianglong Ni (University of California
\, Berkeley\, United StatesUniversity of California\, Berkeley\, CA\, USA)
as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\n
All perfect ideals of codimension two are in the linkage class of a comple
te intersection (licci)\, but in codimension three and beyond this is no l
onger the case. I will share some ongoing work\, joint with Lorenzo Guerri
eri and Jerzy Weyman\, which illustrates how the theory of "higher structu
re maps" originating from Weyman's generic ring may be used to distinguish
licci ideals within the broader class of perfect ideals of codimension th
ree.\n
LOCATION:https://researchseminars.org/talk/VCAS/143/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Erman (University of Wisconsin)
DTSTART;VALUE=DATE-TIME:20221007T130000Z
DTEND;VALUE=DATE-TIME:20221007T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/144
DESCRIPTION:Title: M
atrix factorizations of generic polynomials\nby Daniel Erman (Universi
ty of Wisconsin) as part of IIT Bombay Virtual Commutative Algebra Seminar
\n\n\nAbstract\nI’ll discuss the Buchweitz-Greuel-Schreyer Conjecture on
the minimal size of a matrix factorization\, and my recent proof that the
conjecture holds for generic polynomials.\n
LOCATION:https://researchseminars.org/talk/VCAS/144/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vivek Sadhu (IISER Bhopal\, Madhya Pradesh\, India)
DTSTART;VALUE=DATE-TIME:20221104T120000Z
DTEND;VALUE=DATE-TIME:20221104T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/145
DESCRIPTION:Title: I
njectivity of Brauer groups for valuation rings\nby Vivek Sadhu (IISER
Bhopal\, Madhya Pradesh\, India) as part of IIT Bombay Virtual Commutativ
e Algebra Seminar\n\n\nAbstract\nIn the non noetherian situation\, valuati
on rings often behave like regular rings. We will discuss several such res
ults which are classically known to be true for regular rings\, but also t
rue 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 d
iscuss a more general result in the setting of etale cohomology.\n
LOCATION:https://researchseminars.org/talk/VCAS/145/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ramakrishna Nanduri (IIT Kharagpur\, West Bengal\, India)
DTSTART;VALUE=DATE-TIME:20221111T120000Z
DTEND;VALUE=DATE-TIME:20221111T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/146
DESCRIPTION:Title: O
n regularity of (symbolic) Rees algebra and (symbolic) powers of edge & ve
rtex cover ideals of graphs\nby Ramakrishna Nanduri (IIT Kharagpur\, W
est Bengal\, India) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nIn this talk\, we discuss about the Castelnuovo-Mumford
regularity (or regularity)\nof Rees algebras and symbolic Rees algebras o
f certain ideals associated to finite\nsimple graphs and we give various c
ombinatorial upper bounds. Also we study\nupper bounds for symbolic and or
dinary powers of edge and vertex cover ideals of\nsimple graphs.\n
LOCATION:https://researchseminars.org/talk/VCAS/146/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kohsuke Shibata (Okayama University\, Okayama\, Japan)
DTSTART;VALUE=DATE-TIME:20221125T120000Z
DTEND;VALUE=DATE-TIME:20221125T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/147
DESCRIPTION:Title: B
ounds of the multiplicity of abelian quotient complete intersection singul
arities\nby Kohsuke Shibata (Okayama University\, Okayama\, Japan) as
part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nWata
nabe classified all abelian quotient complete intersection singularities.
Watanabe defined a special datum in order to classify abelian quotient com
plete intersection singularities. In this talk\, I investigate the multipl
icities and the log canonical thresholds of abelian quotient complete inte
rsection singularities in terms of the special datum. Moreover I give boun
ds of the multiplicity of abelian quotient complete intersection singulari
ties.\n
LOCATION:https://researchseminars.org/talk/VCAS/147/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mina Bigdeli (IPM\, Tehran\, Iran)
DTSTART;VALUE=DATE-TIME:20221118T120000Z
DTEND;VALUE=DATE-TIME:20221118T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/148
DESCRIPTION:Title: Q
uadratic monomial ideals with almost linear free resolutions\nby Mina
Bigdeli (IPM\, Tehran\, Iran) as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\n\nAbstract\nThis talk will be about the minimal free reso
lution of quadratic monomial ideals. It is well known that a quadratic mon
omial 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 equivalent
to the linearity of the resolution of all powers of $I$. \n\nIn this talk
we will discuss the case that the resolution of a quadratic monomial ideal
$I$ is linear up to the homological degree $t$ with $t\\geq\\projdim(I)
-2$\, where $\\projdim(I)$ denotes the projective dimension of $I$. As a
n outcome\, we give a combinatorial classification of such ideals and al
so check whether their high powers have a linear resolution.\n\n\nChairper
son: Siamak Yassemi\, University of Tehran\, Iran\n
LOCATION:https://researchseminars.org/talk/VCAS/148/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Utsav Chowdhury (Indian Statical Institute\, Kolkata\, India)
DTSTART;VALUE=DATE-TIME:20221216T120000Z
DTEND;VALUE=DATE-TIME:20221216T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/150
DESCRIPTION:Title: C
haracterisation of the affine plane using $\\mathbb{A}^1$-homotopy theory<
/a>\nby Utsav Chowdhury (Indian Statical Institute\, Kolkata\, India) as p
art of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nChara
cterisation of the affine $n$-space is one of the major problem in affine
algebraic geometry. Miyanishi showed an affine complex surface $X$ is isom
orphic to $\\mathbb{C}^2$ if $\\mathscr{O}(X)$ is a U.F.D.\, $\\mathscr{O}
(X)^*= \\mathbb{C}^∗$ and $X$ has a non-trivial $\\mathbb{G}_a$-action [
3\, Theorem 1]. Since the orbits of a $\\mathbb{G}_a$-action are affine li
nes\, existence of a non-trivial $\\mathbb{G}_a$-action says that there is
a non-constant $\\mathbb{A}^1$ in X. Ramanujam showed that a smooth compl
ex surface is isomorphic to $\\mathbb{C}^2$ if it is topologically contrac
tible and it is simply connected at infinity [5]. Topological contractibil
ity\, in particular pathconnectedness says that there are non-constant int
ervals in X. On the other hand\, $\\mathbb{A}^1$-homotopy theory has been
developed by F.Morel and V.Voevodsky [4] as a connection between algebra a
nd topology. An algebrogeometric analogue of topological connectedness is
$\\mathbb{A}^1$-connectedness. In this talk\, using ghost homotopy techniq
ues [2\, Section 3] we will prove that if a surface $X$ is $\\mathbb{A}^1$
-connected\, then there is an open dense subset such that through every po
int there is a non-constant $\\mathbb{A}^1$ in X. As a consequence using t
he algebraic characterisation\, we will prove that $\\mathbb{C}^2$ is the
only $\\mathbb{A}^1$-contractible smooth complex surface. This answers the
conjecture appeared in [1\, Conjecture 5.2.3]. We will also see some othe
r useful consequences of this result. This is a joint work with Biman Roy.
\n\nReferences\n[1] A. Asok\, P. A. Østvær\; A 1 -homotopy theory and c
ontractible varieties: a survey\, Homotopy Theory and Arithmetic Geometry
– Motivic and Diophantine Aspects. Lecture Notes in Mathematics\, vol 22
92. 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 schemes. Ad
v Math\, Volume 282\, 2016. \n[3] M. Miyanishi\; An algebraic characteriza
tion of the affine plane. J. Math. Kyoto Univ. 15-1 (1975) 19-184.\n
LOCATION:https://researchseminars.org/talk/VCAS/150/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shigeru Kuroda (Tokyo Metropolitan University\, Hachioji\, Japan)
DTSTART;VALUE=DATE-TIME:20221223T120000Z
DTEND;VALUE=DATE-TIME:20221223T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/151
DESCRIPTION:Title: Z
/pZ-actions on the affine space: classification\, invariant ring\, and pli
nth ideal\nby Shigeru Kuroda (Tokyo Metropolitan University\, Hachioji
\, Japan) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nA
bstract\nLet k be a field of characteristic p>0. In this talk\, we conside
r the Z/pZ-actions on the affine n-space over k\, or equivalently the orde
r p automorphisms of the polynomial ring k[X] in n variables over k. For e
xample\, every automorphism induced from a G_a-action is of order p. Hence
\, the famous automorphism of Nagata is of order p. Such an automorphism i
s important to study the automorphism group of the k-algebra k[X].\nWe dis
cuss two topics: (1) classification\, and (2) the relation between polynom
iality of the invariant ring and principality of the plinth ideal. We also
present some conjectures and open problems.\n\ndoi: 10.1007/s00031-022-09
764-2\n
LOCATION:https://researchseminars.org/talk/VCAS/151/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vu Quang Thanh
DTSTART;VALUE=DATE-TIME:20220106T120000Z
DTEND;VALUE=DATE-TIME:20220106T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/152
DESCRIPTION:by Vu Quang Thanh as part of IIT Bombay Virtual Commutative Al
gebra Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/152/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Seccia (University of Genoa)
DTSTART;VALUE=DATE-TIME:20220113T120000Z
DTEND;VALUE=DATE-TIME:20220113T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/153/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A.V. Jayanthan (IIT Madras)
DTSTART;VALUE=DATE-TIME:20220120T120000Z
DTEND;VALUE=DATE-TIME:20220120T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/154/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Caminata (University of Genoa)
DTSTART;VALUE=DATE-TIME:20220127T120000Z
DTEND;VALUE=DATE-TIME:20220127T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/155/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vu Quang Thanh (Hanoi University of Science and Technology\, Hanoi
\, Vietnam)
DTSTART;VALUE=DATE-TIME:20230106T120000Z
DTEND;VALUE=DATE-TIME:20230106T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/156
DESCRIPTION:Title: R
egularity of powers and symbolic powers of squarefree monomial ideals\
nby Vu Quang Thanh (Hanoi University of Science and Technology\, Hanoi\, V
ietnam) as part of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbs
tract\nI will discuss the problem of comparing/computing the regularity of
symbolic powers and regular powers of certain classes of squarefree monom
ial ideals focusing on edge ideals of graphs.\n
LOCATION:https://researchseminars.org/talk/VCAS/156/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Seccia (University of Genoa\, Genoa\, Italy)
DTSTART;VALUE=DATE-TIME:20230113T120000Z
DTEND;VALUE=DATE-TIME:20230113T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/157
DESCRIPTION:Title: W
eakly- closed graphs and F-purity of binomial edge ideals\nby Lisa Sec
cia (University of Genoa\, Genoa\, Italy) as part of IIT Bombay Virtual Co
mmutative Algebra Seminar\n\n\nAbstract\nHerzog et al. characterized close
d graphs as the graphs whose binomial edge ideals have a quadratic Groebne
r basis. In this talk\, we focus on a generalization of closed graphs\, na
mely 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 Knutson id
eals (associated with a certain polynomial f). In doing so\, we re-prove M
atsuda's theorem about the F-purity of binomial edge ideals of weakly-clos
ed graphs in prime characteristic and we extend it to generalized binomial
edge ideals. \nLastly\, we will discuss some open conjectures on the F-pu
rity of binomial edge ideals and on the relation between Knutson ideals an
d compatible ideals.\n
LOCATION:https://researchseminars.org/talk/VCAS/157/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A.V. Jayanthan (IIT Madras\, Chennai\, India)
DTSTART;VALUE=DATE-TIME:20230120T120000Z
DTEND;VALUE=DATE-TIME:20230120T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/158
DESCRIPTION:Title: O
n 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 asymptotic re
surgence of squarefree monomial ideals in $R$. We study the effect on the
resurgence when sum\, product and intersection of ideals are taken. We obt
ain sharp upper and lower bounds for the resurgence and asymptotic resurge
nce of cover ideals of finite simple graphs in terms of associated combina
torial invariants. We also explicitly compute the resurgence and asymptoti
c resurgence of cover ideals of several classes of graphs. We characterize
a graph being bipartite in terms of the resurgence and asymptotic resurge
nce of edge and cover ideals. We also compute explicitly the resurgence an
d asymptotic resurgence of edge ideals of some classes of graphs.\n
LOCATION:https://researchseminars.org/talk/VCAS/158/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessio Caminata (University of Genoa\, Genoa\, Italy)
DTSTART;VALUE=DATE-TIME:20230127T120000Z
DTEND;VALUE=DATE-TIME:20230127T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/159
DESCRIPTION:Title: D
eterminantal varieties from point configurations on hypersurfaces\nby
Alessio Caminata (University of Genoa\, Genoa\, Italy) as part of IIT Bomb
ay Virtual Commutative Algebra Seminar\n\n\nAbstract\nPoint configurations
appear naturally in different contexts\, ranging from the study of the ge
ometry of data sets to questions in commutative algebra and algebraic geom
etry concerning determinantal varieties and invariant theory. In this talk
\, we bring these perspectives together: we consider the scheme X_{r\,d\,n
} parametrizing n ordered points in r-dimensional projective space that li
e on a common hypersurface of degree d. We show that this scheme has a det
erminantal structure and\, if r>1\, we prove that it is irreducible\, Cohe
n-Macaulay\, and normal. Moreover\, we give an algebraic and geometric des
cription of the singular locus of X_{r\,d\,n} in terms of Castelnuovo-Mumf
ord regularity and d-normality. This yields a complete characterization of
the singular locus of X_{2\,d\,n} and X_{3\,2\,n}. This is joint work wit
h Han-Bom Moon and Luca Schaffler.\n
LOCATION:https://researchseminars.org/talk/VCAS/159/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Duarte (University of Genoa\, Genoa\, Italy)
DTSTART;VALUE=DATE-TIME:20230224T120000Z
DTEND;VALUE=DATE-TIME:20230224T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/160
DESCRIPTION:Title: P
erturbations of ideals in local rings\nby Luis Duarte (University of G
enoa\, Genoa\, Italy) as part of IIT Bombay Virtual Commutative Algebra Se
minar\n\n\nAbstract\nLet I be an ideal of a Noetherian local ring R. We st
udy 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 sa
me Hilbert function as R/I\, we show that the Betti numbers of R/J coincid
e with those of R/I. We also compare the local cohomology \nmodules of R/J
with those of R/I.\n
LOCATION:https://researchseminars.org/talk/VCAS/160/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Blickle (Johannes Gutenberg University Mainz\, Rhineland Pa
latinate\, Germany)
DTSTART;VALUE=DATE-TIME:20230217T120000Z
DTEND;VALUE=DATE-TIME:20230217T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/161/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Seceleanu (University of Nebraska-Lincoln\, Lincoln\, NE
\, USA)
DTSTART;VALUE=DATE-TIME:20230331T130000Z
DTEND;VALUE=DATE-TIME:20230331T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/162
DESCRIPTION:Title: P
rincipal symmetric ideals\nby Alexandra Seceleanu (University of Nebra
ska-Lincoln\, Lincoln\, NE\, USA) as part of IIT Bombay Virtual Commutativ
e Algebra Seminar\n\n\nAbstract\nConsider a homogeneous polynomial f in va
riables x_1\,...\,x_n. The set of polynomials obtained from f by permuting
the variables in all possible ways generates an ideal\, which we call a p
rincipal symmetric ideal. What can we say about the Betti numbers of a pri
ncipal 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://researchseminars.org/talk/VCAS/162/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Tucker (University of Illinois Chicago\, IL\, USA)
DTSTART;VALUE=DATE-TIME:20210203T130000Z
DTEND;VALUE=DATE-TIME:20210203T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/163/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vaibhav Pandey (Purdue University\, West Lafayette\, IN\, USA)
DTSTART;VALUE=DATE-TIME:20230303T130000Z
DTEND;VALUE=DATE-TIME:20230303T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/164
DESCRIPTION:Title: L
inkage and F-regularity of generic determinantal rings\nby Vaibhav Pan
dey (Purdue University\, West Lafayette\, IN\, USA) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\n\nAbstract\nWe prove that the gener
ic link of a generic determinantal ring of maximal minors is strongly F-re
gular\, hence it has rational singularities. In the process\, we strengthe
n a result of Chardin and Ulrich. They showed that the generic residual in
tersections of a complete intersection ring with rational singularities ag
ain have rational singularities. We show that they are\, in fact\, strongl
y F-regular. \n\nIn the mid 1990s\, Hochster and Huneke showed that generi
c determinantal rings are strongly F-regular\; however\, their proof is qu
ite involved. The techniques that we discuss will allow us to give a new a
nd simple proof of the strong F-regularity of generic determinantal rings
defined by maximal minors. Time permitting\, we will also share a new proo
f of the strong F-regularity of determinantal rings defined by minors of a
ny size. This is joint work with Yevgeniya Tarasova.\n
LOCATION:https://researchseminars.org/talk/VCAS/164/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Meng (Cheng Meng\, Purdue University\, West Lafayette\, IN\,
USA)
DTSTART;VALUE=DATE-TIME:20220310T130000Z
DTEND;VALUE=DATE-TIME:20220310T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/165/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA)
DTSTART;VALUE=DATE-TIME:20220324T130000Z
DTEND;VALUE=DATE-TIME:20220324T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/166/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Meng (Purdue University\, West Lafayette\, IN\, USA)
DTSTART;VALUE=DATE-TIME:20220310T130000Z
DTEND;VALUE=DATE-TIME:20220310T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/167/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA)
DTSTART;VALUE=DATE-TIME:20220324T130000Z
DTEND;VALUE=DATE-TIME:20220324T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
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://researchseminars.org/talk/VCAS/168/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cheng Meng (Purdue University\, West Lafayette\, IN\, USA)
DTSTART;VALUE=DATE-TIME:20230310T120000Z
DTEND;VALUE=DATE-TIME:20230310T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/169
DESCRIPTION:Title: M
ultiplicities in flat local extensions\nby Cheng Meng (Purdue Universi
ty\, West Lafayette\, IN\, USA) as part of IIT Bombay Virtual Commutative
Algebra Seminar\n\n\nAbstract\nWe introduce the notion of strongly Lech-in
dependent ideals as a generalization of Lech-independent ideals defined by
Lech and Hanes\, and use this notion to derive inequalities on multiplici
ties of ideals. In particular we prove that if (R\,m) and (S\,n) are Noeth
erian local rings of the same dimension\, S is a flat local extension of R
\,and up to completion S is standard graded over a field and I=mS is homog
eneous\, then the multiplicity of R is no greater than that of S.\n
LOCATION:https://researchseminars.org/talk/VCAS/169/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA)
DTSTART;VALUE=DATE-TIME:20230324T120000Z
DTEND;VALUE=DATE-TIME:20230324T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/170
DESCRIPTION:Title: U
niform bounds on symbolic powers in regular rings via closure theory\n
by Takumi Murayama (Purdue University\, West Lafayette\, IN\, USA) as part
of IIT Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nThe cont
ainment problem asks: For a fixed ideal I\, which symbolic powers of I are
contained in an ordinary power of I? We present a closure-theoretic proof
of the theorem which says that for ideals I in regular rings R\, there is
a uniform containment of symbolic powers of I in ordinary powers of I.\n
LOCATION:https://researchseminars.org/talk/VCAS/170/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mitsuyasu Hashimoto (Osaka Metropolitan University\, Sumiyoshi-ku\
, Osaka\, Japan)
DTSTART;VALUE=DATE-TIME:20221230T120000Z
DTEND;VALUE=DATE-TIME:20221230T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/171
DESCRIPTION:Title: A
symptotic behaviors of the Frobenius pushforwards of the ring of invariant
s\nby Mitsuyasu Hashimoto (Osaka Metropolitan University\, Sumiyoshi-k
u\, Osaka\, Japan) as part of IIT Bombay Virtual Commutative Algebra Semi
nar\n\n\nAbstract\nLet k be an algebraically closed field of characteris
tic 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 the symm
etric 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 categor
y {}^*Ref(A)\, the category of Q-graded A-finite A-reflexive A-modules.
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 principle\
, we get some results on the properties of A coming from the asymptotic
behaviors of {}^eA. In this talk\, we talk about the following:\n\nthe g
eneralized F-signature of A (with Y. Nakajima and with P. Symonds).\nExam
ples of G and V such that A is F-rational\, but not F-regular.\nExamp
les of G and V such that (the completion of) A is not of finite F-re
presentation type (work in progress with A. Singh).\nGeneralizing finite
groups to finite group schemes G\, we have that s(A)>0 if and only if G
is linearly reductive\, and if this is the case\, s(A)=1/|G|\, where |G
| is the dimension of the coordinate ring k[G] of G\, provided the acti
on of G on Spec S is ‘small’ (with F. Kobayashi).\n
LOCATION:https://researchseminars.org/talk/VCAS/171/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rajib Sarkar (TIFR\, Mumbai\, India)
DTSTART;VALUE=DATE-TIME:20221209T120000Z
DTEND;VALUE=DATE-TIME:20221209T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/172
DESCRIPTION:Title: L
evel and pseudo-Gorenstein binomial edge ideals\nby Rajib Sarkar (TIFR
\, Mumbai\, India) as part of IIT Bombay Virtual Commutative Algebra Semin
ar\n\n\nAbstract\nGorenstein binomial edge ideals have been completely cha
racterized and they are the paths only. There are two interesting generali
zations of Gorenstein rings: level rings and pseudo-Gorenstein rings. In t
he first half\, we will talk about the behavior of the levelness and pseud
o-Gorensteinness on the decomposable graphs and cone graphs. \nIn the next
half\, we will discuss the characterization of Cohen-Macaulay binomial ed
ge ideals of bipartite graphs and then their levelness and pseudo-Gorenste
inness.\nThis talk is based on the joint work with Giancarlo Rinaldo.\n
LOCATION:https://researchseminars.org/talk/VCAS/172/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christine Berkesh (University of Minnesota\, Minneapolis\, MN\, US
A)
DTSTART;VALUE=DATE-TIME:20230317T120000Z
DTEND;VALUE=DATE-TIME:20230317T130000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/173
DESCRIPTION:Title: D
ifferential operators\, retracts\, and toric face rings\nby Christine
Berkesh (University of Minnesota\, Minneapolis\, MN\, USA) as part of IIT
Bombay Virtual Commutative Algebra Seminar\n\n\nAbstract\nToric face rings
\, introduced by Stanley\, are simultaneous generalizations of Stanley–R
eisner rings and affine semigroup rings\, among others. We use the combina
torics of the fan underlying these rings to inductively compute their ring
s of differential operators. Along the way\, we discover a new differentia
l characterization of the Gorenstein property for affine semigroup rings.
Our approach applies to a more general class of rings\, which we call alge
bras realized by retracts. This is joint work with C-Y. Chan\, P. Klein\,
L. Matusevich\, J. Page\, and J. Vassilev.\n
LOCATION:https://researchseminars.org/talk/VCAS/173/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Saugata Basu (Purdue University\, West Lafayette\, IN\, USA)
DTSTART;VALUE=DATE-TIME:20230210T130000Z
DTEND;VALUE=DATE-TIME:20230210T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/174
DESCRIPTION:Title: H
omology of symmetric semi-algebraic sets\nby Saugata Basu (Purdue Univ
ersity\, West Lafayette\, IN\, USA) as part of IIT Bombay Virtual Commutat
ive Algebra Seminar\n\n\nAbstract\nStudying the homology groups of semi-al
gebraic subsets of $\\mathbb{R}^n$ and obtaining upper bounds\non the Bett
i numbers has been a classical topic in real algebraic geometry beginning
with the\nwork of Petrovskii and Oleinik\, Thom\, and Milnor. In this talk
I will consider semi-algebraic\nsubsets of $\\mathbb{R}^n$ which are defi
ned by symmetric polynomials and are thus stable under the\nstandard actio
n of the symmetric group $\\mathfrak{S}_n$ on $\\mathbb{R}^n$.\nThe homolo
gy groups (with rational coefficients) of such sets\nthus acquire extra st
ructure as $\\mathfrak{S}_n$-modules leading to\npossible refinements on t
he classical bounds. I will also mention some \nconnections with a homolog
ical stability conjecture.\n \nJoint work (separately) with Daniel Perruc
ci and Cordian Riener.\n
LOCATION:https://researchseminars.org/talk/VCAS/174/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Tucker (University of Illinois Chicago\, IL\, USA)
DTSTART;VALUE=DATE-TIME:20230203T130000Z
DTEND;VALUE=DATE-TIME:20230203T140000Z
DTSTAMP;VALUE=DATE-TIME:20240226T013737Z
UID:VCAS/175
DESCRIPTION:Title: T
he Theory of F-rational Signature\nby Kevin Tucker (University of Illi
nois Chicago\, IL\, USA) as part of IIT Bombay Virtual Commutative Algebra
Seminar\n\n\nAbstract\nThe celebrated results of Smith\, Hara\, and Mehta
-Srinivas connect rational singularities in characteristic zero after redu
ction to characteristic p > 0 with F-rational singularities. In recent yea
rs\, a number of invariants defined via Frobenius in positive characterist
ics have been introduced as quantitative measures of F-rationality. These
include the F-rational signature (Hochster-Yao)\, relative F-rational sign
ature (Smirnov-Tucker)\, and dual F-signature (Sannai). In this talk\, I w
ill discuss new results in joint work with Smirnov relating each of these
invariants. In particular\, we show that the relative F-rational signature
and dual F-signature coincide\, while also verifying that the dual F-sign
ature limit converges.\n
LOCATION:https://researchseminars.org/talk/VCAS/175/
END:VEVENT
END:VCALENDAR