BEGIN:VCALENDAR
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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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
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:20210419T094801Z
UID:VCAS/81
DESCRIPTION:by Indranath Sengupta (IIT Gandhinagar) as part of IIT Bombay
Virtual Commutative Algebra Seminar\n\nAbstract: TBA\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:20210419T094801Z
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:Amalendu Krishna (Indian Institute of Science)
DTSTART;VALUE=DATE-TIME:20210521T120000Z
DTEND;VALUE=DATE-TIME:20210521T130000Z
DTSTAMP;VALUE=DATE-TIME:20210419T094801Z
UID:VCAS/83
DESCRIPTION:by Amalendu Krishna (Indian Institute of Science) as part of I
IT Bombay Virtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ravi Rao (Tata Institute of Fundamental Research)
DTSTART;VALUE=DATE-TIME:20210528T120000Z
DTEND;VALUE=DATE-TIME:20210528T130000Z
DTSTAMP;VALUE=DATE-TIME:20210419T094801Z
UID:VCAS/84
DESCRIPTION:by Ravi Rao (Tata Institute of Fundamental Research) as part o
f IIT Bombay Virtual Commutative Algebra Seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/VCAS/84/
END:VEVENT
END:VCALENDAR