December 2020
Public Access Category: All |
5:30pm» Marilina Rossi, University of Genoa- Time:
- 5:30pm
- Description:
- Date and Time: 1 December 2020, 5:30pm IST/ 12:00 GMT/ 7:00am EDT
(joining time: 5:15 pm IST - 5:30 pm IST)
Speaker: Marilina Rossi, University of Genoa
Google meet link: meet.google.com/aum-zrru-xtg
Title: A constructive approach to one-dimensional Gorenstein k-algebras
Abstract: Gorenstein rings are a generalization of complete intersections,
and indeed the two notions coincide in codimension two. Codimension three
Gorenstein rings are completely described by Buchsbaum and Eisenbud's
structure theorem, but despite many attempts, the construction of
Gorenstein rings is an open problem in higher codimension. Gorenstein
rings are of great interest in many areas of mathematics and they have
appeared as an important component in a significant number of problems.
Our task is to give a procedure for constructing all 1-dimensional
Gorenstein k-algebras. Applications to the Gorenstein linkage of
zero-dimensional schemes and to Gorenstein affine semigroup rings are
discussed. The results are based on recent results obtained jointly with
J. Elias.
- Time:
- 7:00pm
- Description:
- Speaker: Ezra Miller.
Time: 7pm IST (gate opens 6:45pm IST).
Google meet link: meet.google.com/rxx-wfnm-kkq.
By phone: (US) +1 402-277-8494 (PIN: 677936904).
Title: Minimal resolutions of monomial ideals.
Abstract: It has been an open problem since the 1960s to construct
closed-form, canonical, combinatorial minimal free resolutions
of arbitrary monomial ideals in polynomial rings. This
talk explains how to solve the problem, in characteristic 0
and almost all positive characteristics, using sums over
lattice paths of combinatorial data from simplicial
complexes, one simplicial complex for each lattice point.
Any minimal free resolution of any monomial ideal must --
either implicitly or explicitly -- produce homomorphisms
between various homology groups of these simplicial
complexes. Therefore an important aspect of the solution
is an explicit way to write down canonical homomorphisms
between these homology groups without choosing bases.
Joint work with Jack Eagon and Erika Ordog.
- Time:
- 6:30pm
- Description:
- Date and Time: 4 December 2020, 6:30pm IST/ 1:00pm GMT/ 8:00am EDT
(joining time: 6:15 pm IST - 6:30 pm IST)
Speaker: Ian Aberbach, University of Missouri
Google meet link: meet.google.com/aum-zrru-xtg
Title: On the equivalence of weak and strong F-regularity
Abstract: Let $(R, m, k)$ be a (Noetherian) local ring of positive prime
characteristic $p.$ Assume also, for 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 closure of an ideal sits between the ideal itself
and its integral closure. When the tight closure of an ideal $I$ is $I$
itself we call $I$ tightly closed. For particularly nice rings, e.g.,
regular rings, every ideal is tightly closed. We call such rings weakly
$F$-regular. Unfortunately, tight closure is known not to commute with
localization, and hence this property of being weakly $F$-regular is not
known to localize. To deal with this problem, Hochster and Huneke defined
the notion of strongly $F$-regular (assuming $R$ is $F$-finite), which
does localize, and implies that $R$ is weakly $F$-regular. It is still an
open question whether or not the two notions are equivalent, although it
has been shown in some classes of rings. Not much progress has been made
in the last 15-20 years. I will discuss the problem itself, the cases that
are known, and also outline recent progress made by myself and Thomas
Polstra.
- Time:
- 5:00pm
- Description:
- Title: Mathematics of deep learning.
Speaker: Anirbit Mukherjee
Time: 5:00 pm
Joining link : https://meet.google.com/kyz-exxu-hvw
ABSTRACT :
One of the paramount mathematical mysteries of our times is to be able to
explain the phenomenon of deep-learning. Neural nets can be made to paint
while imitating classical art styles or play chess better than any machine
or human ever and they seem to be the closest we have ever come to
achieving "artificial intelligence". But trying to reason about these
successes quickly lands us into a plethora of extremely challenging
mathematical questions - typically about discrete stochastic processes.
Some of these questions remain unsolved for even the smallest neural nets!
In this talk we will give a brief introduction to neural nets and describe
two of the most recent themes of our work in this direction.
Firstly we will explain how under certain structural and mild
distributional conditions our iterative algorithms like ``Neuro-Tron"
which do not use a gradient oracle can often be proven to train nets using
as much time/sample complexity as expected from gradient based methods but
in regimes where usual algorithms like (S)GD remain unproven. Our theorems
include the particularly challenging regime of non-realizable data. Next
we will briefly look at our first-of-its-kind results about sufficient
conditions for fast convergence of standard deep-learning algorithms like
RMSProp, which use the history of gradients to decide the next step. In
the second half of the talk, we will focus on the recent rise of the
PAC-Bayesian technology in being able to explain the low risk of certain
over-parameterized nets on standardized tests. We will present our recent
results in this domain which empirically supersede some of the existing
theoretical benchmarks in this field and this we achieve via our new
proofs about the key property of noise resilience of nets.
This is joint work with Amitabh Basu (JHU), Ramchandran Muthukumar (JHU),
Jiayao Zhang (UPenn), Dan Roy (UToronto, Vector Institute), Pushpendre
Rastogi (JHU ->Amazon), Soham De (DeepMind, Google), Enayat Ullah (JHU),
Jun Yang (UToronto, Vector Institute) and Anup Rao (Adobe).
About the speaker :
Anirbit Mukherjee finished his Ph.D. in applied mathematics at the Johns
Hopkins University advised by Prof. Amitabh Basu. He is now a post-doc at
Wharton (UPenn) with Prof. Weijie Su. He specializes in deep-learning
theory.
- Time:
- 7:00pm
- Description:
- Title: Standard monomials, matroids, and lattice paths.
Time: 7pm IST, gate opens 6:45pm IST.
Google meet link: meet.google.com/cbv-twjd-vno.
Phone: (US) +1 401-764-4238 PIN: 316 692 499#
Speaker Raman Sanyal, Goethe-Universität Frankfurt.
Abstract: Every finite collection of points is the set of solutions to
some system of polynomial equations. This is a (computationally)
reasonable representation, in particular when writing down defining
equations is easier than the actual points. Motivated by Grobner basis
theory for finite point
configurations, I will discuss standard complexes of 0/1-point
configurations. For a matroid basis configuration, the corresponding
standard complex is a subcomplexes of the independence complex, which is
invariant under matroid duality. For the lexicographic term order, the
standard complexes satisfy a deletion-contraction-type recurrence. For
lattice path matroids these complexes can be explicitly described in terms
of lattice path combinatorics. The talk is based on work with Alexander
Engstrom and Christian Stump.
- Time:
- 5:15pm - 6:30pm
- Description:
- Speaker:* Rajendra V. Gurjar, IIT Bombay *
Date/Time: *15 December 2020, **5:30pm IST/ 12:00 GMT/ 7:00am EST* (joining
time: 5:15 pm IST - 6:30 pm IST)
Google meet link: meet.google.com/pwt-vxdm-gbc
Title:* Zariski-Lipman Conjecture for Module of Derivations*
Abstract: Zariski conjectured that if the module of derivations of a local
ring $R$ at a point on an algebraic variety defined over a field of
characteristic $0$ is a free $R$-module then $R$ is regular. In these two
talks we will survey most of the interesting results proved affirming the
conjecture.
Results of Lipman, Scheja-Storch, Becker, Hochster, Steenbrink-van Straten,
Flenner, Kallstrom, Biswas-Gurjar-Kolte, and some general results which can
be deduced by combining some of these results will be discussed. An
interesting proposed counterexample due to Hochster will be introduced.
Some unsolved cases in the paper of Biswas-Gurjar-Kolte will be mentioned.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar/
- Time:
- 10:45am
- Description:
- Speaker Name: Prof.Satoshi Murai, Waseda University,
Title: Betti tables of S_n-invariant monomial ideals.
Time: 10:45 am IST (gate opens 10:35 am IST), Wednesday, 16 December.
Google meet link: meet.google.com/qtw-ibbr-qkz.
Phone: (US) +1 505-738-3123 (PIN: 572892305).
Abstract: Let $R_n=K[x_1,\dots,x_n]$ be a polynomial ring with $n$
variables. A monomial ideal in $R_n$ is said to be $S_n$-invariant if it
is fixed by the natural action of the $n$-th symmetric group $S_n$ to
$R_n$. In this talk, I will discuss Betti numbers of $S_n$-invariant
monomial ideals of $R_n$. In particular, I will mainly talk about recent
results relating to the following problem: Fix a sequence of monomials
$u_1,\dots,u_r$ and let $I_n$ be the $S_n$-invariant monomial ideal of
$R_n$ generated by the set of $\{\sigma(u_k):k=1,2,\dots,r, \sigma \in
S_n\}$. In this setting, what can be said about Betti numbers of $I_n$
when $n$ increases?
- Time:
- 5:15pm - 6:30pm
- Description:
- Speaker: *Rajendra V. Gurjar, IIT Bombay *
Date/Time: *18 December 2020, **5:30pm IST/ 12:00 GMT/ 7:00am EST* (joining
time: 5:15 pm IST - 6:30 pm IST)
Google meet link: meet.google.com/pwt-vxdm-gbc
Title:* Zariski-Lipman Conjecture for Module of Derivations*
Abstract: Zariski conjectured that if the module of derivations of a local
ring $R$ at a point on an algebraic variety defined over a field of
characteristic $0$ is a free $R$-module then $R$ is regular. In these two
talks we will survey most of the interesting results proved affirming the
conjecture.
Results of Lipman, Scheja-Storch, Becker, Hochster, Steenbrink-van Straten,
Flenner, Kallstrom, Biswas-Gurjar-Kolte, and some general results which can
be deduced by combining some of these results will be discussed. An
interesting proposed counterexample due to Hochster will be introduced.
Some unsolved cases in the paper of Biswas-Gurjar-Kolte will be mentioned.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar/
- Time:
- 5:30pm
- Description:
- Speaker: Ken-ichi Yoshida, Nihon University, Japan.
Date/Time: *22 December 2020, 5:30pm IST/ 12:00 GMT/ 7:00am EST* (joining
time: 5:15 pm IST - 5:30 pm IST).
Google meet link: https://meet.google.com/bje-tjog-rkh
Title: Lower bound on Hilbert-Kunz multiplicities and some related results
Abstract: In my talk, we introduce some results of lower bounds on
Hilbert-Kunz multiplicities for non-regular local rings. In the later
half,
we will discuss the upper bound on F-signature.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar/
- Time:
- 6:30pm
- Description:
- Talk 1*Speaker: *Anthony Iarrobino, Northeastern University, Boston, MA *
Date/Time: *29 December 2020, 6:30pm IST/ 1:00pm GMT/ 8:00am EST* (joining
time: 6:15 pm IST - 6:30 pm IST).
Google meet link: https://meet.google.com/tih-ghos-zzg
Title: *Jordan type and Lefschetz properties for Artinian algebras*
Abstract: The 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 the 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 Sperner number of A – the highest value
of the Hilbert function H(A). We say that (A,x) is strong Lefschetz if P_x
is the conjugate of the Hilbert function.
Weak and strong Lefschetz properties of A for a generic choice of x have
been studied, due to the connection with topology and geometry, where A is
the cohomology ring of a topological space or a variety X. We discuss some
of the properties of Jordan type, and its use as an invariant of A, its
behavior for tensor products and free extensions (defined by T. Harima and
J. Watanabe).
If there is time, we will discuss an application to the study of local
Artinian Gorenstein algebras of fixed Hilbert function H; in recent work
with Pedro Macias Marques we show that in codimension three the properties
of Jordan type and of symmetric decompositions show that certain families
Gor(H) in codimension three or greater have several irreducible components.
The first part of the talk is based on work with Chris McDaniel and Pedro
Marques (arXiv:math.AC/1802.07383, to appear JCA).