November 30, 2020 - December 4, 2020
Public Access Category: All |
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- Time:
- 4:00pm-5:00pm
- Description:
- Speaker: Pranav Pandit, ICTS
Time: Monday 30th November 4 to 5pm (joining time 3.45 pm IST)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Noncommutative geometry of Landau-Ginzburg models
Abstract: I will outline a categorical framework for studying the
symplectic geometry of Landau-Ginzburg models and the algebraic geometry
of Tyurin degenerations. The main ingredients in this story are sheaves of
categories and spherical functors. I will explain how this framework can
be used to construct Calabi-Yau structures on categories of branes, and
(shifted) symplectic structures on certain moduli spaces of branes. This
talk is based on joint work with L. Katzarkov and T. Spaide.
- 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.