Seminar Categories

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Wed Sep 05

Commutative Algebra Seminar

10:10am - Vincent Hall 313
Using mixed Gauss--Manin systems to project, restrict, and dualize $A$-hypergeometric systems
Avi Steiner, Purdue

Let $A$ be an integer matrix, and assume that its semigroup ring $\mathbb{C}[\mathbb{N} A]$ is normal. I will discuss how to use mixed and dual mixed Gauss--Manin systems, a notion I introduced recently, to compute the holonomic dual of an $A$-hypergeometric system; and to compute, for $F$ a face of the cone of $A$, the projection and restriction of an $A$-hypergeometric system to the coordinate subspace corresponding to $F$.

Thu Sep 20

Commutative Algebra Seminar

1:25pm - Ford Hall 170
Random Monomial Ideals
Jay Yang, University of Minnesota

I will discuss the contents of my joint paper with Daniel Erman, Random Flag Complexes and Asymptotic Syzygies. In this paper we use the Stanley-Reisner ideals of random flag complexes to construct new examples of Ein and Lazarsfeld's non-vanishing for asymptotic syzygies, and of Ein, Erman, and Lazarsfeld's conjecture on the asymptotic normal distribution of Betti numbers. I will also discuss some work in progress related to the Random Monomial Ideals paper by De Loera, Petrovic, Silverstein, Stasi, and Wilburne.

Thu Oct 18

Commutative Algebra Seminar

1:25pm - Ford Hall 170
Computations in Local Rings using Macaulay2
Mahrud Sayrafi, University of Minnesota

Local rings are ubiquitous in commutative algebra and algebraic geometry. In this talk I will describe two avenues for computing in local rings with respect to prime ideals, first using the associated graded algebra and then using only Nakayama's lemma. Time permitting, I will demonstrate various examples and applications, such as computing the Hilbert-Samuel multiplicity, using Macaulay2.

Thu Nov 29

Commutative Algebra Seminar

1:25pm - Ford Hall 170
On invariant theory for "coincidental" reflection groups
Victor Reiner, University of Minnesota

(joint work with A. Shepler and E. Sommers)
Complex reflection groups W are the finite subgroups of GL_n(C) with the following property: when they act on polynomials in n variables, their invariant ring is again a polynomial algebra. It is also known by
a result of Eagon and Hochster that, for any of their W-representations U, the U-isotypic polynomials form a _free_ module over the W-invariant subalgebra. In a few cases, we know an explicit basis for these U-isotypic polynomials. This talk will discuss a class of complex reflection groups W, sometimes called the "coincidental types", where conjecturally we know explicit bases for the U-isotypic component when U is any tensor product of the exterior powers of the reflection representation and its dual. This conjecture would explain pleasant product formulas for combinatorial objects, such as face numbers for finite type cluster complexes.

Thu Feb 28

Commutative Algebra Seminar

1:25pm - Vincent Hall 301
Virtual Resolutions of Monomial Ideals
Jay Yang, University of Minnesota

Virtual resolutions as defined by Berkesch, Erman, and Smith,
provide a more geometrically meaningful generalization of free
resolutions in the case of subvarieties of a toric variety. In this
setting I prove an analog of Hilbert's syzygy theorem for virtual
resolutions of monomial ideals in toric varieties subject to some mild
conditions.

Thu Mar 14

Commutative Algebra Seminar

1:25pm - Vincent Hall 301
Towards Free Resolutions Over Scrolls
Aleksandra Sobieska-Snyder, Texas A&M

Free resolutions over the polynomial ring have a storied and active
record of study. However, resolutions over other rings are much more
mysterious; even simple examples can be infinite! In these cases, we
look to any combinatorics of the ring to glean information. This talk
will present a minimal free resolution of the ground field over the
semigroup ring arising from rational normal 2-scrolls, and (if time
permits) a computation of the Betti numbers of the ground field for all
rational normal k-scrolls.

Thu Mar 28

Commutative Algebra Seminar

1:25pm - Vincent Hall 301
Bernstein-Sato polynomials in positive characteristic and Hodge theory
Thomas Bitoun, University of Toronto

Bernstein-Sato polynomials are fundamental in D-module theory. For
example, they are the main finiteness ingredient in the construction of
nearby cycles. We will present a positive characteristic analogue of the
Bernstein-Sato polynomials. After diving in the world of characteristic
p D-modules, we shall consider how our construction varies with the
prime p. This turns out to be related to questions of Hodge theory and
Poisson homology.

Thu Apr 04

Commutative Algebra Seminar

1:25pm - Vincent Hall 301
An Analogue of the Hartshorne-Polini Theorem in Positive Characteristic
Zhang Wenliang, University of Illinois at Chicago

Recently, Hartshorne and Polini proved a theorem to characterize the
dimension of certain de Rham cohomology groups of a holonomic D-module
over complex numbers as the number of specific D-linear maps associated
with the holonomic D-module. In this talk, I will explain an analogue of
this result in positive characteristic for F-finite F-modules. This is a
joint work with Nicholas Switala.

Thu Apr 11

Commutative Algebra Seminar

1:25pm - Vincent Hall 301
Semi-Ample Asymptotic Syzygies
Juliette Bruce

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.