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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.