The Schedule

David Eisenbud

Title: Free Resolutions, Finite and Infinite

There have recently been some remarkable new results on both finite and infinite resolutions, leading to conjectures that go far beyond what we can prove. I'll talk about a sample of these, and pose some open problems stemming from observations of examples from Macaulay2 in work of mine with Hai Long Dao and with Bernd Ulrich.

Laura Cranton Heller

Title: Regularity of powers of multigraded ideals

Abstract: In the case of a singly graded ring S, the Castelnuovo-Mumford regularity of the powers I^n of an ideal I in S is eventually a linear function in n.  When S is multigraded, the regularity of each power is a region rather than an integer.  I will present bounds on the regularity of the powers I^n of a multigraded ideal that translate linearly in n.

Laura Matusevich

Title: Differential operators and toric face rings

Abstract: Rings of differential operators are easy to define, but hard to compute. Even the ring of differential operators for the polynomial ring has some surprises, if one works in positive characteristic. 

In the combinatorial setting, rings of differential operators are known for the usual suspects: Stanley--Reisner rings, and affine semigroup rings (the latter in characteristic 0). The goal of this talk is to describe the rings of differential operators of toric face rings.

Toric face rings are a common generalization of Stanley--Reisner rings, and affine semigroup rings. Intuitively, such rings are made out of a bunch of affine semigroup rings stuck together. In this way, the affine semigroup rings involved turn out to be algebra retracts of the toric face ring. Our methods for computing rings of differential operators actually apply to any ring that has "enough" retracts, mimicking the situation for toric face rings. All of these words will be defined (and hopefully make sense) in the talk.

This is joint work with Berkesch, Chan, Klein, Page and Vassilev; it was started at a WICA (Women in Commutative Algebra) workshop, and continued in an ongoing AIM Square.

Srishti Singh

Title: Wilf’s Conjecture and More (and Less)

Abstract: Wilf’s conjecture establishes an inequality that relates three fundamental invariants of a numerical semigroup: the minimal number of generators (or the embedding dimension), the Frobenius number, and the number of gaps. Based on a preprint by Srinivasan and me, the talk will discuss the past, present, and future of this conjecture. We prove that this Wilf inequality is preserved under gluing of numerical semigroups.  If the numerical semigroups minimally generated by A = { a_1,..., a_p} and B = { b_1, ... , b_q} satisfy the Wilf inequality, then so does their gluing which is minimally generated by C =k_1A ⊔ k_2B. We discuss the extended Wilf's Conjecture in higher dimensions and prove an analogous result.

A.V. Jayanthan

Title: Binomial expansion for saturated and symbolic powers of sums of ideals.

Abstract: There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordinary powers. We prove a binomial expansion formula for saturated powers of sums of ideals. This gives a uniform treatment to an array of existing and new results on both notions of symbolic powers of such sums: binomial expansion formulas, computations of depth and regularity, and criteria for the equality of ordinary and symbolic powers. This is a joint work with Huy Tai Ha, Arvind Kumar and Hop D. Nguyen.

Reception @ Dale and Hema's house (address will be provided on Saturday)

Day 2

September 24

Hema Srinivasan

Title: Affine semigroup rings

Abstract: Affine semigroup rings are rings of the form k[x1,...,xp]/IA where IA is a toric ideal associated with an n × p matrix of non-negative integers. The columns of A generate ⟨A⟩, the submonoid of N^n. When n = 1, these are called the numerical semigroups and numerical semigroups rings. We will discuss an operation called "gluing" of semigroups which generalizes the notion in the numerical case. A semigroup ⟨C⟩ in N^n is a gluing of ⟨A⟩ and ⟨B⟩ if the set of minimal generators of C splits into two parts, C = k1A ⊔ k2B with k1, k2 ≥ 1, and the defining ideals of the corresponding semigroup rings are such that IC is generated by IA + IB and one special binomial straddling both A and B. The glued semigroup C and the semigroup ring k[C] inherits much of the properties of the two parts that are glued and the numerical invariants of C can be derived from those of A and B. We will give necessary and sufficient conditions for gluing two semigroups ⟨A⟩ and ⟨B⟩. In general A and B cannot be glued if they both have maximal ranks. We will show how one can remedy this situation in the homogeneous case by suitably embedding them in higher dimensions.

Jakub Witaszek

Title: Splinters in mixed characteristic via the Riemann-Hilbert correspondence

Abstract: In my talk, I will discuss new developments on the openness of the splinter locus and the localisation of the mixed characteristic test ideals. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.

Monalisa Dutta

Cancelled because of Covid-19.

Andrew Soto-Levins

Title: An Auslander Bound for Complexes 

Abstract: The Auslander bound of a module can be thought of as a generalization of projective dimension. For a finitely generated module M with finite projective dimension over a Noetherian local ring R, Ext_R^n(M,N) = 0 for n >pdim_R(M) and Ext_R^{pdim_R(M)}(M,N)  =\= 0 for all finitely generated modules N. We say that the Auslander bound of M is finite if for all finitely generated modules N, there exists an integer b that only depends on M so that Ext_R^n(M,N) = 0 for n > b whenever Ext_R^n(M,N) = 0 for n >> 0. The Auslander bound is the least such b. In this talk we give an introduction to the Auslander bound of a module and share some related results. We will then see that we can define an Auslander bound for complexes and extend the module results to complexes.

Satyagopal Mandal

Title: K-theory of the category of (perfect) CM-modules

Abstract: I plan to report on what I found in K-theory, over last ten years. (Gossips are not ruled out.) The point of the talk would be, how much commutative algebra you can cultivate and harvest in K-theory?

For a commutative noetherian ring A (X = Spec(A)), a finitely generated A-module M is called a perfect A-module if grade(M) = dim_V (X) M, the projective dimension. I like to call them CM-modules. Such CM-modules work like modules of finite length and finite projective dimension.

Variety of categories of such CM modules seemed to have hit a chord, in my work in K-theory. My methods work for quasi projective schemes X. All that lead to long exact sequences of K-groups and Gersten complexes.