### Abstracts

Ian Aberbach

**Title:** Uniform Artin-Rees bounds for Syzygies

**Abstract:** Let (R,m) be a
Noetherian local ring of dimension d, M a finitely generated R-module, and F_{•} a finite
free resolution of M. Let Z_{i} be the i^{th} syzygy module in the resolution.
Eisenbud and Huneke raised the following uniform Artin-Rees style question:
Given an ideal I, is there a uniform k such that for all i > 0 and for
all n > 0 the containment Z_{i} ∩ I^{n} F_{i-1} ⊆ I^{n-k}Z_{i}?
They were able to show that this is true if M has both constant rank and finite projective dimension on the punctured spectrum.
We show, in fact, that not only is the general answer yes, but that the uniformity
is true to a much greater extent. While the *beginning* of the resolution may be quite badly behaved,
once we get to the d^{th} syzygy, we get uniform behavior for all modules and all ideals in R.
Specifically, we show that there exists k ≥ 0 such that for all finitely generated modules M,
for all ideals I, and for i > d, Z_{i} ∩ I^{n} F_{i-1} ⊆ I^{n-k}Z_{i}.
The proof relies on Huneke's uniform Artin-Rees theorem, and can also be viewed as an extension of that theorem.
This work is joint with Aline Hosry and Janet Striuli.

Tom Marley

**Title:** Rigidity of Ext and Tor with coefficients in residue fields

**Abstract:** This is joint work with Lars Christensen and Srikanth Iyengar.
Let R be a commutative Noetherian ring and p a prime ideal. Let k(p) denote the residue field of R_{p}.
We say that Ext^{*}_{R}(k(p), -) is * rigid at n* if whenever Ext^{n}_{R}(k(p),M)=0
for any R-module M (not necessarily finitely generated) then Ext^{i}_{R}(k(p), M)=0
for all i ≥ n. We define Tor^{R}_{*}(k(p),-) to be rigid at n in an analogous manner.
We show that Ext^{*}_{R}(k(p), -) is rigid at n = dim R + dim R_{p} + 1 and
Tor^{R}_{*}(k(p),-) is rigid at n = dim R_p +1.

Jonathan Montano

**Title:** Local cohomology of powers of monomial ideals

**Abstract:** Let (R,m) be a commutative local ring of dimension d
and I an R-ideal. Let L(M) be the length of the module M. The asymptotic behavior of the sequence L(H^{0}_{m}(R/I^{n})) for n > 0 has
been studied by several authors. For example, Dale Cutkosky proved that if R is analytically unramified and d > 0, the limit of the
sequence L(H^{0}_{m}(R/I^{n}))/n^{d} exists for any I. In this work, we focus on the sequence
L(H^{i}_{m}(R/I^{n})) for i>0. We are able to show that
for a monomial ideal I, if L(H^{i}_{m}(R/I^{n})) is finite for n >> 0, then this sequence coincides with a quasi-polynomial.
Moreover, for a family
of square-free quadratic monomial ideals we show that the limit of the sequence L(H^{i}_{m}(R/I^{n}))/n^{d} exists.
We also prove the existence of a related
limit that involves the minimal degree of the modules H^{i}_{m}(R/I^{n}). This is joint work with Hailong Dao.

Claudia Pollini

**Title:** Composition series of local cohomology modules

**Abstract:** A long standing problem in algebraic geometry and
commutative algebra is to determine whether every irreducible curve in projective three-space is the intersection of
two hypersurfaces. One way to approach this problem is via the study of local cohomology modules. As modules over the ring, local
cohomology modules are huge (neither finitely generated nor Artinian), hence intractable. However, as modules over the Weil algebra
D they have a finite composition series and become manageable. Hence an important task is to understand the D-module structure of local
cohomology modules. In this talk we describe their composition series. This is joint work with Robin Hartshorne.

Steven Sam

**Title:** Questions about Boij-Söderberg theory

**Abstract:** Boij-Söderberg theory is concerned with numerical invariants of graded modules and coherent
sheaves up to scalar multiple. There is a rich theory behind the case of polynomial rings and projective spaces,
respectively, and a number of generalizations have appeared. I'll highlight some questions and possible research directions.
This is based on http://arxiv.org/abs/1606.01867

Karl Schwede

**Title:** Fundamental groups of F-regular singularities via F-signature

**Abstract:** Inspired by the analogy between F-regular singularities in characteristic p > 0 and
KLT singularities in characteristic zero, it is natural to hope that the punctured spectrum of an
F-regular singularity has finite etale fundamental group. I will talk about joint work with Carvajal and
Tucker which proves exactly this. In fact, we prove more, we bound the size of the fundamental group by the
reciprocal of the F-signature. As another consequence of this work, we obtain new purity of the branch locus results for rings with "mild" singularities.