**Thursday, September 15, 11:15 a.m. - 12:05 p.m.**

**Lukas Katthaen, **Frankfurt

**Abstract: **
The minimal free resolution of a quotient of the polynomial ring
admits a (generally non-associative) multiplication which satisfies
the Leibniz rule. This multiplication is far from being unique and
in favorable cases it can be chosen to be associative, which makes
the resolution into a DGA. In this talk, I consider these
multiplicative structures in the setting of monomial ideals. On the
one hand, I will present some structure theorems about these
multiplications, in particular in the associative case. On the other
hand, I will show that the presence of an associative multiplication
has implications on the possible Betti numbers of the ideal.

**Thursday, September 22, 11:15 a.m. - 12:05 p.m.**

**Vic Reiner, **Minnesota

**Abstract: **
For any (finite-dimensional, faithful, complex) representation of a
finite group G, we define a new invariant, which one might call its
"sandpile group." This invariant is a finite abelian group, but
also has a natural commutative product: it is the augmentation
ideal for a certain quotient of the commutative ring of virtual
characters of G. It also bears a close relation to the McKay
correspondence.
The goal in this talk will be to define this invariant, compute a
few examples, and advertise an example where it is an open
combinatorial challenge to compute its structure completely. It is
our hope that more commutative algebraic technique might help in
resolving this. This is joint work with G. Benkart and C. Klivans;
arXiv:1601.06849.

**Thursday, September 29, 11:15 a.m. - 12:05 p.m.**

**Christine Berkesch Zamaere, **Minnesota

**Abstract: **
Given a module M over the Cox ring S of a smooth toric variety, one
can consider free complexes that are acyclic modulo irrelevant
homology, which we call a free Cox complex for M. These complexes
have many advantages over minimal free resolutions over smooth toric
varieties other than projective spaces. We develop this in detail
for products of projective spaces. This is joint work with Daniel
Erman and Gregory G. Smith.

**Thursday, October 27, 11:15 a.m. - 12:05 p.m.**

**Eric Ramos, **Wisconsin

**Abstract: **
Much of the work in homological invariants of FI-modules has
been concerned with properties of certain right exact
functors. One reason for this is that the category of coherent
FI-modules over a commutative ring very rarely has sufficiently
many injectives. In this talk, we consider the (left exact)
torsion functor on the category of coherent FI-modules, and show
that its derived functors exist. Properties of these derived
functors, which we call the local cohomology functors, can be
used in reproving well known theorems relating to the depth,
regularity, and stable range of a module. We will see that
various facts from the local cohomology of modules over a
polynomial ring have analogs in our context. We will also see
that there is a way to make sense of a kind of local duality for
FI-modules. This is largely joint work with Liping Li.

**Thursday, November 17, 11:15 a.m. - 12:05 p.m.**

**Lukas Katthaen, **Frankfurt

**Abstract: **
A graded (or local) ring is called Golod if the Betti numbers of
the residue field grow as fast as possible. This is equivalent
to the vanishing of all Massey products on the Koszul
homology. In this talk, I discuss the latter condition for
Stanley--Reisner rings, where one has a geometric interpretation
of the product on Koszul homology. I will present several
examples to show how one can use geometry to effectively compute
these products. In particular, I will present an example where
the Golod property depends on the underlying field, and an
example where the Koszul homology has a trivial product but a
nontrivial ternary Massey product.

**Thursday, December 8, 11:15 a.m. - 12:05 p.m.**

**Robert Walker, **Michigan

**Abstract: **
More recently, my dissertation (e.g., arxiv.org/1510.02993,
arxiv.org/1608.02320) attempts to devise--and establish affirmative
results towards--a "toric" variant of a short-lived conjecture of
Brian Harbourne (2009) that says: For N \ge 2, the symbolic power
I^{(N(r-1)+1)} lies in I^r for all r>0 and all graded ideals I in
the coordinate ring R of a projective N-space over a field
(sometimes arbitrary, sometimes not). This talk will discuss
criteria for ideal containments of type I^{(E(r-1)+1)} \subseteq I^r
for all r>0: at the expense of focusing rigidly on a very specific
type of ideal, I can give you a duly explicit slope E. These
criteria already apply to a fairly prodigious class of normal
domains (e.g., several with European-honorific names).

2015-2016 Schedule

2014-2015 Schedule