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).