Seminars are held in **Avery 351** twice a week:

- Wednesdays
3:30-4:20 pm
- Thursday
2:30-3:20 pm

**Seminar schedule**

January 16, 2019

Ralf Schiffler (University of Connecticut)

**Title***: *Frieze varieties: A characterization of the
finite-tame-wild trichotomy for acyclic quivers

**Abstract**: Given a quiver (=oriented
graph) without oriented cycles, one can construct its frieze variety. It is
defined in an elementary recursive way by constructing a set of points in
affine space. From a more conceptual viewpoint, the coordinates of these points
are specializations of cluster variables in the cluster algebra associated to
the quiver. We give a new characterization of the finite-tame-wild trichotomy
for acyclic quivers in terms of their frieze varieties. We show that an acyclic
quiver is representation finite, tame, or wild, respectively, if and only if
the dimension of its frieze variety is 0,1 or >1,
respectively. This is a joint work with Lee, Li, Mills and Seceleanu.

January 30, 2019

Matt Mills (Michigan State University)

**Title**: Cluster and Poisson
structures on quasi-affine varieties* *

**Abstract**: One large appeal of the study
of cluster algebras is that the ring of regular functions for many varieties
can be equipped with a cluster algebra structure. In this talk we will describe
sufficient conditions for a quasi-affine variety to admit a cluster algebra
structure. Furthermore, we will follow the work of Gekhtman,
Shapiro, and Vainshtein to show when these cluster
structures can be realized as Poisson algebras with a Poisson bracket
compatible with the cluster structure. Finally, we present an open question
about cluster variables in Poisson-Lie groups.* *

February 6, 2019

**Title**: Cohomologically full
rings

**Abstract**:** **The class of cohomologically full rings
has been recently introduced by Hailong Dao, Linquan Ma and myself. This class includes nice
singularities such as Cohen-Macaulay rings, F-pure rings in characteristic p,
Stanley Reisener rings, etc. The goal of this talk is
to present some of the motivations behind the definition, and some of the
properties that cohomologically full rings enjoy.
Among them, upper bounds on projective dimensions, and on the Castelnuovo-Mumford regularity in the graded case.

February 13, 2019

**Title**: Weak and strong F-regularity for Gorenstein
rings

**Abstract**: I will present a proof due to
Huneke and Leuschke of the
fact that weak and strong F-regularity are equivalent for Gorenstein
rings. Whether the same statement holds in the general case is one of the major
open problems in the area of singularities in characteristic p>0. The aim is
to make the talk largely accessible to graduate students (in particular those
who have been showing up for Math 918 this semester).

February 20-21, 2019

Hailong
Dao (University of Kansas)

**Title 1**: Mysterious ideals, part 1: Golod
ideals

**Abstract 1**: In these talks I will introduce some joint work with recent
Nebraska postdocs. One common theme is that the problems reduce to very
concrete, but surprisingly stubborn problems about ideals, even in polynomial
rings of few variables. In the first talk, I will describe my work with
Alessandro De Stefani, focusing on the question: when is the product of two
homogenous ideals Golod?

**Title 2**: Mysterious ideals, part 2: linearly presented ideals

**Abstract
2:** A homogenous ideal is linearly presented if the
presentation matrix of it has only linear entries. They have attract increasing
attention in recent years, due to connections to regularity, Hirsch-type
bound on diameter of polytopes, and so on. I will introduce these questions and
then focus on new results, obtained with Thanh Vu, on regularity bound for such
linearly presented monomial ideals.

February 27, 2019

Ben Drabkin

**Title**: Growth of the Symbolic Defect

**Abstract**: Let R be a commutative Noetherian ring, and let I be an
ideal in R. The symbolic defect is a numerical measurement of the difference
between the symbolic and ordinary powers of I. In the
case that I has sufficiently well-behaved symbolic powers (i.e. its symbolic
Rees algebra is finitely generated) we prove that the symbolic defect grows
eventually quasi-polynomially.

March 1, 2019 at 10:30 am in 351 Avery (**please note** **unusual date/time**)

June Huh (IAS Princeton)

**Title**: Standard conjectures for finite vector spaces

**Abstract**: I will build a commutative ring that satisfies "standard
conjectures", starting from a finite field. What is this ring? What does
it say about the finite field? This talk will be elementary: No background
beyond the first year graduate algebra will be necessary. Joint with Mats Boij, Bill Huang, and Greg Smith.

March 6, 2019

William Trok (University of Kentucky)

**Title:** The
Geometry of Point Configurations and Hyperplane Arrangements

**Abstract**: Given a finite set of points, Z, in
a projective space, P(V), we can associate to Z a dual set, A(Z), of hyperplanes
in the dual space, P(V^*). We ask, "How does the geometry of Z relate to
the geometry of A(Z)?". In this talk, building
off of work of Faenzi and Valles, as well as Cook, Harbourne, Migliore and Nagel, we
relate the module of derivations, D(A(Z)) of A(Z), to the homogeneous ideal
I(Z). For instance, we show that D(A(Z)), gives bounds
on the regularity of I(Z). Along the way we study ideals, I(Z),
whose intersection with powers of ideals of general linear subspaces have
larger than expected intersection.

March 13-14, 2019

Daniel Smertnig (University of Graz and Waterloo)

**Title 1**: The role of transfer Krull monoids in studying non-unique factorizations

**Abstract 1**: In studying factorizations of elements in domains (or cancellative monoids; e.g. when studying direct sum
decompositions of modules), surprisingly often one is able to reduce the
problem to one over a Krull monoid. In this case,
factorization questions typically reduce to problems in combinatorial and
additive number theory, many of which have been well studied. I will recall
this technique and survey some recent results.

**Title 2**: Locally free
cancellation for definite quaternion orders

**Abstract 2**: Steinitz's theorem can be generalized to describe
finitely generated locally free modules over an order in a central simple
algebra B (over a number field). However, when B is a definite quaternion
order, generally locally free cancellation fails, and one only obtains a
description up to stable isomorphism. I will talk about the classification of
the finitely many definite quaternion orders that still possess the
cancellation property. This property is also reflected in the factorization of
elements. This is joint work with John Voight.

March 27-28, 2019

Janina Letz (University of Utah)

**Title 1**: Generation time and
(co)ghost maps

**Title 2: **Local to
global principles for generation time over commutative rings

**Abstract**: In the derived category of modules over a noetherian ring a complex G is said to generate a complex X
if the latter can be obtained from the former by taking finitely many summands
and cones. The number of cones needed in this process is the generation time of
X. I will present some

local to global type results for computing this invariant, and also discuss
some applications.

April 3, 2019

Taran Funk

**Title**: Frobenius and Homological Dimensions of Complexes

**Abstract**: The Frobenius endomorphism has
proven to be an effective tool for characterizing when a given finitely
generated module over a commutative Noetherian local ring of prime
characteristic has certain homological properties. In particular, many
have used this endomorphism to detect when such a module has finite
projective dimension. I will be discussing some results by Tom Marley
and myself in our paper with the same title as this talk. We show, among other
things, how to extend a few of these results to arbitrary modules.

April 10-11, 2019

Claudia
Miller (Syracuse University)

**Title**: Resolutions and
possible dg-algebra structures for compressed Artinian algebras

**Abstract**: We construct free resolutions of compressed
Artinian graded algebra quotients of polynomial rings and give a method to
reduce them to a minimal resolutions. Our result generalizes results of El Khoury and Kustin for Gorenstein algebras of even socle
degree with a different proof.

Then we use this
to show current progress towards constructing dg-algebra structures in the Gorenstein case. For this we will discuss two general
homological tools less known in the commutative algebra world, namely of
transferring A^{∞} structures (and dg-algebra structures in nice
situations) along homotopy equivalences and a tool
for creating new homotopy equivalences from old
ones.

This is joint work with
Hamid Rahmati.

April 18, 2019

Claudia
Miller (Syracuse University)

**Title**: Resolutions and
possible dg-algebra structures for compressed Artinian algebras (continued from
the previous week)

April 24-25, 2019

Lorenzo Guerrieri
(University of Catania and the Ohio State University)

**Title**: Directed unions of
local monoidal transform of regular local rings

**Abstract**: Let R be a regular local ring of dimension d >1.
Recently, several authors studied the rings obtained as infinite directed union
of iterated local quadratic transforms of R, and call them quadratic Shannon
extensions. Here we study features of directed union of local monoidal
transforms of a regular local ring (monoidal Shannon extensions) and more
generally of directed unions of GCD domains. In particular we are interested in
understanding when these rings are still GCD domains.

Claudia
Miller (Syracuse University)

**Title**: Resolutions and
possible dg-algebra structures for compressed Artinian algebras (continued from
the previous week)

May 1st, 2019 at 3:30 pm in 351 Avery (**finals week**)

Kurt Herzinger
(United States Air Force Academy)

**Title:** Using Numerical Semigroups to
Study the Game of Sylver Coinage

**Abstract:** Sylver Coinage is a two-player
game played on the positive integers described by John Conway in the book *Winning Ways for Your Mathematical Plays*.
The game is connected in a natural way to submonoids
of the non-negative integers, known as numerical semigroups. We will
examine the connections between these two topics and demonstrate how numerical
semigroups can be used to analyze winning and losing positions in Sylver Coinage.

May 20, 2019 at 3:30 pm in 11 Avery

Hamid Rahmati (Miami
University)

**Title:** Free Resolutions of Frobenius powers of the maximal ideal over a generic
hypersurface in 3 variables

**Abstract:** We discuss the asymptotic behavior of the free
resolutions of the bracket powers of the maximal ideal in the hypersurface ring
$R=k[x,y,z]/(f)$, where $k$
is a field of positive characteristic and $f$ is chosen generically. We show
that high enough Frobenius powers of the maximal
ideal have identical Betti numbers. We also compute
the Hilbert-Kunz function of such rings. This is joint work with Claudia Miller
and Rebecca R.G.