Commutative Algebra Seminar at UNL

Spring 2019

The seminar organizers are Alexandra Seceleanu (aseceleanu at unl.edu) and Mark Walker (mark.walker at unl.edu).

Seminars are held in Avery 351 twice a week:

 

Seminar schedule

January 16, 2019

          Ralf Schiffler (University of Connecticut)

          TitleFrieze 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

          Alessandro De Stefani

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

          Alessandro De Stefani

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.