Seminars are held in **Avery 351** and on Zoom on
Wednesdays 3:30-4:45 pm. The Zoom link will be sent through the seminar mailing
list.

**Seminar schedule**

March 24, 2021

Robert Walker
(Wisconsin)

**Title:** Uniform Asymptotic Growth of Symbolic Powers of Ideals

March 31, 2021

Paolo Mantero (Arkansas)

**Title:** Interpolation
problems.

**Abstract:**** **Homogeneous polynomial interpolation problems are interesting
problems in Algebraic Geometry. They ask for information regarding
hypersurfaces of given degrees passing with given multiplicity through a given
set X of points in a projective space. A theorem by Zariski and Nagata allows
one to translate these problems into questions about symbolic powers of the
ideal I_X defining the set of points. In the present talk we will state
classical and recent results about some interpolation problems, and discuss open
problems and conjectures.

April 14, 2021

Erica Musgrave (UNL)

**Title:** Free complexes over the exterior
algebra with small homology

April 21, 2021

Luis Núñez-Betancourt (CIMAT, Mexico)

**Title:** TBA

October 28, 2020

Laila Awadalla (UNL)

**Title:** Levels, ghost maps, and Gorenstein dimension

October 14, 2020

Eloísa Grifo
(UC Riverside)

**Title:** Constructing non
proxy-small modules

October 7, 2020

Jack Jeffries (UNL)

**Title:** Bernstein’s
inequality for some singular rings

September 23, 2020

Amadeus Martin (UNL)

**Title:** Curved BGG
correspondence

September 16, 2020

Eric Hopkins (UNL)

**Title:** N-fold matrix
factorizations

September 9, 2020

Alexandra Seceleanu (UNL)

**Title:** Surprising
examples involving Nagata idealizations

August 19, 2020

Mark Walker (UNL)

**Title:** On lim-Ulrich modules

March 11-12, 2020

Hamid Rahmati (UNL)

**Title:** Weak Complete
intersection ideals

February 26-27, 2020

Justin Lyle
(University of Kansas)

**Title
1:** Extremal growth of Betti
numbers and rigidity of (co)homology

**Abstract 2:** We say that a Cohen-Macaulay local ring has
finite CM_{+}-representation type if there exist only finitely many
isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are
not locally free on the punctured spectrum. In this talk, we consider
finite CM_{+}-representation type from various points of view, relating
it with several conjectures on finite/countable Cohen-Macaulay representation
type. We prove in dimension one that the Gorenstein
local rings of finite CM_{+}-representation type are exactly the
local hypersurfaces of countable CM_{+}-representation type; under mild
hypotheses these are exactly the hypersurfaces of type A_{∞} and
D_{∞}. We also consider partial results and obstacles in higher
dimension. This is joint work with Toshinori
Kobayashi and Ryo Takahashi.

February 5-6, 2020

Alexandra Seceleanu (UNL)

**Title:** Symmetric
shifted ideals

January 23, 2020

Jan Trlifaj (Charles University, Prague)

**Title:** Test
sets for factorization properties of modules, and transfinite extensions of
simple artinian rings

November 20-21, 2019

Rachel Diethorn (Syracuse University)

**Title 1:** Koszul homology of quotients
by edge ideals

**Abstract 1:** We show that the Koszul homology
algebra of a quotient by the edge ideal of a forest is generated by the
lowest linear strand. This provides a class of Koszul
algebras whose Koszul homology algebras satisfy
this property. We also recover a result of Roth and Van Tuyl
on the graded Betti numbers of quotients of edge
ideals of trees.

**Title 2:** Generators of Koszul homology with
coefficients in a J-weak complete intersection module

**Abstract 2:** We introduce a class of modules, called J-weak complete
intersection modules, inspired by the weak complete intersection ideals studied
by Rahmati, Striuli, and Yang. We present explicit
formulas for the generators of Koszul homology with
coefficients in such modules. This generalizes work of Herzog and of Corso, Goto, Huneke, Polini,
and Ulrich. We use these formulas to study connections between J-weak complete
intersection ideals and weak complete intersection ideals.

November 7, 2019

Alessandra Costantini (University
of California Riverside)

**Title:** Cohen-Macaulay
property of special fiber rings

**Abstract:** Let
R be a Noetherian local ring. The special fiber ring, or fiber cone, F(E),
of a module E is defined as the tensor product of the Rees algebra with the
residue field. In this talk, I will discuss a general technique to study
the Cohen-Macaulay property of F(E), and provide sufficient conditions
for F(E) to be Cohen-Macaulay. The latter generalize previous work of
Corso-Ghezzi-Polini-Ulrich
and Montaño for the special fiber ring of an ideal.

October 30, 2019

Ben Drabkin

**Title:** Freiman
cover ideals

**Abstract:** For a monomial ideal I in a polynomial ring,
it is difficult in general to find lower bounds on the number of generators of
I^2. However, Freiman's Theorem, a result in
additive number theory, gives rise to such a lower bound for certain types of
monomial ideals in terms of the number of generators of I and the analytic
spread of I. Ideals which meet this bound are called Freiman.
This talk covers an attempt to determine when cover ideals (a class of squarefree monomial ideals arising from graphs) are Freiman.

October 23-24, 2019

Mark Walker

**Title:** How
short can a module of finite projective dimension be?

**Abstract**: This is very much in progress
work, done in collaboration with Srikanth Iyengar and Linquan
Ma. I’ll describe what we know so far about the question in the title.

October 9-10, 2019

Lucho Avramov

**Title:** Monoids
of Betti tables over graded algebras. The case of
short Gorenstein algebras

**Abstract**: These talks, based joint work
with Courtney Gibbons and Roger Wiegand, I will explore a reflection of
representation theory in homological algebra. Some general approaches
will be discussed in the first talk. In the case of graded Gorenstein algebras R with R_i=0
for i>2 the picture is complete; it will be
presented in the second talk.

October 3, 2019

Roger Wiegand

**Title**: Vanishing
of Tor over fiber products

**Abstract**: Let (S,m,k) and (T,n,k)
be local rings, and let R denote their fiber product
over their common residue field k. Inspired by work of Nasseh and Sather-Wagstaff, we explore consequences of
vanishing of Tor^R_m(M,N) for various values of m, where
M and N are finitely generated R-modules. For
instance, assume that neither S nor
T is a discrete valuation domain and that Tor^R_m(M,N)
= 0 for some m > 5. Then at least one of
M, N has projective dimension at most one. This is
joint work with Thiago Freitas, Victor Hugo Jorge Pérez, and Sylvia Wiegand.

September 26, 2019

Tomasz Szemberg (Pedagogical University of Cracow)

**Title**:
Unexpected hypersurfaces and their geometry

**Abstract**: In the ground-breaking
article of Cook II, Harbourne, Migliore
and Nagel, the notion of unexpected curves has been introduced. It has been
generalized in the subsequent article to unexpected hypersurfaces. The subject
is dynamically growing ever since and shows unexpected ties to other areas of
algebra and geometry. I will report on some of recent developments.

September 25, 2019

Justyna Szpond (Pedagogical University of Cracow)

**Title**:
Fermat-type configurations in geometry and algebra

**Abstract**: A Fermat arrangement of lines
in the complex projective plane is given by linear factors of the polynomial (x^n-y^n)(y^n-z^n)(z^n-x^n) for some n≥3.
Singular points of these arrangements have appeared recently in commutative
algebra, more precisely in the containment problem between symbolic and
ordinary powers of homogeneous polynomials and in algebraic geometry in the
theory of linear systems and unexpected hypersurfaces. I will explain these two
appearances and present considerable generalizations which led to substantial,
new results in both fields.

September 18-19, 2019

Eloísa Grifo (University of
California Riverside)

**Title 1**:
Symbolic powers and the containment problem

**Abstract 1**: The n-th
power of an ideal is easy to compute, though difficult to describe
geometrically; in contrast, symbolic powers are difficult to compute while
having a natural geometric description. The Containment problem tries to
compare these two notions, and it relates to other questions such as the degree
of polynomials vanishing on a given variety. Harbourne
conjectured that a famous answer to the containment problem, by Ein--Lazersfeld--Smith, Hochster--Huneke
and Ma--Schwede, could be improved. In this talk, we
will introduce the containment problem and Harbourne's
Conjecture, and discuss how despite the fact that the conjecture does not hold
in full generality, many interesting versions of it do hold.

**Title 2**: Two versions of Harbourne's Conjecture

**Abstract 2**: Harbourne's
Conjecture is a statement depending on n that unfortunately has been disproved
for particular values of n. In this talk, we will discuss two different
versions of this conjecture that do hold. The first is joint work with Linquan Ma and Karl Schwede,
showing that even over certain singular rings, Harbourne's
Conjecture holds for a large class of ideals. The second is joint work with
Craig Huneke and Vivek Mukundan,
towards a stable version of Harbourne's Conjecture
that may hold for all ideals -- where we replace all n with n large enough.

September 5, 2019

**Title***: *An open problem motivated by algebra, geometry and combinatorics

**Abstract**: I'll discuss how a problem involving the Strong Lefschetz Property (algebra) is equivalent to a vanishing
problem involving points in the plane (geometry), which in a special case
concerns open problems involving line arrangements (combinatorics) that recent
progress has been made on (by Ben Green and Terence Tao a few years ago, and
Krishna Hanumanthu, me and Alex Dimca
more recently).

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.