Thursday's Schedule of Talks
Principal Speaker - Bernd Ulrich, Purdue University - About the Speaker
Title: Linkage, Residual Intersections, and Applications
Abstract: Linkage, or liaison, is a tool for classifying and studying varieties and ideals
that has its origins in 19th century algebraic geometry. Its generalization, residual
intersection, has broad applications in enumerative geometry, intersection theory,
the study of Rees rings, and multiplicity theory. After surveying basic properties
of linkage, we will focus on the computation of Picard groups and divisor class groups
and on the structure of rigid algebras in the linkage class of a complete intersection.
We will also discuss recent work on ideals in the linkage class of a complete intersection
and connections with the study of Hilbert schemes. We will describe applications of
residual intersections and explain the techniques used to determine their Cohen-Macaulayness,
canonical modules, duality properties, and defining equations. An emphasis will be
on weakening the hypotheses classically required in this subject.
Claudiu Raicu, University of Notre Dame - About the Speaker
Title: Polynomial Functors and Stable Cohomology
Abstract: The theory of polynomial representations of the general linear group goes back to
the thesis of Issai Schur at the turn of the 20th century. Such representations include
the tensor, symmetric, and exterior powers of a vector space, and have been completely
classified in the work of Schur when the underlying field is the complex numbers.
While there has been significant progress since the work of Schur, the story over
a field of positive characteristic remains largely unknown. In my talk I will describe
some novel stabilization results for sheaf cohomology, and explain their connection
to the study of polynomial representations / functors.
This is based on joint work with Keller VandeBogert.
Keller VandeBogert, University of Notre Dame - About the Speaker
Title: The Extension Problem for Hooks
Abstract: An important open problem in polynomial functor theory is to understand extension
groups between Schur functors, a distinguished class of polynomial functors. One of
the main limitations regarding this problem is the lack of computational methods,
which leads to the more general question: how can we compute such extension groups
at all? In this talk, we will see how Ext in the category of polynomial functors can
instead be computed as the ”stable” cohomology of certain vector bundles on projective
space. This, combined with a remarkable invariance property, allows us to give a full
solution to the extension problem for hook Schur functors.
This is based on joint work with Claudiu Raicu.
Jiamin Li, University of Oklahoma - About the Speaker
Title: Nodal Domains of Homogeneous Caloric Polynomials
Abstract: In this talk we will discuss the structures of local cohomology modules, especially
those of Segre products of graded modules supported at Segre products of ideals. We
will discuss some important questions of local cohomology modules, and we will apply
our results to study these questions in our setting.
This is joint work with Wenliang Zhang.
Jonathan Montaño, Arizona State University - About the Speaker
Title: Presburger Modules: Quasi-polynomials and Tameness
Abstract: In commutative algebra, functors such as local cohomology, Ext, and Tor applied to
sequences of modules often grow quasi-polynomially, i.e., they grow periodically along
finitely many polynomials. In this work we use the theory of tame modules from persistent
homology and Presburger arithmetic to provide an explanation for this quasi-polynomial
behaviour. Our methods extend directly to the multigraded setting for families of
multigraded modules over affine semigroup rings.
This is joint work with Hailong Dao, Ezra Miller, Christopher O’Neill, and Kevin Woods.
Patricia Klein, Texas A&M University - About the Speaker
Title: An Application of Gorenstein Liaison to Frobenius Splitting
Abstract: In 2009, Knutson showed that certain kinds of Frobenius splittings descend to new
splittings under Groebner degeneration. Using a connection between Groebner degeneration
and Gorenstein liaison, we will be able to give a partial converse to this result.
In this talk, we will review key facts about Frobenius splitting, Groebner degeneration,
and elementary G-biliaison before stating our theorem and discussing examples.
This talk is based on work in progress with Emanuela De Negri, Elisa Gorla, Jenna
Rajchgot, Lisa Seccia, and Mayada Shahada.
Fridays's Schedule of Talks
Steven Dale Cutkosky, University of Missouri - About the Speaker
Title: Rees Valuations and Divisorial Filtrations
Abstract: Let Ɨ = {In} be a divisorial filtration of mR-primary ideals in a (Noetherian) local ring R. Each ideal In has associated Rees valuations. If Ɨ is Noetherian (i.e. ⊕n ≥ 0In is a finitely generated R-algebra), then the set of Rees valuations of all the In is finite. We discuss asymptotic properties of the set of Rees valuations of the
In as n goes to infinity, giving illustrative examples, in the case that Ɨ is not Noetherian.
This is mostly joint work with Jonathan Montaño.
Alessandra Costantini, Tulane University - About the Speaker
Title: Rees Algebras of Linearly Presented Ideals
Abstract: Rees algebras represent an essential algebraic tool in the study of singularities
of algebraic varieties, as they arise, for instance, as homogeneous coordinate rings
of blowups or graphs of rational maps. In this talk, I will discuss the problem of
finding the defining equations of Rees algebras. Although this is wide open in general,
the problem becomes treatable in the case of height-two perfect ideals with a linear
presentation, where one can use a combination of homological methods and linear algebra,
inspired by classical elimination theory.
This is part of joint work with E. Price and M. Weaver (arxiv:2308.16010 and arxiv:2409.14238).
Xianglong Ni, University of Notre Dame - About the Speaker
Title: Codimension 4 Gorenstein Ideals and the Lie Algebras En
Abstract: Recent developments suggest a relationship between (1) codimension three perfect
ideals with type two, (2) codimension four Gorenstein ideals, and (3) representation
theory of the exceptional Lie algebras En. In the first half of the talk, we give a concrete construction relating (1) and
(2), spiritually similar to Ferrand’s construction for linkage of perfect ideals.
In the second half, we use our construction to extend a recent theorem relating (1)
and (3) to classify codimension four Gorenstein ideals on up to n = 8 generators over a field of characteristic zero. As a consequence, we deduce that
all such ideals are in the linkage class of a complete intersection.
Claudia Polini, University of Notre Dame - About the Speaker
Title: Licci Ideals
Abstract: Linkage has been used for over a century to study and classify curves in projective
three-space and, more generally, varieties in projective space or homogeneous ideals
in polynomial rings. Of particular importance have been licci ideals, ideals that
can be linked to a complete intersection in a finite number of steps. It is known
that the Castelnuovo-Mumford regularity of a licci ideal forces a very strict upper
bound for the initial degree of the ideal. Now, in joint work with Craig Huneke and
Bernd Ulrich, we conjecture that it also bounds the number of generators of the ideal,
and we prove this conjecture in many cases. In addition, we provide new sufficient
conditions for an ideal to be licci, for classes of ideals of height three and for
ideals containing a maximal regular sequence of quadrics.
The talk will also explain connections with recent work by Guerrieri, Ni, Weyman and
by Jelisiejew, Ramkumar, Sammartano.
Saturday's Schedule of Talks
Louiza Fouli, New Mexico State University - About the Speaker
Title: Generalized Hamming Weights and the Algebraic Structure of Matroids
Abstract: In this talk, we explore the relationship between the Generalized Hamming weights
of a linear code and the structure of the symbolic powers of the Stanley-Reisner ideal
of the matroid associated with the dual code. We extend this connection to arbitrary
matroids, highlighting how generalized Hamming weights can be understood through the
initial degrees of these symbolic powers.
This is joint work with Michael DiPasquale, Arvind Kumar, and Ştefan O. Tohăneanu.
Vinh Nguyen, University of Arkansas - About the Speaker
Title: Symbolic Powers of Matroids
Abstract: Matroids are abstract structures that encode the combinatorics of linear independence
among vectors. They can be seen as special classes of abstract simplicial complexes,
hence we can study their Stanley-Reisner and cover ideals. Herzog, Hibi, and Trung
pointed out a way to view generators of symbolic powers of square-free monomial ideals
as corresponding to vertex covers of simplicial complexes. Using this description,
we can describe the structure of the minimal generators of symbolic powers of matroids.
With this structure result we are able to answer many questions concerning their symbolic
powers.
Yairon Cid-Ruiz , North Carolina State University - About the Speaker
Title: Noetherian Operators in Commutative Algebra
Abstract: The Fundamental Principle of Ehrenpreis and Palamodov is a celebrated result that
describes the solutions of a linear system of PDE with constant coefficients. Most
surprisingly, at the core of this analytic result is a description of primary ideals
in terms of differential operators. We will discuss how we have borrowed these analytic
techniques to develop a robust framework of Noetherian operators in commutative algebra.
For instance, we will discuss a notion of differential primary decomposition and present
a version of the Briançon-Skoda theorem for nonreduced rings.
Joint works with Jack Jeffries and Bernd Sturmfels.
Yevgeniya "Jonah" Tarasova , University of Michigan - About the Speaker
Title: Bounds for the F-Signature of Determinantal Hypersurfaces
Abstract: The F-signature for strongly F-regular rings is known to be a value between 0 and 1, however, in the majority of
cases not much is known beyond that. In particular, even the classical case of determinantal
hypersurfaces, the F-signature has not been computed beyond the size 2 minors. In this talk, by using
Grobner degeneration to a toric hypersurface, we compute a lower bound for the the
F-signature of determinant hypersurfaces. We also provide an upper bound.
The talk is based on joint work with Hang (Amy) Huang, Cheng Meng and Suchitra Pande.
Ritvik Ramkumar , Cornell University - About the Speaker
Title: Hilbert Scheme of Points and Linkage
Abstract: will introduce the Hilbert scheme of points in three-space, which parameterizes ideals
of finite colength in k[x, y, z]. I will describe some recent progress in their study and focus on the connections
to linkage and licci ideals.
Uwe Nagel , University of Kentucky - About the Speaker
Title: Graded Betti numbers of Jacobian Ideals of Hyperplane Arrangements
Abstract: hyperplane arrangement A is a finite union of hyperplanes in Pn. Thus, it is defined by a product fA of linear forms. Its Jacobian ideal JA is generated by the partial derivatives of fA, and A is said to be free if JA is Cohen-Macaulay. Terao’s conjecture posits that freeness is a combinatorial property
of a hyperplane arrangements, that is, it is determined by the intersection lattice
of A. Extending earlier results, we discuss results in the spirit of this conjecture.
In particular, we identify mild conditions on A, which imply that the graded Betti numbers of the top-dimensional part JAtop are combinatorially determined.
For line arrangements, we have additional results. Note that in this case JtopA is the same as the saturation of JA. We show for an arbitrary line arrangement A that the graded Betti numbers of JAsat determine the graded Betti numbers of JA. We also obtain a new freeness criterion. It highlights the fact that free line arrangements
are special by proving that a related codimension two ideal has the least possible
number of generators, namely two, if and only if A is free.
This is based on ongoing joint work with Juan Migliore.
The public lecture will take place in Hillside Auditorium (HILL) room 206. The street address for HILL is 902 West Dickson Street.
Conference Banquet will take place in the Arkansas Union Ballroom. The street address for the Union is 435 North Garland Avenue. Check U of A Parking and Dickson Street Parking for paid parking details. For local dining options, visit Experience Fayetteville's Restaurant Guide.
