Abstracts

Invited Speakers

Updated 10/5/2026

Prashanth Sridhar
University of Alabama

Reconstruction Results for Projective Varieties
I will discuss new reconstruction results for projective varieties using sheaves of differential graded algebras, generalizing results of Bondal–Orlov and Ballard. In particular, I will describe natural examples of dg-categories (and n-stable categories) associated to projective varieties from which the varieties can be recovered. These constructions allow reconstruction beyond the classical ampleness hypotheses on the (anti) canonical bundle.

Olivia Strahan
University of New Mexico

Local Cohomology in t-Stanley-Reisner Rings
There is a profound relationship between the algebraic invariants of Stanley-Reisner rings and the topological data of the associated simplicial complex. Hochster’s formula, which describes the local cohomology modules in terms of the simplicial cohomology of links, is one of many famous theorems built on this relationship. This talk will introduce a mixed-characteristic analogue of Stanley-Reisner theory and the adaptation of Hochster’s formula to this new setting. These and related results are from a joint paper with Mel Hochster.

Fix a discrete valuation ring (V,t) with fraction field L of characteristic zero and residue field K of characteristic p>0. In a t-Stanley-Reisner ring, the uniformizer t plays the role of one variable. These rings correspond to simplicial complexes with a distinguished vertex v_0. As is the case for classical Stanley-Reisner rings, this correspondence induces many deep connections between algebra and topology, but the relationship in our setting takes on a distinctly mixed-characteristic flavor. The original Hochster’s formula describes local cohomology in terms of the simplicial cohomology of the links of faces; in our setting, we must replace simplicial cohomology with “cellular sheaf cohomology,” where the coefficients vary from face to face rather than remaining fixed.

Keller VandeBogert
University of Kentucky

Carlsson's Conjecture and the Tate-valued Frobenius
Rank conjectures are ubiquitous in commutative algebra and topology. At a conceptual level, they predict that objects constrained by many independent equations, or endowed with many independent symmetries, must have "large" derived invariants. In 2018, Iyengar and Walker showed that many of these rank conjectures are false in odd characteristic, leaving the status unknown but precarious in characteristic 2. In this talk, I will explain recent work showing that the algebraic and topological forms of these conjectures are true in characteristic 2. The key idea is to use the so-called Tate construction as an intermediary to compare tensor and Frobenius powers.

Rahul Ajit
University of Kansas

Test Ideal Can Be Computed by a Boundary
Suppose that R is a normal reduced F-finite ring of char p > 0. We prove that there exist an effective Q-divisor Δ on Spec R such that K_R + Δ is Q -Cartier with index not divisible by p, that computes the test ideal \tau_b (R) = \tau_b (R Δ). This answers a question of Schwede, Smith and other experts. Time permitting, we will answer two related questions.

Dale Cutkosky
University of Missouri

Multiplicity of Graded Families of m-primary Ideals
We define multiplicity for a graded family of m-primary ideals on an arbitrary Noetherian local ring R so that they exist as a natural limit. This multiplicity agrees with the volume when R is analytically unramified. We show that many classical theorems about multiplicity for ideals, such as the Rees theorem and the Minkowski inequality and equality, generalize to this multiplicity.

Tom Marley
University of Nebraska

Tor Rigidity and Frobenius
We survey the many known "rigidity" theorems for Tor and the Frobenius functor for modules over Noetherian local rings of prime characteristic, starting with the classical theorem of Kunz. This will be followed by a discussion of several remaining open questions in this area. We then present some new rigidity results (in joint work with Anna Brosowski and Victor D. Mendoza Rubio) for modules of finite complete intersection dimension.

Invited Graduate Student Speakers

Kara Fagerstrom
University of Nebraska

Tor Algebra Classification for Rings from General Points
Any ring that comes as a quotient of a regular local or graded ring has an associated Tor Algebra, which in some cases has a unique multiplication structure that can be classified. We will consider rings constructed from sets of general points and classify their Tor Algebra structures.

Ryan Hunter
University of Kansas

Frobenius Stability as Measure of Singularity
In positive characteristic algebra, the singularities of a ring may be understood via properties of the Frobenius morphism on certain local cohomology modules. We study a property called Frobenius stability of certain local cohomology modules, emphasizing a connection with the structure of related local cohomology modules as modules over the ring of differential operators. Further, we make this connection explicit by studying a few classes of well-known singular hypersurfaces.

Zachary Nason
University of Nebraska

Quasi-Gorenstein Morphisms of DG-rings and Exact Sequences
A quasi-Gorenstein homomorphism of commutative noetherian local rings, as first defined by Avramov and Foxby, naturally characterizes the Gorenstein property in commutative algebra. In particular, the canonical surjection of a noetherian local ring R onto its residue field is quasi-Gorenstein if and only if R is Gorenstein, and if a local homomorphism is quasi-Gorenstein, then its source is Gorenstein if and only if its target is Gorenstein. Extending the classical definition of a quasi-Gorenstein homomorphism, we introduce quasi-Gorenstein morphisms between DG-rings (which are special DG-algebras that behave similarly to noetherian local rings). We will then demonstrate that a sequence of elements in a local ring is exact if and only if certain maps between Koszul complexes formed with respect to the sequence are quasi-Gorenstein (as DG-rings). This result generalizes a previous theorem on exact sequences by Avramov, Henriques, and Şega. This presentation is based on joint work with Andrew Soto Levins and Ryan Watson.

Poster Presentations

Ana Podariu
University of Nebraska

A Multigraded Analog to the Lefschetz Properties
This poster will recall the Lefschetz Properties and Bruzzo, Gondim, Holanda and Montoya's definition of the Toric Lefschetz Properties from a 2024 paper. We will mention some results that hold in the standard graded case and their analogs in a multigraded case.

Aniketh Sivakumar
Tulane University

The Multiplicity Sequence of Monomial Ideals
We establish a convex-geometric formula for the multiplicity sequence of monomial ideals in terms of mixed volumes of polytopes constructed from the Newton polyhedron of the ideal. We also give a counterexample to a conjecture of Achilles and Manaresi regarding a proposed formula for this invariant.

Benjamin Mudrak
Purdue University

Linkage and the Watanabe Yoshida Conjecture
We study the Watanabe Yoshida conjecture through the lens of linkage theory. Specifically, we show that licci ideals satisfy this conjecture along with other classes.

David Crosby
University of Arkansas

Symbolic Rees Algebra Generators of Matroidal Ideals
We refine a combinatorial description of the minimal square-free monomial generators of a matroidal ideal and use this refinement to determine many algebraic properties of matroidal ideals and of there symbolic Rees algebras.

Dinesh Limbu
University of South Carolina

Monomial Basis for Koszul Homology
For a monomial ideal I, we will give a combinatorial description of the multigraded Koszul homology. Using this, we will show that a specific class of ideal have monomial Koszul basis. We will also recover some previously known results on monomial Koszul basis. 

Haoxi Hu
Tulane University

Graded Families of Ideals and Convex Regions
We study the interplay between graded families of ideals in K-domains and their associated convex regions. These regions, called Newton-Okounkov regions, arise naturally from graded families of ideals associated to a valuation with one-dimensional leaves. Our main focus is to compute asymptotic resurgence number of a pair of graded families of ideals. By combining techniques from Attouch-Wets topology and convex-geometric properties of Newton-Okounkov regions, we characterize the asymptotic resurgence number through containment relations between the pair of corresponding Newton-Okounkov regions.

Jonathan Feigert
Baylor University

Syzygies of Modules of Covariants
A fundamental setting in classical invariant theory, dating back to at least Weyl, is a classical group H acting on a space W of several vectors and covectors. The module of covariants for a partition is the space of all H-equivariant polynomial functions from W into a fixed irreducible representation of H labeled by that partition. In the special case of the trivial representation (i.e., when the module of covariants is the ring of invariants), Weyl gave generators and first-order syzygies; ultimately higher-order syzygies were given by the minimal free resolutions of Lascoux and Józefiak–Pragacz–Weyman. Our main result generalizes this program to arbitrary partitions: we construct minimal free resolutions for all such modules of covariants, expressing the maps in the resolutions as explicit matrices of syzygies. We also interpret these maps in terms of covariant differential operators.

Julianne Faur
University of Nebraska

Cohomological Properties of Koszul Algebras: An Example
We present an example of a Koszul algebra R (in the classical sense in which the residue field admits a linear resolution over R) satisfying the following properties: (1) The cohomological support variety of R is a hyperplane; and (2) The homotopy Lie algebra of R has no nilpotent elements of degree 2. We will give an idea of the proofs and gadgets involved, in addition to providing a discussion as to why the existence of such a ring is interesting and somewhat surprising.

Manav Batavia
Purdue University

Vanishing of Local Cohomology in Unramified Mixed Characteristic
Given an ideal I in a regular local ring A, the cohomological dimension of I in A is the index of the highest non-vanishing local cohomology of A supported at I. Determining effective upper bounds on the cohomological dimension in terms of topological invariants of Spec(A/I) is a central problem in commutative algebra: foundational results include the Hartshorne-Lichtenbaum Vanishing Theorem and the Second Vanishing Theorem. In equal characteristic, Faltings established in 1980 a general bound on the cohomological dimension of an ideal in terms of its “big height”. In this work, I extend Faltings’ results to the unramified mixed characteristic setting and show that the resulting bound is sharp.

Marc Abi Aad
University of Missouri

Chow Decompositions of Cubic Surfaces.
Given the homogeneous equation of a cubic surface f, we aim to describe an algorithm that will generate all 120 Chow-decompositions of f (ie, all possible ways of writing f as al_1l_2l_3+bl_4l_5l_6 where l_i is the equation of a plane). This problem is intimately related to the problem of 27 lines on a cubic surface, and has applications in the field of Computer Vision.

Noah Walker
University of Nebraska

The Minimum Number of Generators of Symmetric Ideals
We study symmetric ideals in polynomial rings and the minimum number of polynomials required to generate them up to permutations of the variables. We give a representation-theoretic formula for this number. We also study extremal symmetric ideals and construct explicit examples of these ideals for every admissible number of generators. As an application, the principal case gives the sharp stable range for a theorem of Harada-Seceleanu-Sega on general principal symmetric ideals.

Raneeta Dutta
University of Kansas

Cohomology Vanishing, Koszul Cohomology and Multigraded Regularity on Projective Varieties
We prove a general cohomology vanishing theorem on arbitrary projective varieties within the framework of multigraded Castelnuovo--Mumford regularity. In particular, we apply this vanishing theorem to prove the vanishing of Koszul cohomology groups for product of base point free line bundles.

Reid Buchanan
Pittsburg State University

Characteristic-Dependent Monomial Resolutions with Few Generators
Let R be a polynomial ring over the base field k and n ≥ 3. We construct a family of monomial ideals with 3n-1 generators and projective dimension equal to 2n if and only if char(k) divides n. This improves upon previously known topological constructions with the same property, which require at least 7n-2 generators. Our construction is algebraic, starting from an extremal ideal and obtained after setting carefully-selected variables to 1.

Saad Awan
University of Kansas

First-Derivative Chromatic Symmetric Reconstruction for Proper Trees
In 1995 Richard Stanley asked whether every tree $T$ is determined up to isomorphism by its chromatic symmetric function $X_T$. We study this open problem by regarding $X_T$ as a polynomial in the power-sum symmetric functions $p_1, p_2, \dots$ and studying the invariant $\Phi_T = \left.\frac{\partial {X_T}}{\partial p_1}\right\rvert_{p_1 = 0}$. This strictly weaker invariant allows us to reconstruct an infinite subclass of trees showing us that even weaker invariants contain enough information to reconstruct trees entirely. We are further able to give an equivalent formulation of Stanley's question in terms of $\Phi_T$. Finally, we ask how much information is required to distinguish between trees in general. We find that the minimum number of edges required to distinguish between the isomorphism classes of $k-$vertex trees for all $k\ge 4$ is bounded below by $\left\lfloor k/2 \right\rfloor$.

Sabrina Fowler
University of Nebraska

Counting the Number of Generators of a Polarized Neural Ideal
The study of neural codes involves viewing sets of binary vectors as patterns of neural activity. To such a set we can associate a squarefree monomial ideal called a polarized neural ideal (PNI). In this poster we aim to describe restrictions on invariants of such ideals, in particular the number of elements in the minimal monomial generating set. We characterize how to build such a generating set for a quadratic PNI and use this to enumerate all possibilities for its size. Then we explore how to generalize this framework to larger degrees and describe what is known and what is still open in this setting.

Souvik Dey
University of Arkansas

On Complexities of Pairs of Cohen-Macaulay Modules and Some Applications
The complexity of a module, introduced by Avramov, measure the growth of Betti and Bass numbers of a module, and distinguish the modules of infinite homological dimension. The notion of complexity was extended by Avramov-Buchweitz to pairs of modules that measure the growth of Ext modules. The related notion of Tor complexity was first studied by Dao. Inspired by this, we study (Ext and Tor) complexity of pairs of certain CM (Cohen-Macaulay) modules, and establish lower bounds of complexity of pairs of modules in terms of that of a single module. As an upshot, under some additional hypotheses, we characterize complete intersection and regular local rings via injective complexity of the ring and that of the module of Kähler differentials respectively. Thus, we make partial progress towards a question of Jorgensen-Leuschke.

William Clark
Baylor University

Solutions to the Wave Equation and Generalized Dirac Equations via Unitary Highest Weight Modules
The classical wave equation and the generalized Dirac equations can be viewed as maps from one function space to another. Associated to these function spaces are linear operators that change one function into another, and the essence of Representation Theory is to study how these linear operators interact with each other and with the function space they are defined on. The collection of these linear operators form a Lie algebra. Using the properties of the Lie algebra, we can take one solution of the wave equation and change it into another solution that is independent from the original, and can do likewise for solutions to the generalized Dirac equations. What would normally need to be described through an iterative process is presented here as an explicit formula using certain decompositions of the function space.