Date: March 2nd, 2026
Time: 3:30-4:30 PM
Location: SCHM 114
Speaker: Siva Somasundaram (Purdue University)
Title: An Exposition of Linkage Theory and the Linkage of Zero-Dimensional Monomial Ideals
Abstract: "Linkage theory (also known as liaison theory) originates in ideas dating back to the late 1930s, but it gained significant momentum following the influential work of Peskine and Szpiro in 1974. Beyond being an active area of study in its own right, linkage theory has become an important tool in commutative algebra, providing methods for constructing examples and understanding structural obstructions in various contexts. In recent decades, techniques from linkage have played an increasingly prominent role in proving results across the subject.
In this talk, I will present an introduction to both complete intersection linkage and Gorenstein linkage, emphasizing their similarities and differences through illustrative examples and basic structural results. I will then discuss developments from the past two decades, with particular attention to zero-dimensional monomial ideals. A primary focus will be an overview of the work of Huneke and Ulrich on monomial liaison (2005). The talk will conclude with a discussion of several open problems in the area.
The presentation is intended to be accessible to graduate students who have completed MA 557 (Commutative Algebra I). "
Date: February 23rd, 2026
Time: 4:30-5:30 PM
Location: SCHM 114
Speaker: Anna Natalie Chlopecki (Purdue University)
Title: Infinite Matrix Schubert Varieties are Cohen-Macaulay
Abstract: "I will show that the Stanley-Reisner ring of an infinite simplicial complex is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. I will then introduce the concept of an infinite matrix Schubert variety and apply the aforementioned result to show that these varieties are Cohen-Macaulay. This is joint work with Nathaniel Gallup and Jason Meintjes."
Date: January 26th, 2026
Time: 3:30-4:30 PM
Location: SCHM 114
Speaker: Tyler Dunaisky (Purdue University)
Title: Cosmological Correlators and Triangulating the Dual Cosmological Polytope
Abstract: "A cosmological correlator is a multi-dimensional Mellin integral, associated to a graph G, which encodes information about the state of the early universe. Evaluation of these integrals is extremely challenging, even in simple cases. However, it turns out the integrand can be identified with the so-called canonical form of the cosmological polytope, allowing the application of techniques from combinatorial commutative algebra. I'll sketch my contribution to this story and advertise the up-and-coming field of positive geometry, which seeks to provide a general framework for canonical forms, of geometric objects more exotic than polytopes."
Date: November 19th, 2025
Time: 12:30 - 1:30 PM
Location: LWSN B134
Speaker: Ryan Watson (University of Nebraska - Lincoln)
Title: Cohomological Support Varieties Over Complete Intersections
Abstract: "Cohomological support varieties give a way to assign a variety to a finitely generated module over a local ring R. This variety encodes homological information about the module (and the ring). These were first introduced by Avramov in 1989 when the ring is a complete intersection and only until recently was the theory expanded to encompass all noetherian local rings. In this talk, I will stick to the classical setting of working over a complete intersection, and go over the definition of these objects, some useful results involving them, and time permitting, prove a theorem that I’ll use in my talk in the commutative algebra seminar (in the CI case)."
Date: November 11th, 2025
Time: 2:30 - 3:30 PM
Location: Zoom
Speaker: Kesavan Mohana (University of Nebraska - Lincoln)
Title: Lifting Systems for Finite Length Modules
Abstract: "Suppose R -> S is a surjective map of Noetherian local rings. In this talk, I will discuss a classical notion of lifting S-modules to R-modules along this map. We say an S-module M lifts to an R-module L if L is isomorphic to M upon extending scalars via R -> S, and Tor_i(L,R) = 0 for all i > 0. It turns out, it is interesting to consider a much weaker and more geometric notion of lifting modules, introduced by Nawaj KC and Andrew Soto Levins, called Serre liftable modules. Instead of asking the higher Tors to vanish, we require that the lift L of M is of the 'correct dimension', that is, dim(R) - dim(L) = dim(S) - dim(M).
A finitely generated R-module L is a naive lift of an S-module M if L is isomorphic to M upon extending scalars via R -> S. The problem is to determine the maximum depth and dimension among all naive lifts of M, which we term the liftable depth and liftable dimension of M along R -> S. We approached this problem via a notion of lifting systems. We then provide a characterization of the classical notion of liftability and the recent notion of Serre liftability to a regular local ring for finite length modules. This work is joint with Ben Katz (UNL), Nawaj KC (U of U), Andrew Soto Levins (TTU), and Ryan Watson (UNL)."
Date: October 28th, 2025
Time: 2:30-3:30 PM
Location: BRNG 1206
Speaker: Tyler Dunaisky (Purdue University)
Title: How to Find the Area of a Polygon
Abstract: "Combinatorial objects tend to give rise to nice algebraic structures. Today, we'll see how polytopes (bounded polyhedra) correspond to toric ideals, which will allow us to use Gröbner bases to compute their volume."
Date: October 21st, 2025
Time: 2:30-3:30 PM
Location: BRNG 1206
Speaker: RJ Barnes (Purdue University)
Title: D-modules and b-functions
Abstract: "D-modules provide an algebraic perspective of linear PDE's. Bernstein-Sato polynomials, or b-functions serve as an invariant that contain information about singularities on varieties as well as the behavior of the solutions to PDE's near those singularities. We will introduce and discus basic properties of D-modules and the information we can extract from b-functions."
Date: October 7th, 2025
Time: 2:30 - 3:30 PM
Location: BRNG 1206
Speaker: Joel Castillo-Rey (Basque Center for Applied Mathematics)
Title: F-Invariants and F-Singularities. Some connections
Abstract: "In this talk, we will discuss some classic results on the relationship between F-singularities and two important F-invariants – F-signature and Hilbert–Kunz multiplicity. For the F-signature, we will discuss some ideas going into the proof of Kevin Tucker's proof of existence, as well as the characterisation of regularity and strong F-regularity. For Hilbert–Kunz multiplicity, we will describe Watanabe–Yoshida conjecture and present some results illustrating the guiding philosophy that 'low Hilbert–Kunz multiplicity implies mild singularities.'"
Date: September 23rd, 2025
Time: 2:30 - 3:30 PM
Location: BRNG 1206
Speaker: Manav Batavia (Purdue University)
Title: Algebras with a Straightening Law in Invariant Theory
Abstract: "Algebras with a straightening law (ASLs) provide a striking illustration of how combinatorial methods can simplify ring-theoretic computations. In this talk, I will introduce the concept of ASLs, outline their key structural properties, and present natural examples arising as invariant rings of specific group actions."
Date: September 16th, 2025
Time: 2:30 - 3:30 PM
Location: BRNG 1206
Speaker: Benjamin Mudrak (Purdue University)
Title: Licci and Glicci Monomial Ideals
Abstract: “We will discuss what is known about the linkage and G-linkage classes of monomial ideals. We will particularly focus on the results contained in the paper ‘Liaison of Monomial Ideals’ by Craig Huneke and Bernd Ulrich.”
Date: September 10th, 2025
Time: 12:30 - 1:15 PM
Location: LWSN B134
Speaker: Vignesh Jagathese (University of Illinois Chicago)
Title: An Introduction to F-Pure Singularities
Abstract: "In this talk we will provide a broad overview of singularities in positive characteristic. In particular, we will discuss tools to measure singularities via the Frobenius morphism, how they relate to each other, and how they recover results in characteristic 0. There will be a particular focus on F-pure singularities, or the class of rings for which the Frobenius map is a "pure" map of R-modules. Time permitting, we will discuss some asymptotic invariants in positive characteristic that measure failure of F-purity, such as the F-pure Threshold, and relate it to well known characteristic 0 invariants."
Date: September 2nd, 2025
Time: 2:30 - 3:30 PM
Location: BRNG 1206
Speaker: Aryaman Maithani (University of Utah)
Title: Invariant Theory of Commutative Rings
Abstract: "Given a group G acting on a ring R, we consider the subring R^G, the subring of elements fixed by G. It's a natural question to ask what "good" properties of R are inherited by R^G. Some of these questions were considered by Hilbert and Noether, and were a motivation to study noetherian rings. We will discuss some of these results. This talk should be accessible to someone who's done a first course in module theory."