CARS
This semester CARS was organized by Abbey Moos, Uyen Tran, and Stephen Stern
Kara Fagerstrom
An Introduction to Macaulay Inverse Systems
Oct 28
Abstract: In this talk we will define Macaulay Inverse Systems, go through several examples, and discuss what properties are known. The goal is to be an accessible primer on the subject with the hope that afterwards everyone will be able to understand the correspondence and how the explicit computation works.
Ben Katz
Ulrich modules and lim Ulrich sequences
Oct 21
Abstract: In this talk we'll explore Ulrich modules and lim Ulrich sequences. We will also discuss conjectures related to these topics. Finally, we will go through a construction of Ma's detailing how to build lim Ulrich sequences over standard graded rings.
Abbey Moos
When do local cohomology modules have finitely many associated primes?
Oct 14
Abstract: Although local cohomology modules are typically not finitely generated, there are conditions in which local cohomology modules may have finitely many associated primes. It has been conjectured by Lyubeznik that for any regular local ring R and any ideal I in R, the local cohomology module of R with support in I has finitely many primes associated primes. We will discuss some work that has been made to show under which conditions that this conjecture is true, but we will also talk about a recent example by Linquan Ma which proves that the conjecture is not true for all regular local rings.
Danny Anderson
Formal verification, Lean, and commutative algebra
Oct 7
Abstract: Lean is a proof assistant, i.e., a programming language that lets you encode mathematical structures and prove theorems about those structures. In this talk, we will discuss how you can try Lean for yourself—without needing to download anything—as well as provide information on how to learn more about Lean. We will also have a demonstration of what it is like to use Lean, and we will conclude by showing what has already been proven about commutative algebra within Mathlib, a Lean library for many mathematical structures and proofs. Time permitting, we will also see some proofs from sources other than Mathlib.
Ana Podariu
Some Results on A Multigraded Analog of The Lefschetz Properties
Sep 30
Abstract: In this talk we will recall the definitions of the Lefschetz Properties on standard graded Artinian rings, as well as some nice results in this original case. We will then introduce Bruzzo, Gondim, Holanda, and Montoya's 2024 definition of the Toric Lefschetz Properties, an analog of the Lefschetz Properties that applies to G-graded algebras. We'll see what results transfer from an N graded case to an N^d grading, and what damage they sustain along the way.
Noah Walker
Products and Symbolic Powers of Determinantal Ideals
Sep 23
Abstract: Given a polynomial ring in mn variables, we can arrange the variables into an (m x n)-matrix and consider its minors. The ideals generated by the minors of this matrix of variables are called determinantal ideals. In this talk, we will look at products and symbolic powers of these ideals and see how they are connected. We will also introduce some of the combinatorial ideas that are useful for working with and understanding these ideals.
Dorian Kalir
Koszul Algebras and Koszul Duality
Sep 16
Abstract: Koszul duality was originally defined for graded algebras over a field and has seen many different generalizations. This is in part due to the many different ways Koszulness can be understood. In this talk we will examine a perspective guided by the twisted tensor construction which points to a generalization of Koszulness by Briggs, Cameron, Letz, and Pollitz from the field setting to morphisms of local rings.
Isaiah Martinez
The singularity category of a Gorenstein ring
Sep 9
Abstract: Given a Noetherian ring R, Ragnar-Olaf Buchweitz introduced the notion of the singularity category of R (under a different name). Buchweitz showed that when R is Gorenstein, the singularity category can be better understood through maximal Cohen-Macaulay modules and complete resolutions. We will briefly discuss the history of Buchweitz's manuscript, present the main result that brings the singularity category of a Gorenstein ring "down to earth," and (if time permits) discuss the significance and impact of Buchweitz's manuscript.
Zach Nason
Quasi-Gorenstein Morphisms II: Electric Boogaloo
Sep 2
Abstract: On Sunday, I talked about joint work with Andrew Soto Levins and Ryan Watson that characterized quasi-Gorenstein morphisms of DG-rings using the virtually G-small property. In CARS today, I'm going to go over the other half of our work, which deals with exact sequences in noetherian local rings, and how to detect exact sequences by measuring whether certain maps of Koszul complexes are quasi-Gorenstein.
Kesavan Mohana Sundaram
On the annihilators of local cohomology modules
Aug 26
Abstract: In 2011, Lynch conjectured that the Krull dimension of the top local cohomology module supported at an ideal I in a Noetherian local ring R, is same as the dimension of R/T, where T is the I-torsion submodule of R. In 2019, Hochster raised a question that, is the top local cohomology module of a local Noetherian domain with support in a given ideal faithful? In the recent years, some cool progress and counterexamples have been made on these questions. So, why not check them out?
