This semester CARS is organized by Shalom Echalaz Alvarez and Kesavan Mohana Sundaram
David Lieberman
Making Sandwiches of Functional Equations: Bernstein's inequality, Bernstein-Sato polynomials, and Sandwich Bernstein Sato Polynomials
February 7
For a polynomial ring over a field K, the ring of differential operators is known as the Weyl algebra: the non-commutative K-algebra whose generators are the "multiply by a variable" maps and the partial derivatives with respect to each variable. In this setting a powerful result is Bernstein's Inequality, which puts bounds on the dimension of modules over the Weyl Algebra. Another closely related object of study is the Bernstein-Sato polynomial for an element of the polynomial ring. In this talk, I will share some recent developments in proving Bernstein's Inequality for the ring of differential operators over some singular rings, as well as a "Sandwich" version of Bernstein-Sato polynomials that gives a new method for showing the inequality in novel settings. This work is a collaboration with my advisor Jack Jeffries.
Michael P DeBellevue from the Syracuse University
The classical and change-of-rings bar resolution
February 14
The bar resolution is a type of universal resolution whose use dates back to computing group homology. First, we will see the motivation behind the original construction and work through some examples. Then, we will discuss a relative version for transferring data of resolutions along ring maps R -> S. This requires either differential graded algebra (DG) structures (for the version due to Iyengar) or more recently, an A_infinity ("associative up to an infinite family of homotopies") algebra. As an application, we'll prove the Golod-Serre bound on the growth of Betti numbers of modules over any ring.
Stephen Stern
The Persistence Property
February 21
In this talk I’ll share some of what is known about ideals I such that {Ass(R/In)}n forms an ascending chain, i.e. ideals that have the persistence property (or in some cases, ideals that don’t). Part of the story is a theorem that gives a (pretty ugly) combinatorial description of Ass(R/In) when I is a square-free monomial ideal. In the second half, I will avoid stating this theorem by generalizing it for monomial ideals I.
Kesavan Mohana Sundaram
Adams Operations in Commutative Algebra
February 28
Recently I started learning about Adams Operations and I think it is interesting. In this talk I will define adams operations and go over the first few facts that I learned.
Jordan Barrett
Toric Varieties & Zariski-Nagata Type Theorems
March 6
The Zariski-Nagata theorem is a classical result which expresses the nth symbolic power of a radical ideal I in a polynomial ring over a perfect field in terms of the nth regular powers of the maximal ideals in mSpec(I). In this 25-minute practice talk, I will state a well-known Zariski-Nagata type theorem for projective varieties, give a brief crash course on toric varieties, and discuss my work on characterizing which toric varieties satisfy a Zariski-Nagata result.
Spring Break
March 13
Julianne Faur
DG Algebra Resolutions
March 20
In this talk, I will define a differential graded ("DG") algebra over a fixed commutative ring Q, give some examples of DG algebras one often sees out in the wild, and describe a process---due to Tate (1956)---of creating a DG algebra extension of a fixed DG algebra A by adjoining variables for the purpose of killing specific cycles of A. This construction of adjoining variables to kill cycles is extremely useful for constructing a DG algebra resolution of a given Q-algebra of the form R = Q/I, as I will demonstrate with two illustrative examples. Finally, I will describe some results that connect the existence of certain DG algebra resolutions to the Buchsbaum-Eisenbud-Horrocks ("BEH") conjecture.
Wolfgang Allred
Beyond Abelian Categories
March 27
In 915 we become intimately familiar with R-Mod, the category of modules over a ring. So much so in fact that if we were only interested in commutative algebra, we might be tempted to ask ourselves if we even need any other categories beyond R-Mod (and maybe Set and Ring). Today, I will hopefully disabuse us of any such notions. More specifically, in the first half of the talk I will gently introduce some categorical machinery that goes beyond what we see in 915, and in the second half I'll try to convince you as to why we should care about these constructions.
Abraham Pascoe from the University of Kansas
Local cohomology of linear subspace varieties and simplicial homology
April 3
In this talk, I will define the local cohomology modules of a ring R with respect to an ideal I of R. The vanishing of local cohomology modules determine useful properties about a ring such as dimension and depth. Cohomological dimension of R in I is the maximum index where the local cohomology modules are nonzero. A classical problem in local cohomology is determining when cohomological dimension is bounded by the depth of R/I. I will define a family of simplicial complexes whose simplicial homology determines the vanishing of local cohomology modules. In particular, I provide a way to calculate the cohomological dimension and in some cases relating it to the depth of R/I. Additionally, I will define numerical invariants of local cohomology modules, called Lyubeznik numbers. These invariants are an analog of Betti numbers for injective resolutions. We will see alternate calculations of the Lyubeznik numbers, again utilizing the homology of this family of simplicial complexes. Finally, I will give a brief introduction on spectral sequences in order to explain how these results came about, specifically using the Mayer-Vietoris spectral sequence in local cohomology.
Robert Ireland
Lower Bounds on Betti Numbers
April 10
When studying finitely generated, graded modules over a polynomial ring, there are many invariants one could look at, with the Betti numbers being among some of the richest. In this talk, I will present what Betti numbers are and how they are found, and a few historical conjectures and theorems pertaining to them.
Ben Katz
A brief Introduction to Hilbert Polynomials and Multiplicities
April 17
In this talk I will go over graded rings and filtrations and then look at how we can define Hilbert polynomials in both of those settings. I will go through some theorems relating to the size of Hilbert polynomials. I will then define multiplicities and show a formula for how one can calculate them.
Ana Podariu
The Jordan Type Theorem + An Anecdote
April 24
In this talk I will introduce the Weak + Strong Lefschetz Properties and some other relevant properties in order to prove the Jordan Type Theorem. If I manage to do this in a timely fashion, I will show an example of the theorem making an accurate prediction and share an anecdote in which Kara and I wanted to see a different feature of the theorem and became extremely confused.
Zach Nason
Cohen-Macaulay Differential Graded Algebras
May 1
In commutative algebra, Cohen-Macaulay rings are ubiquitous. Indeed, as those who are taking MATH 906 now know, "life is really worth living in a Cohen-Macaulay ring." In my talk today, I'll present a generalization of Cohen-Macaulay local rings to differential graded algebras and will compare properties of Cohen-Macaulay rings with Cohen-Macaulay differential graded algebras. I'll end by going over a problem that makes Cohen-Macaulay differential graded algebras not nearly as nice as Cohen-Macaulay rings: the failure of the Cohen-Macaulay property to localize.
Sabrina Klement
Applied Abstract Algebra: Not an Oxymoron
May 8
In this talk, I will demonstrate an application of abstract algebra to coding theory. Anyone who has taken Math 818 will be familiar with all the algebra terms I will use. I will discuss some basics of cyclic codes and error detection, then we will construct a BCH code and use the Peterson-Gorenstein-Zierler algorithm to detect and correct an error.
