CARS
This semester, CARS was organized by Michael DeBellevue and Frank Zimmitti
Nawaj KC
Rings of Invariatns: A finitely generated algebra
February 3
Suppose an n-dimensional k-vector spaceV is a k-linear representation of some group G . This induces a G-representation on the k-algebra k[x1, ... , xn]. The ring of invariants is the set of polynomials that are stable under this indued action of G. An important question in the so-called invariant theory is the following: is the ring of invariants a finitely generated k-algebra? Given some mild assumptions on the group G, Hilbert answered the question in the affirmative in his grounbreaking paper in 1890. I discuss this and other related results in my talk.
Matthew Bachmann
MCM Modules over Gorenstein Rings
February 10
There are many reasons to study maximal Cohen Macaulay (MCM) modules over commutative Noetherian Gorenstein rings. I have recently been interested in their relationship to the singularity category of R. In some sense, MCM modules describe "all of the stable homological features" of Gorenstein rings (Buchweitz). A crucial piece to understanding this relationship involves characterizing MCMs by a certain duality property. During my talk, I will prove this characterization, but the primary goal is to provide a gentle introduction to Gorenstein rings and MCM modules.
Andrew Soto Levins
The Injective Hull
February 17
Do you think that injective modules are a mystery? When I was first learning about projective and injective modules, I thought "wow projective modules are a lot nicer than injective modules". The main reason for thinking this was that it is easy to write down a projective module (since every free module is projective), but difficult to write down an injective module.
David Lieberman
What happens when you turn a ring upside-down?
February 24
The Matlis Duality Functor. Who is she? What does it do? What can we use it for? These questions will be answered today! The Matlis duality is a duality between Noetherian and Artinian modules. In essence, you take a module and turn it upside-down. This duality has important applications in and uses when it comes to local cohomology. We will define the Matlis duality functor, and discuss some of its properties and uses.
Su Ji Hong
Understanding path algebras and root systems through quivers of type A_n.
March 3
The directed graphs, or quivers, appear in many theories in mathematics including graph theory (duh), cluster algebras, representation theory, Lie algebras, and, etc. In this talk, we focus on path algebras of quivers and root systems. We will define basic definitions for path algebras and root systems. We will look at an example of type A_n quiver.
Frank Zimmitti
The Geometry of Varieties in Codimension 1
March 10
We start the first part of a two week talk by discussing invariants of a variety determined by its subvarieties of codimension 1. In particular, we will discuss the Picard group and divisor class group and when these groups are isomorphic.
Jake Kettinger
Exploring the Wonderful World of Divisors
March 17
We pick up where Frank left off and talk about the canonical divisor for a curve and the adjunction formula. We then talk about Riemann-Roch and how divisors can be used in geometry and algebra, including in the study of elliptic curves. We then finish by talking about linear systems of divisors and Cayley-Bacharach.
The above is an illustration of Pascal's theorem, which can be proven using the Cayley-Bacharach theorem. No matter where you move the six points A1, A2, A3, B1, B2, and B3, the points X, Y, and Z will always be collinear!
Michael DeBellevue
Computing Ext 3 Different Ways
March 24
A common theme in algebra is to first try to understand simple objects, and then understand how the fit together to form more complicated ones. More precisely, modules M and N "fit together" if there is an exact sequence between them. In this talk, I'll introduce the ext functor as equivalence classes of exact sequences. I will then discuss how ext is typically computed in practice, using projective resolutions. Finally, specializing to the case of ext(k,k) for k the residue field of a local ring, I will discuss how ext(k,k) is generated by a certain Lie subalgebra known as the homotopy lie algebra.
Erica Hopkins
BGG Correspondence
April 7
Today we start the first part of a two week talk about the BGG correspondence which is an equivalence of bounded derived categories of graded modules of the polynomial ring and that of graded modules over the exterior algebra. That sounds really technical, but essentially the important thing to keep in mind is that under this correspondence we can relate a complex of graded modules over the polynomial ring to a complex of graded modules over the exterior algebra and talk about some nice properties of this relationship. In this first part of the two part series, we will define the functors that make up this correspondence and spend most of our time doing examples.
Eric Hopkins
BGG Correspondence (Part 2)
April 14
After a brief foray into derived category land, we'll wrap up our discussion of the BGG correspondence and some important related result.
20 Minute Talk Day: Michael DeBellevue
Algebras with Finite Off-Diagonal Deviations
April 21
I will introduce Koszul algebras, which are characterized by their graded deviations vanishing off the diagonal. Avramov and Peeva proved that if R's deviations vanish off the diagonal above 1, then R must be a Koszul algebra tensored with a regular ring. Ferraro conjectured that if R's deviations vanish off the diagonal above 2, then R must be a Koszul algebra tensored with a complete intersection. In this talk I will discuss a counterexample of this conjecture that I discovered, and my proof of a modified form of this conjecture.
20 Minute Talk Day: David Lieberman
Do You Believe in Magic?
April 21
A magic square is a square array of non-negative integers such that the rows and columns all sum to the same value. In this talk, using the power of Frobenius, Cohen-Macauly, and the a-invariant, we show that the number of magic squares for a fixed row/column size r and a fixed row/column sum n is a polynomial in n.
