CARS
This semester, CARS was organized by Erica Hopkins and Amadeus Martin
Eric Hopkins
Matrix Factorizations
December 11
A matrix factorization is a way of writing one matrix as a product of others. We'll investigate a particular class of matrix factorizations and a classical result from Eisenbud, then generalize his result. And if you don't like matrices, you're in luck -- never will an actual matrix appear in the talk.
Amadeus Martin
Construction of the Derived Category
December 4
I'll be doing some review of the last two CARS talks. When you calculate Exti(M,N), you use a projective resolution P, of M as an intermediate object. However M isn't equal to P, it's not even the same type of object. When attempting to resolve these issues by stepping into the homotopy category, you no longer have an abelian category. There is however a triangulated structure with several interesting tools. Today we'll be using some of these tools to develop the idea of a Verdier quotient. With said quotient we can create the derived category where most of the issues involving M and its resolutions vanish. This will also allow us to gain a different perspective on what exactly is Exti(M,N).
Rachel Diethorn (Special Guest)
Koszul Homology of Quotients by Edge Ideals
November 20
We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides a class of Koszul algebras whose Koszul homology algebras satisfy this property. We also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.
Laila Awadalla
The Category of Chain Complexes
November 13
Last week we tried to study a module using derived functors. For this we needed to use the module's (injective and projective) resolutions in the category of chain complexes, where they are not unique. An equivalence relation that fixed this took us to the homotopy category. This category is not abelian, but it is triangulated; in this talk we describe its exact triangles and how to work with them.
Jake Kettinger
Abelian Categories and Derived Functors
November 6
Abelian categories are important for their many useful properties, such as having kernels, cokernels, and short exact sequences. But when we want to study derived functors from the category of R-modules (such as Ext and Tor), we need to pass over to a category that is not abelian. We will discuss the motivation for studying these triangulated categories and set up the discussion for what useful tools these categories have.
Michael DeBellevue
Lattice Polytopes Part III: Brion's Formula
October 30
We finish the series on lattice polytopes by proving Brion's formula, a remarkable identity on infinite series. The original proof required some complicated algebraic geometry, but we will present a proof using injective resolutions and some counting arguments.
Ben Drabkin
Canonical Modules, Injective Resolutions, and the Dualizing Complex for Semigroup Rings: Not as Horrifying as it Sounds
October 23
Today's talk will delve into some commutative algebra topics which, in the context of semigroup rings, are extremely pretty. In order to set up for next week's talk, this talk will introduce canonical modules, injective resolutions, and the dualizing complex in the context of semigroup rings.
Su Ji Hong
Ehrhart Polynomial
October 16
I will continue my discrete talk from earlier this month in CARS by talking more about the lattice polytopes (this statement is mostly false. I'm not using anything from that talk...). A (convex) lattice polytope is a polytope (higher dimensional polygon) whose vertices are in ℤd. The Ehrhart polynomial EP(m) of a lattice polytope P counts the number of lattice points in the mth dilation of P. We'll prove that EP(m) is a polynomial of degree dim(P) using the Hilbert function of a semigroup ring.
Erica Hopkins
A Link Between Worlds
October 9
We will be finishing up our sequence of linkage talks today by linking together all of the tools we have been learning. We will use Koszul homology and linkage to show a few more lemmas. Finally we will discuss a result that involves saying some nice things about the algebras associated to an ideal if that ideal is in the linkage class of a complete intersection.
Matthew Bachmann
It's Dangerous to Go Alone
October 2
Take Koszul homology with you. Koszul homology is a useful tool for studying the different algebras associated to an ideal. In this talk, I will give an introduction to the construction of the Koszul complex. Then we will prove some results about when the Koszul homology is Cohen-Macaulay. By the end of next week, we will use Koszul homology and linkage to say a great deal about the algebras associated to an ideal.
David Lieberman
A Link to the Past
September 25
(This talk will be pretty gosh darn accessible) In this talk, we will give a brief reminder of Cohen-Macaulay and Gorenstein rings. Then we present the main focus of the talk: the concept of ideal linkage. We present the notion of two ideals being linked, explore some examples, and present some useful qualities and interesting questions about the deeper relation between two linked ideals.
Taran Funk
Non-Noetherian Cohen-Macaulay Rings
September 18
(This talk should be accessible to everyone) We will start with a reminder of the definition and properties of a (Noetherian) Cohen-Macaulay ring. We will then see how some of our basic intuition is off when we step into the wild land of Non-Noetherian rings. Once in this abyss we will see the attempts to make a workable definition for a (non-Noetherian) Cohen-Macaulay ring, and the pitfall the current definition has.
Frank Zimmitti
Projective Varieties
September 11
We will expand on last week's talk by introducing projective varieties and some ways to study them. We will discuss some invariants of varieties and introduce birational equivalence, a central notion in algebraic geometry.
Andrew Connor
Algebraic Geometry and Powers of Ideals
September 4
Today I'll kick off CARS for the semester with a talk aimed at our newer folks. We'll discuss some algebraic geometry 101, including the ideal-variety correspondence and how that lets us unify techniques from algebra, geometry, and topology. We'll also give an application to taking powers of ideals -- the ideal-variety correspondence is wonderfully useful in general, but it's often a stumbling block when taking powers of ideals in polynomial rings.
