CARS
This semester, CARS was organized by Ben Drabkin and Josh Pollitz
Andrew Conner
Asymptotic Invariants of Symbolic Powers, Part I: Resurgence
April 18
This week and next I'll be speaking on two invariants of symbolic powers introduced by Bocci and Harbourne in 2009. This week, we'll look at a quantity called the resurgence, which has an upper bound stemming from the Ein-Lazarsfeld-Smith containment that Ben and Eric presented a few weeks ago. We'll give a lower bound on the resurgence and use the two bounds to examine whether the Ein-Lazarsfeld-Smith containment can be improved.
Nick Packauskas
Homotopy Lie Algebras
April 11
After covering a couple more results in basic theory of Lie algebras, we will construct an Ext algebra of a local ring using these techniques. We will then explore the structure of the Lie algebra and the homological information it contains.
Josh Pollitz and Nick Packauskas
Tate Resolutions and Lie Algebras
April 4
Part 1: Last week, we introduced Ext-algebras as originating from the cohomology of a DG algebra. In the first part of the talk, we introduce Tate resolutions which give us a handle of identifying the algebra structure for the Ext-algebra of the residue field for a local ring. Part 2: The technique of using noncommutative structures to study free resolutions has led to a vast array of research. In this part of the talk, we will introduce some noncommutative machinery that will provide a different perspective on the Ext algebra of a local ring. This construction will be exploited in next week’s talk to provide some hands-on computations and connections to more traditional commutative algebra.
Josh Pollitz
DG Algebras and the Algebra Structure on Ext
March 28
The main goal of this talk is to discuss one way Ext can be realized as a graded algebra. In the following weeks, Nick (and possibly I, time permitting) will discuss the algebra structure of certain Ext modules. To do this, we first will introduce DG algebras and give several examples of them. Then we will use this framework to view certain complexes as DG algebras. This DG algebra structure will be exploited to obtain the graded algebra structure on Ext, in general, that will be investigated in the coming weeks.
Ben Drabkin
Birational Geometry and Symbolic Powers - Part 3: Multiplier Ideals and Uniform Bounds
March 14
This talk will introduce multiplier ideals and present Ein, Lazarsfield, and Smith's proof of uniform bounds on the containment of symbolic powers of ideals in ordinary powers.
Eric Canton and Ben Drabkin
A Lil' Sketch of Birational Geometry - Part 2
March 7
In this talk, we will discuss divisors and the resolutions of singularities, and then give a brief introduction to symbolic powers.
Eric Canton
A Lil' Sketch of Birational Geometry - Part 1
February 28
Birational geometry is a central branch of algebraic geometry with numerous applications to commutative algebra, representation theory (e.g. Milen Yakimov's talks last week), and particle physics (e.g. mirror symmetry and the geometry of Calabi-Yau manifolds). Working towards Ein, Lazarsfeld, and Smith's application of birational geometry to the containment of symbolic powers, in this talk I will attempt to build intuition about two central tools in birational geometry: blowing-up and canonical bundles.
Mohsen Gheibi
Dimension for Subcategories of Modules
February 21
In this talk, I will go over the section 5 (the last section) of Takahashi's lecture. More precisely, we will define the dimension of a subcategory of modules and then show that if R has finite representation type then the dimension of the category of maximal CM modules is zero. Also, we will show that the converse holds if R is complete and Gorenstein.
Amadeus Martin
Improving theorem 3.7
February 14
I will be going over section 4 of the notes by Takahashi. It will be a continuation of Erica's talk, in particular we will improve a result from last week.
Erica Musgrave
Finite Generation in the Module Category
February 7
I will be going over section 3 of the notes by Takahashi. I will introduce some notation and definitions, and then prove some more results about Tor and Ext and some results about CM local rings.
Taran Funk
Annihilators and Nonfree Loci
January 31
We will continue on in the notes by Takahashi. I'll try to remind everyone of any results/definitions given by Eric last week as I use them.
Eric Hopkins
Uniform Annihilation of Tor (and Ext)
January 24
I'll kick things off in the notes, talking about Annihilation (no, not the Natalie Portman movie) of some Tor (and Ext) modules over Cohen-Macaulay Rings.
Nick Packauskas
An Introduction to MCM Modules
January 17
Over the past several decades, there has been a lot of work dedicated to the study of Maximal Cohen Macaulay (MCM) modules. In this talk, I hope to give some background about why they are an important area of research and set the groundwork for our group to read through Ryo Takahashi’s notes on MCM modules.
