CARS

Corona Virus

How to derail a semester of CA talks?

March 18-End of Semester

School closed and CARS did not continue via Zoom.

Jake Kettinger

Automorphism Groups of Curves and Surfaces

March 11

Automorphism groups are great! They tell us about symmetry and let us draw lots of pretty pictures. We explore how the automorphism groups of curves change when we punch more and more holes in them. Then we become demolitions experts and blow up points on surfaces! How does that change the automorphism group? Our pictures will tell us! (Root systems will also tell us.)

Michael DeBellevue

Homological Tools to Detect Complete Intersections

March 4

A complete intersection ring corresponds to the geometrical situation of a d-dimensional algebraic surface sitting in n-dimensional space and cut out by n−d polynomial equations. In this talk I'll introduce the deviations of a ring, which is a numerical invariant that can be used to detect when a ring is a complete intersection. Then I will introduce acyclic closures and minimal models, which are two kinds of differential-graded algebras that can be used to compute the deviations of a ring.

David Lieberman

What You Should Know About Integer-Valued Polynomials

February 26

In the year 2000, Rings of Integer-Valued Polynomials was added to the AMS Mathematics Subject Classification. In this talk, we present the concept of integer-valued polynomials. Most of this talk will be concerned with the algebraic properties of the ring of polynomials with rational coefficients which take integer values on integer arguments. We will show some introductory results for this ring, and generalize to the case of an arbitrary integral domain.

Amadeus Martin

Chain Complexes, DGS and CGAs

February 19

This talk has 3 parts. We'll begin with a quick review chain complexes and some of the tools of homological algebra, in particular homology and mapping cones. The second part, we'll introduce differentially graded algebras (DGAs) and their modules. We'll sketch the construction of its derived category. Finally, if time allows, we'll simultaneously generalize DGAs and chain complexes through Curved DGAs. Modules over these don't have homology, yet a sort of a derived category can still be constructed.

Eric Hopkins

Gorenstein Rings - Part 2

February 12

Ubiquity -- the fact of appearing everywhere or of being very common. In 1997, Craig Huneke gave a talk celebrating Hyman Bass, then published a paper based on this talk, "Hyman Bass and Ubiquity: Gorenstein Rings". Today we talk about the Ubiquity of Gorenstein Rings, and some of why they are nice and/or important to study.

Eric Hopkins

Gorenstein Rings - Part 1

February 5

Ubiquity -- the fact of appearing everywhere or of being very common. In 1997, Craig Huneke gave a talk celebrating Hyman Bass, then published a paper based on this talk, "Hyman Bass and Ubiquity: Gorenstein Rings". Today we'll go through the historical portion of the paper, with many definitions, classical problems, and tying ideas like canonical modules and Matlis duality from Taran's talk to the notion of a Gorenstein ring. This will set us up to tackle the Ubiquity of Gorenstein rings in the following week.

Taran Funk

Canonical Module

January 29

We will discuss what the canonical module is and why this is an object people study. This will involve a lot of setup, so get ready for some local cohomology.

Su Ji Hong

Lee^2 Conjecture on Geometric Description of C-Vectors.

January 22

It has been shown that for an acyclic quiver, the set of d-vectors, the set of c-vectors, and the set of real Schur roots coincide. None of these objects are geometric though. In an effort to give a geometric description, Kyu-Hwan Lee and Kyungyong Lee conjectured that the set of roots obtained from a non-self-crossing admissible curve coincide with the set of c-vectors for an acyclic quiver. I almost proved this conjecture for acyclic quivers of finite type. In this talk, I will describe these objects and talk about the case I have not yet figured out.