Student Seminar

Algebraic Geometry, Commutative Algebra, and Number Theory

Summer 2015

Seminar Info

Meetings

Friday, 12:00 - 1:00
Room: LeConte 312

Organizer: Blake Farman
Faculty Sponsor: Matt Ballard

Abstracts

Speaker

Tyler Lewis

Infinite Free Resolutions

Since Hilbert introduced the idea of associating a free resolution to a module it has become an integral part of the study of modules. Many properties of a module are encoded in its resolution. Infinite free resolutions are very common; and compared to finite free resolutions we know very little about the properties of infinite free resolutions. Many of the techniques used to study finite free resolutions do not apply to the infinite case. One approach to studying infinite free resolutions is to look at the sequence of ranks of the free modules in the resolutions, i.e. the sequence of Betti numbers. This talk will focus on results and open questions based on the sequence of Betti numbers and the generating function for the Betti numbers.

Cameron Atkins

DG-Algebras

DG-algebras are a powerful tool in commutative algebra. In this seminar we will cover topics discussed in "A Somewhat Gentle Introduction to Differential Graded Commutative Algebra" by Kristen Beck and Sean Sather-Wagstaff.

Cameron Atkins

Stanley-Reisner Rings and Totally Reflexive Modules

Totally reflexive modules have attractive homological properties. However, examples are difficult to find, and in some cases might not even exist. Stanley-Reisner rings are a nice family of rings in the sense that most of the algebraic invariants of these rings are coded combinatorically, and thus easy to compute. In this seminar I will present my findings on totally reflexive modules in the case of Stanley-Reisner rings.

Blake Farman

Fully Faithful Abstract Nonsense: Representable Functors and The Yoneda Embedding

Named for Nobuo Yoneda (1930 - 1996), a Japanese mathematician and computer scientist, the Yoneda Lemma is the starting point for the view that objects in a category can readily be determined by the morphisms into that object. In particular, it is the cornerstone of Grothendieck's method of studying schemes by way of the so called Functor of Points. We will go through the Yoneda Lemma and some applications.

Harsh Mehta

Binomial coefficients mod p

We will look at some properties of binomial coefficients mod primes . We prove two theorems by Lucas and Kummer and then about their generalizations using results by Granville.