Algebra and Number Theory Seminar Winter 2012

January 17, 2012

How often is a subgroup of GLn(Z) thin?

Elena Fuchs, Berkeley

Given a finitely generated subgroup G of GLn(Z), Salehi-Golsefidy-Varju have recently given necessary and sufficient conditions for the family of graphs associated to finite quotients of G to be an expander family. These results have found unexpected applications in number theory. Their most interesting applications involve groups which are thin, or of infinite index in their Zariski closure. It is therefore of interest to understand both how generic such a group is, and how to determine whether a group is thin in the first place. In this talk we will discuss our approach to these questions. This is joint work with I. Rivin.

January 24, 2012

An arithmetic introduction to K3 surfaces

Alex Beloi, UCSC

People encounter K3 surfaces in a variety of ways. In this talk we'll spend some time on K3 surfaces that arise as resolutions of Kummer surfaces associated to hyperelliptic curves. We'll try to give some reasons as to why these objects are of arithmetic interest, as well as touch on some ways in which they appear in other fields.

January 31, 2012

Fourier-Mukai partners of K3 surfaces in positive characteristic

Martin Olsson, Berkeley

I will discuss my recent paper with Max Lieblich in which we extend classical results on Fourier-Mukai partners of K3 surfaces to positive characteristic. This work is based on a derived category version of the classical Torelli theorem, and a study of the so-called Mukai motive of a K3 surface. Some arithmetic applications will also be discussed.

February 14, 2012

Introduction to Hilbert modular forms

Mitchell Owen, UCSC

In this talk we will give an introduction to Hilbert modular forms following Bruinier's chapter in "The 1-2-3 of modular forms".

February 16, 2012

Numerical experiments with ATR points

Marc Masdeu, Columbia University

In his book on rational points on modular elliptic curves, Henri Darmon gives a construction of a supply of algebraic points predicted by the Birch and Swinnerton-Dyer conjecture, in cases where the Heegner point construction does not work. One of these cases arises with "Almost Totally Real" (ATR) extensions, and Darmon and Logan gathered some numerical evidence supporting the conjecture. However, all the curves for which they construct algebraic points are isogenous to their Galois conjugates, and in that situation one might hope for a variation of the Heegner point construction to still work. In a joint project with Xavier Guitart (Universitat Politecnica de Catalunya), we set ourselves the goal of testing the conjecture in an example for which no other construction of points is available: an elliptic curve of conductor 1 defined over Q(509). In this talk I will describe the construction given by Darmon and explain the problems -- both theoretical and practical -- that we have had to overcome to compute the conjectural points in this "simplest" example.

February 21, 2012

Short Gaps Between Primes and the Selberg Sieve

Daniel Goldston, San Jose State University

I will describe the method of Goldston-Pintz-Yildirim for finding small gaps between consecutive primes. Ultimately the method depends on the Selberg sieve to find weights that are small except in regions rich in numbers with very few prime factors. These sieve weights do an excellent job, but our ability to untangle the primes from these weights is very limited.

March 6, 2012

Introduction to Frobenius Splitting

Cameron Franc, UCSC

In this talk we'll give a short introduction to Frobenius splitting. We will cover some material from Allen Knutson's paper "Frobenius splitting, point-counting, and degeneration", focusing on the connection presented there between Frobenius splitting and point-counting of certain affine hypersurfaces. This talk is aimed at graduate students and is intended as background for Jenna Rajchgot's upcoming lecture in this seminar series.

March 13, 2012

Frobenius splitting and an application to stratifying the Hilbert scheme of points in the plane

Jenna Rajchgot, Cornell

The theory of Frobenius splitting has proven to be a useful tool in representation theory, commutative algebra, and algebraic geometry. I'll introduce the basic notions of Frobenius splitting and present a range of applications. Following this, I'll focus my attention on the Hilbert scheme of points in the plane and discuss its stratification by all "compatibly Frobenius split" subvarieties.