# Algebra and its Applications (Fall 2010)

Welcome to the Algebra and its Applications seminar home page, which has been copied from Ettore's old webpage.

The seminar is organized by Amod Agashe. Please email agashe@math.fsu.edu to contact me. All entries below with a question mark around the names of the speakers are tentative.

## Schedule

Sept 9 Sept 16 Robert Underwood (Auburn University) Haar Measure on the Additive Group of Adeles Saikat Biswas (FSU) Neron Models, Flat Cohomology and Visibility Saikat Biswas (FSU) Neron Models, Flat Cohomology and Visibility II Randy Heaton (FSU) Congruences Between Spaces of Cuspidal Modular Forms Chris Sinclair (University of Oregon) Random Matrices and Heights of Polynomials Nelson Colon (FSU) Modular forms/functions and their relation with arithmetic functions in number theory Nelson Colon (FSU) Modular forms/functions and their relation with arithmetic functions in number theory II Ugo Bruzzo (SISSA and U. Penn) Uhlenbeck-Donaldson Compactification for Framed Sheaves James Fullwood (FSU) Geometrization of the Weak Coupling Limit of F-theory No seminar: Veteran's day Gregory (Ivan) Dungan (FSU) The Classical Approach to Stacks Followed by the Modern Homotopic Approach No seminar: Thanksgiving Vivek Pal (FSU) Periods of quadratic twists of elliptic curves

## Abstracts

### Randy Heaton: Congruences Between Spaces of Cuspidal Modular Forms

A prime p is said to be a congruence prime linking two spaces of cuspidal modular forms X and Y if there exist two cusp forms, $f \in X$ and $g \in Y$, such that the Fourier coefficients of f and g are all congruent modulo p. We describe research to date on the problem and discuss a novel approach.

### Chris Sinclair: Random Matrices and Heights of Polynomials

A height of a polynomial is a measure of its complexity. Examples of heights include norms of coefficient vectors and Hardy norms on the unit circle. Taking p to 0, the Hardy p-norms converge to another height function: the Mahler measure. Mahler measure is convenient to work with since it is multiplicative. The set of coefficient vectors of degree N real (complex) polynomials whose Mahler measure is at most 1 is a compact subset of R^{N+1} (C^{N+1}). In this talk I will show how choosing a polynomial uniformly from this region induces a Pfaffian (determinantal) point process and provide and demonstrate how techniques from random matrix theory produce an asymptotic estimate for the number of integer polynomials of fixed degree and bounded Mahler measure as this bound approaches infinity.

### Ugo Bruzzo: Uhlenbeck-Donaldson Compactification for Framed Sheaves

We introduce the moduli spaces of framed sheaves. A reason for being interested in these objects is that under suitable conditions they are nice moduli spaces (quasi-projective, smooth and fine) and provide desingularizations of moduli spaces of framed instantons. We shall study a "partial compactification" for these moduli space, called the Uhlenbeck-Donaldson compactification, and relate it to the moduli space of "ideal" framed instantons.

### James Fullwood: Geometrization of the Weak Coupling Limit of F-theory

F-theory compactified on an elliptically fibered (complex dimensional)fourfold is conjecturally equivalent to type IIB string theory on the base. The realization of this duality is via the weak coupling limit, which recently has been distilled into a problem of pure geometry. In this talk we will explore the geometry of the weak coupling limit of F-theory, as well as Chern class identities inspired by the tadpole relations coming from the equivalence of type IIB and F-theory.

### Gregory (Ivan) Dungan: The Classical Approach to Stacks Followed by the Modern Homotopic Approach

Sheaves have been a tremendous catalyst in broadening the field of Algebraic Geometry and in an effort to enrich the field even further, a cohomology theory for sheaves was developed. Unfortunately, due to the coarseness of the Zariski topology in which sheaves are defined, the cohomology theory produced a lot of trivialities. In order for a cohomology theory of sheaves to be of any use, a finer topology would need to be defined in which a coherent sheaf theory could exist. This can be done and has a nice generalization to categories that we call sites which have a type of topological structure that resembles open coverings. Now, some of these "generalized" sheaves not only have sections that can be constructed locally, but also the objects of the site can be constructed locally. In order to keep track of this data we will define stacks via descent theory. This descent theory can be overwhelming, especially when one wants to generalize stacks to n-stacks, so there is a much simpler homotopic description that we will discuss.

### Vivek Pal: Periods of quadratic twists of elliptic curves

In this talk I will prove a relation between the period of an elliptic curve and the period of its real and imaginary quadratic twists. This relation is often misstated in the literature.