Keller 403

Abstract:

formalization of multivariable calculus in Mizar. We are now formalizing

multivariable calculus with the aim of formalizing differential

manifolds, partial differential equations, and numerical analysis. In

the formalization of multivariable calculus, the handling of

higher-order partial derivatives tends to be more troublesome than in

the one variable case, but this can be alleviated somewhat by

introducing the Fréchet derivative in Banach space. I would like to give

an overview of the Mizar project and discuss topics ranging from the

definition of higher-order partial derivatives in normed spaces to

formalization of the implicit and inverse function theorems.</p>

Speaker: Daniel Erman (University of Wisconsin)

Title: Ultraproducts, Hilbert’s Syzygy Theorem, and Stillman’s

Conjecture

When and where: 3-3:50pm, December 7, in Keller 403

Abstract: Hilbert’s Syzygy Theorem is a classic finiteness result about

a construction in algebra known as a free resolution. Stillman once

proposed an analogue of Hilbert’s result, which involved potentially

considering polynomials in infinitely many variables. Stillman’s

Conjecture was recently solved, and perhaps the simplest proof is based

upon a novel use of an ultraproduct. I’ll give an expository overview

of the history of Stillman’s Conjecture (very little algebraic

background will be assumed), and then explain how and why ultra products

came to play such a key role.

Title: Klingen Eisenstein series and symmetric square $L$-functions

Abstract: It is well-known in number theory that some of the deepest results come in connecting complex analysis in the form of $L$-functions with algebra/geometry in the form of Galois representations/motives. In this talk we will consider this for a particular case. Let $f$ be a newform of weight $k$ and full level. Associated to $f$ one has the adjoint Galois representation and the symmetric square $L$-function. The Bloch-Kato conjecture predicts a precise relationship between special values of the symmetric square $L$-function of $f$ with size of the Selmer groups of twists of the adjoint Galois representation. We will outline a result providing evidence for this conjecture by lifting $f$ to a Klingen Eisenstein series and producing a congruence between the Klingen Eisenstein series and a Siegel cusp form with irreducible Galois representation. time permitting, we will discuss a modularity result for a 4-dimensional Galois representation that arises from the congruence and studying a particular universal deformation ring. This is joint work with Kris Klosin.