Logic seminar: Kazuhisa Nakasho (Yamaguchi University)
Nov 16 @ 3:00 pm – 4:00 pm

<p>Title: Formalization of multivariable calculus in Mizar</p>


<p>In this presentation, I will report on the progress of the
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>

Logic seminar: Daniel Erman
Dec 7 @ 3:00 pm – 4:00 pm

Speaker: Daniel Erman (University of Wisconsin)
Title: Ultraproducts, Hilbert’s Syzygy Theorem, and Stillman’s
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.

First Day of Instruction
Jan 9 all-day
Number Theory Seminar – Jim Brown @ Keller 301
Feb 2 @ 4:30 pm – 5:30 pm
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.


Last day of instruction
May 3 all-day