Join us for a colloquium presentation this Friday (09/23) at 3:30pm in Keller 302.
The speaker will be our colleague Yash Lodha.
—-Title: Adventures in the land
Abstract: The groups of homeomorphisms of the real line and the circle exhibit a remarkable subgroup structure, and consequently, a plethora of topological, algebraic and combinatorial phenomena. In this talk I shall provide some historical motivation to study such groups, and describe some concrete examples that illustrate the theory. If time permits, I will present some recent advances in the area.
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
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.