An elliptic curve over the complex numbers can be expressed as modulo a lattice . The problem of finding meromorphic functions on becomes the same as looking for meromorphic functions on which are periodic with respect to .
We will outline these ideas emphasizing the roles of the Weierstrass special functions , , the Riemann function and the Weierstrass -function. We will then introduce how this story ports to the -adic setting, due to Tate.
Correction: The uniformization theory of elliptic curves seminar is Thursday, Feb 22nd
Feb. 22, 2018 11am (MATH 350)
Geometry/Analysis
Tam Nguyen Phan (Max Planck)
X
Aspherical manifolds are manifolds that have contractible universal covers, or equivalently, those that have trivial higher homotopy groups. The homotopy types of such manifolds are determined by their fundamental groups. Examples include nonpositively curved manifolds (by Cartan-Hadamard), and in particular, the rich class of locally symmetric spaces of non compact type. However, it is in general not easy to construct aspherical manifolds. e.g. the obvious connect sum operation usually ruins asphericity. One way to construct aspherical manifolds is to use the "reflection group trick". I will describe how to construct examples of aspherical manifolds by applying the reflection group trick to the Borel-Serre compactifications of locally symmetric spaces. This will be an example-oriented talk and everything will be explained via examples.
Examples of aspherical manifolds, the reflection group trick and Borel-Serre compactifications Sponsored by the Meyer Fund
Feb. 22, 2018 2pm (Math 350)
Functional Analysis
Dana Williams (Dartmouth College)
X
An important way to construct examples of -algebras is to form a groupoid -algebra from a locally compact groupoid, in analogy to the classical group--algebra construction. To do this, we need an analogue of a Haar measure called a “Haar system”. In contrast to the group case, there are a number of interesting open questions surrounding the existence and uniqueness of Haar systems. I’ll briefly discuss these and focus on the relation to the important notion of “groupoid equivalence”. In particular, I’ll talk about a recent result showing that if a groupoid is equivalent to one with a Haar system, then it also possesses a Haar system.
Haar Systems on Equivalent Groupoids Sponsored by the Meyer Fund
Feb. 22, 2018 3pm (MATH 350)
Probability
Koushik Ramachandran (Oklahoma State University)
X
A polynomial lemniscate is a curve in the complex plane defined by . Erdos, Herzog, and Piranian posed the extremal problem of determining the maximum length of a lemniscate when p is a monic polynomial of degree n. In this talk, we will look at the length and topology of a random lemniscate whose defining polynomial has independent Gaussian coefficients. When the polynomial is sampled from the Kac ensemble we show that the expected length approaches a nonzero constant as . Concerning the connected components of a random lemniscate, we prove that the average number of them is asymptotically n and that there is a positive probability (independent of n) of a giant component. Based on joint work with Erik Lundberg.
Correction: The uniformization theory of elliptic curves seminar is Thursday, Feb 22nd