Department of

Mathematics


Seminar Calendar
for events the day of Thursday, March 14, 2019.

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Thursday, March 14, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 14, 2019

Extremal primes for elliptic curves without complex multiplication

Ayla Gafni (Rochester Math)

Abstract: Fix an elliptic curve $E$ over $\mathbb{Q}$. An ''extremal prime'' for $E$ is a prime $p$ of good reduction such that the number of rational points on $E$ modulo $p$ is maximal or minimal in relation to the Hasse bound. In this talk, I will discuss what is known and conjectured about the number of extremal primes $p\le X$, and give the first non-trivial upper bound for the number of such primes when $E$ is a curve without complex multiplication. The result is conditional on the hypothesis that all the symmetric power $L$-functions associated to $E$ are automorphic and satisfy the Generalized Riemann Hypothesis. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in recent work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes have a very small Sato-Tate measure.

12:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

SL(3,C), SU(2,1) and the Whitehead link complement

Pierre Will (Institut Fourier)

Abstract: In this talk, I will explain how it is possible to construct interesting geometric structures modelled on the boundary at infinity of the complex hyperbolic 2-space. In particular, I will describe examples of hyperbolic 3-manifolds that appear this way. This talk is based on joint works with Antonin Guilloux, and John Parker.

12:30 pm in 464 Loomis,Thursday, March 14, 2019

A proposal for nonabelian mirrors in two-dimensional theories

Eric Sharpe (Virginia Tech)

Abstract: In this talk we will describe a proposal for nonabelian mirrors to two-dimensional (2,2) supersymmetric gauge theories, generalizing the Hori-Vafa construction for abelian gauge theories. By applying this to spaces realized as symplectic quotients, one can derive B-twisted Landau-Ginzburg orbifolds whose classical physics encodes quantum cohomology rings of those spaces. The proposal has been checked in a variety of cases, but for sake of time the talk will focus on exploring the proposal in the special case of Grassmannians.

2:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

Generalized Derivatives

Alastair Fletcher (Northern Illinois University)

Abstract: Quasiregular mappings are only differentiable almost everywhere. There is, however, a satisfactory replacement for the derivative at points of non-diffferentiability. These are generalized derivatives and were introduced by Gutlyanskii et al in 2000. In this talk, we discuss some recent results on generalized derivatives, in particular the question of how many generalized derivatives there can be at a particular point, and explaining how versions of the Chain Rule and Inverse Function Formula hold in this setting. We also give some applications to Schroeder functional equations.

3:00 pm in 243 Altgeld Hall,Thursday, March 14, 2019

To Be Announced

Satya Mandal (University of Kansas)

Abstract: Title: Splitting property of projective modules, by Homotopy obstructions Speaker: Satya Mandal, U. of Kansas \noindent{\bf Abstract:} Follow the link: http://mandal.faculty.ku.edu/talks/abstractIllinoisMarch19.pdf Alternate version: The theory of vector bundles on compact hausdorff spaces $X$, guided the research on projective modules over noetherian commutative rings $A$. There has been a steady stream of results on projective modules over $A$, that were formulated by imitating existing results on vector bundles on $X$. The first part of this talk would be a review of this aspects of results on projective modules, leading up to some results on splitting projective $A$-modules $P$, as direct sum $P\cong Q\oplus A$. % Our main interest in this talk is to define an obstruction class $\varepsilon(P)$ in a suitable obstruction set (preferably a group), to be denoted by $\pi_0\left({\mathcal LO}(P) \right)$. Under suitable smoothness and other conditions, we prove that $$ \varepsilon(P)\quad {\rm is~trivial~if~and~only~if}~ P\cong Q\oplus A $$ Under similar conditions, we prove $\pi_0\left({\mathcal LO}(P) \right)$ has an additive structure, which is associative, commutative and has n unit (a "monoid").