Department of

Mathematics


Seminar Calendar
for events the day of Thursday, March 5, 2020.

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Thursday, March 5, 2020

11:00 am in 241 Altgeld Hall,Thursday, March 5, 2020

Potential automorphy of Galois representations into general spin groups

Shiang Tang (Illinois Math)

Abstract: Given a connected reductive group $G$ defined over a number field $F$, the Langlands program predicts a connection between suitable automorphic representations of $G(\mathbb A_F)$ and geometric $p$-adic Galois representations $\mathrm{Gal}(\overline{F}/F) \to {}^LG$ into the L-group of $G$. Striking work of Arno Kret and Sug Woo Shin constructs the automorphic-to-Galois direction when $G$ is the group $\mathrm{GSp}_{2n}$ over a totally real field $F$, and $\pi$ is a cuspidal automorphic representation of $\mathrm{GSp}_{2n}(\mathbb A_F)$ that is discrete series at all infinite places and is a twist of the Steinberg representation at some finite place: To such a $\pi$, they attach geometric $p$-adic Galois representations $\rho_{\pi}: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GSpin}_{2n+1}$. In this work we establish a partial converse, proving a potential automorphy theorem, and some applications, for suitable $\mathrm{GSpin}_{2n+1}$-valued Galois representations. In this talk, I will explain the background materials and the known results in this direction before touching upon the main theorems of this work.

3:00 pm in 347 Altgeld Hall,Thursday, March 5, 2020

Fulton's Conjecture, Extremal Rays, and Applications to Saturation

Joshua Kiers   [email] (University of North Carolina, Chapel Hill)

Abstract: We begin by recalling a conjecture of Fulton on Littlewood-Richardson coefficients and discussing two generalizations. With a little lemma from algebraic geometry, we find ourselves on the way to naming the extremal rays of the "eigencone'' of asymptotic solutions to the branching question in representation theory of semisimple Lie groups. After giving an algorithm for finding all such extremal rays, generalizing some prior work joint with P. Belkale, we report on progress on the saturation conjecture for types $D_5$, $D_6$, and $E_6$.

4:00 pm in 245 Altgeld Hall,Thursday, March 5, 2020

Plane Trees and Algebraic Numbers

George Shabat (Russian State University for the Humanities and Independent University of Moscow)

Abstract: The main part of the talk will be devoted to an elementary version of the deep relations between the combinatorial topology and the arithmetic geometry. Namely, an object defined over the field of algebraic numbers, a polynomial with algebraic coefficients and only two finite critical values, will be associated to an arbitrary plane tree. Some applications of this construction will be presented, including polynomial Pell equations and quasi-elliptic integrals (going back to N.-H. Abel). The relations with finite groups and Galois theory will be outlined. At the end of the talk the possible generalizations will be discussed, including the dessins d'enfants theory initiated by Grothendieck.