Department of

Mathematics


Seminar Calendar
for events the day of Thursday, November 16, 2017.

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Thursday, November 16, 2017

11:00 am in 241 Altgeld Hall,Thursday, November 16, 2017

Kloosterman sums and Siegel zeros

James Maynard (Institute For Advanced Study)

Abstract: Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field $F_p$, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

12:30 pm in 222 Loomis,Thursday, November 16, 2017

Quantum gravity and quantum chaos

Stephen Shenker (Stanford)

Abstract: One hallmark of chaos is sensitive dependence to initial conditions, the “butterfly effect.” We will discuss recent advances in our understanding of the quantum butterfly effect and its connection to the quantum physics of black holes. We will discuss a universal bound on the rate of development of quantum chaos motivated by these developments. Then we will briefly describe recent work on a connection between the late time behavior of black holes and the dynamics of random matrices.

2:00 pm in 243 Altgeld Hall,Thursday, November 16, 2017

Entropic uncertainty relations via Noncommutative $L_p$ norms

Li Gao (UIUC Math)

Abstract: The Heisenberg uncertainty principle states that it is impossible to prepare a quantum particle for which both position and momentum are sharply defined. The first formulation of uncertainty principle using entropy was proved by Hirschman in 1957. Since then entropic uncertainty relations have been obtained for many different scenarios, including some recent advances about generalizations with quantum memory. In this talk, I will present an unified approach to entropic uncertainty relations via noncommutative Lp spaces and von Neumann algebra. The connections to Jones’ subfactor index theory and quantum information theory will also be mentioned. This is a joint work with Marius Junge, and Nicholas LaRacuente.

3:00 pm in 243 Altgeld Hall,Thursday, November 16, 2017

A Counterexample to the Weitzenböck Conjecture in Characteristics p > 2

Stephen Maguire (UIUC Math)

Abstract: Weitzenböck’s Theorem states that a representation $\mu: \mathbb{G}_a \to \mathrm{GL}(V_n)$ has a finitely generated ring of invariants $k[X]^{\mathbb{G}_a}$ if the field $k$ is an algebraically closed field of characteristic zero. In this talk, we produce a representation $\mu : \mathbb{G}_a \to \mathrm{GL}(V_6)$ over an algebraically closed field $k$ of characteristic $p > 2$ such that the ring of invariants $k[x_1, \dots , x_6]^{\mathbb{G}_a}$ is not a finitely generated $k$-algebra. In order to do this, we reduce this problem to a curve counting problem, and then use this reduction to further reduce this problem to a problem about the support of a bi-graded ring.

4:00 pm in 245 Altgeld Hall,Thursday, November 16, 2017

Primes with missing digits

James Maynard (Institute For Advanced Study, Princeton)

Abstract: Many famous open questions about primes can be interpreted as questions about the digits of primes in a given base. We will talk about recent work showing there are infinitely many primes with no 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most $X^{1-c}$ elements less than $X$) which is typically very difficult. The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.