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Thursday, October 17, 2019

11:00 am in 241 Altgeld Hall,Thursday, October 17, 2019

A new approach to bounds for L-functions

Jesse Thorner (University of Florida)

Abstract: Let $L(s)$ be the $L$-function of a cuspidal automorphic representation of $GL(n)$ with analytic conductor $C$. The Phragmen-Lindelof principle implies the convexity bound $|L(1/2)| \ll C^{1/4+\epsilon}$ for all fixed $\epsilon>0$, while the generalized Riemann hypothesis for $L(s)$ implies that $|L(1/2)|\ll C^{\epsilon}$. A major theme in modern number theory is the pursuit of subconvexity bounds of the shape $|L(1/2)| \ll C^{1/4-\delta}$ for some fixed constant $\delta>0$. I will describe how to achieve (i) an unconditional nontrivial improvement over the convexity bound for all automorphic $L$-functions (joint work with Kannan Soundararajan), and (ii) an unconditional subconvexity bound for almost all automorphic $L$-functions (joint work with Asif Zaman).

1:00 pm in 464 Loomis Laboratory ,Thursday, October 17, 2019

Sphere packing, modular bootstrap and extremal functionals

Dalimil Mazac

Abstract: I will prove a new theorem about 2D CFTs: Every unitary 2D CFT must contain a non-trivial Virasoro primary of scaling dimension at most c/8 + 1/2, where c is the central charge. At large c, this is an improvement of the Hellerman bound c/6 + O(1), and is relevant for constraining the spectrum of gravitational theories in AdS3. The proof follows from the modular bootstrap and uses analytic extremal functionals, originally developed in the context of four-point SL(2) conformal bootstrap. In the second part of the talk, I will discuss a surprising connection between modular bootstrap and the sphere-packing problem from discrete geometry. In particular, the above bound on the gap becomes a bound on the sphere-packing density. In 8 and 24 dimensions, this bound is sharp and leads to a solution of the sphere-packing problem in these dimensions, as originally proved by Viazovska et al. The talk will be based on arXiv:1905.01319.

2:00 pm in 347 Altgeld Hall,Thursday, October 17, 2019

Branching Processes Part 1

Peixue Wu (UIUC Math)

Abstract: The first part of this talk is very introductory, I will talk about the basic ideas of branching mechanism (originated from random walk) and some generalizations of the simple branching process, e.g., age-dependent processes, multi-type branching process. ​I will focus on the limit theorem of branching processes.​ ​In the second part, I will talk about the superprocess (which is measure-valued branching processes) and some recent works about it.

4:00 pm in 245 Altgeld Hall,Thursday, October 17, 2019

Orbit Equivalence and Entropy

Hanfeng Li   [email] (University at Buffalo)

Abstract: Entropy is one of the most important numerical invariants for probability-measure-preserving (pmp) actions of countable infinite groups. Orbit equivalence is a fairly weak equivalence relation between pmp actions. In general orbit equivalence may not preserve entropy. A few years ago Tim Austin showed that integrable orbit equivalence between pmp actions of finitely generated amenable groups does preserve entropy. I will introduce a notion of Shannon orbit equivalence, weaker than integrable orbit equivalence, and a property SC for pmp actions. The Shannon orbit equivalence between pmp actions of sofic groups with the property SC preserves the maximal sofic entropy. If a group G has a w-normal subgroup H such that H is amenable and neither locally finite nor virtually cyclic, then every pmp action of G has the property SC. In particular, if two Bernoulli shifts of such a sofic group are Shannon orbit equivalent, then they are conjugate. This is joint work with David Kerr.