## Log lower bound on the number of nodal domains on some surfaces of negative curvature

Steven Zelditch
Northwestern University
February 17, 2017
Abstract: An open problem is to prove that for any (or at least any generic) Riemannian metric, there is some sequence of eigenfunctions of the Laplacian for which the number of nodal domains tends to infinity. It sounds easy but as yet there are almost no examples of such metrics except for separation of variables situations, and the results of Ghosh-Reznikov-Sarnak and of Junehyuk Jung and myself. This talk is about a quantitative improvement in which we give a log lower bound for the number of nodal domains for negatively curved `real Riemann surfaces'.

## Equidistribution of random waves on shrinking balls

Melissa Tacy
Australian National University
February 17, 2017
Abstract: In the 1970s Berry conjectured that the behavior of high energy, quantum-chaotic billiard systems could be well modeled by random waves. That is random combinations of the plane waves e^{ik ·x}. On manifolds it is more natural to randomize over the eigenfunctions of the Laplace-Beltrami operator. In this talk I will present results showing that such random waves equidistribute on balls that shrink with the eigenvalue. This is joint work with Xiaolong Han.

## Mirror symmetry via Berkovich geometry III: log Calabi-Yau surfaces

Tony Yue Yu
Visitor, School of Mathematics
February 17, 2017
I will explain the case of log Calabi-Yau surfaces, based on my previous works and my work in progress with Sean Keel.

## $C^\infty$ closing lemma for three-dimensional Reeb flows via embedded contact homology

Kei Irie
Kyoto University
February 16, 2017
$C^r$ closing lemma is an important statement in the theory of dynamical systems, which implies that for a $C^r$ generic system the union of periodic orbits is dense in the nonwondering domain. $C^1$ closing lemma is proved in many classes of dynamical systems, however $C^r$ closing lemma with $r > 1$ is proved only for few cases. In this talk, I'll prove $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds. The proof uses recent developments in quantitative aspects of embedded contact homology (ECH).

## Functional inequalities and gradient flow for quantum evolution

Eric Carlen
Rutgers University
February 15, 2017
We present joint work with Jan Maas showing that Quantum Markov semigroups satisfying a detailed balance condition are gradient flow for quantum relative entropy, and use this prove some conjectured inequalities arising in quantum information theory.

## Mirror symmetry via Berkovich geometry II: the non-archimedean SYZ fibration

Tony Yue Yu
Visitor, School of Mathematics
February 15, 2017
I will explain the construction and the geometry of the non-archimedean SYZ fibration.

## An improvement of Liouville theorem for discrete harmonic functions

Eugenia Malinnikova
Norwegian University of Science and Technology
February 14, 2017
Abstract: The classical Liouville theorem says that if a harmonic function on the plane is bounded then it is a constant. At the same time for any angle on the plane, there exist non-constant harmonic functions that are bounded outside the angle.
The situation is different for discrete harmonic functions on Z^2. We show that the following improved version of the Liouville theorem holds. If a discrete harmonic function is bounded on 99% of the plane then it is constant. It is a report on a joint work (in progress) with L. Buhovsky, A. Logunov and M. Sodin.

## A unified duality-based approach to Bayesian mechanism design

Matt Weinberg
Princeton University
February 14, 2017

We provide a duality framework for Bayesian Mechanism Design. Specifically, we show that the dual problem to revenue maximization is a search over virtual transformations. This approach yields a unified view of several recent breakthroughs in algorithmic mechanism design, and enables some new breakthroughs as well. In this talk, I'll:

1) Provide a brief overview of the challenges of multi-dimensional mechanism design.

2) Construct a duality framework to resolve these problems.

## Harmonic functions - positivity and convexity

Dan Mangoubi
Hebrew University
February 14, 2017
Abstract: We present a sequence of positive quadratic forms associated with harmonic functions on Abelian groups. We show how the positivity property recovers the polynomial Liouville property and we prove a three spheres theorem in terms of random walks.
The talk is based on a joint work with Gabor Lippner.