## 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.

## Friends Lunch with a Member

Reuben Jonathan Miller, Friends of the Institute for Advanced Study Member
School of Social Science
February 10, 2017

Reuben Jonathan Miller, Friends of the Institute for Advanced Study Member

## Mirror symmetry via Berkovich geometry I: overview

Tony Yue Yu
Visitor, School of Mathematics
February 13, 2017
Berkovich geometry is an enhancement of classical rigid analytic geometry. Mirror symmetry is a conjectural duality between Calabi-Yau manifolds. I will explain (1) what is mirror symmetry, (2) what are Berkovich spaces, (3) how Berkovich spaces appear naturally in the study of mirror symmetry, and (4) how we obtain a better understanding of several aspects of mirror symmetry via this viewpoint. This member seminar serves also as an overview of my minicourses in the same week.

## Nearest neighbor search for general symmetric norms via embeddings into product spaces

Ilya Razenshteyn
Massachusetts Institute of Technology
February 13, 2017
I will show a new efficient approximate nearest neighbor search (ANN) algorithm over an arbitrary high-dimensional *symmetric* norm. Traditionally, the ANN problem in high dimensions has been studied over the $\ell_1$ and $\ell_2$ distances with a few exceptions. Thus, the new result can be seen as a (modest) step towards a "unified theory" of similarity search.

## Discrete harmonic analysis and applications to ergodic theory

Mariusz Mirek
University of Bonn; Member, School of Mathematics
February 8, 2017
Given $d, k\in\mathbb N$, let $P_j$ be an integer-valued polynomial of $k$ variables for every $1\le j \le d$. Suppose that $(X, \mathcal{B}, \mu)$ is a $\sigma$-finite measure space with a family of invertible commuting and measure preserving transformations $T_1, T_2,\ldots,T_{d}$ on $X$. For every $N\in\mathbb N$ and $x \in X$ we define the ergodic Radon averaging operators by setting \[ A_N f(x) = \frac{1}{N^{k}}\sum_{m \in [1, N]^k\cap\mathbb Z^k} f\big(T_1^{ P_1(m)}\circ T_2^{ P_2(m)} \circ \ldots \circ T_{d}^{ P_{d}(m)} x\big).