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.

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).