VIASM ANNUAL MEETING 2023  

Sunday, July 30th, 2023

• 1. Prof. Alex Lubotzky, The Hebrew University, Israel

Title: Good locally testable codes

Abstract: An error-correcting code is locally testable (LTC) if there is a random tester that reads only a small number of bits of a given word and decides whether the word is in the code, or at least close to it.
A long-standing problem asks if there exists such a code that also satisfies the golden standards of coding theory: constant rate and constant distance. Unlike the classical situation in coding theory, random codes are not LTC, so this problem is a challenge of a new kind.
We construct such codes based on what we call (Ramanujan) Left/Right Cayley square complexes. These objects seem to be of independent group-theoretic interest. The codes built on them are 2-dimensional versions of the expander codes constructed by Sipser and Spielman (1996).
The main result and lecture will be self-contained. But we also hope to explain how the seminal work of Howard Garland ( 1972) on the cohomology of quotients of the Bruhat-Tits buildings of p-adic Lie group has led to this construction ( even though it is not used at the end).
Based on joint work with I. Dinur, S. Evra, R. Livne, and S. Mozes.

• 2. Prof. Nguyen Huu Hoi, The Ohio State University, USA

Title: Random matrices: universality of the spectra and cokernels

Abstract: Random Matrix Theory is a rich area with many applications. In this talk I will give a brief survey on some recent exciting developments and useful techniques in the field, focusing mainly on the universal aspects of the spectra and cokernels.

• 3. Prof. Nguyen Trong Toan, Pennsylvania State University, USA

Title: To surf Langmuir waves

Abstract: Of great physical and mathematical interest is to establish the large time dynamical behavior of excited electrons and their possible final states in a non-equilibrium setting, classically modeled by the meanfield Vlasov equations used in plasma physics or by the meanfield Hartree equations used in quantum mechanics. Their rich physical behavior includes trapped trajectories, phase mixing, oscillations known as Langmuir waves, and Landau damping. This talk is to precise the survival threshold of spatial frequencies at which the purely oscillatory Langmuir waves are damped due to their resonant interaction with excited electrons, a classical decaying mechanism known as Landau damping, as well as to give an overview on the nonlinear problem.
Reference: https://arxiv.org/pdf/2305.08672.pdf

• 4. Prof. Sergiu Klainerman, Princeton University, USA

Title: Are black holes real?

Abstract: Black holes are specific solutions of the Einstein field equations. They exist as “real’’ rich and beautiful mathematical objects, which deserve to be studied for their own sake. They are also consistent with many indirect astrophysical observations. But are they real?

• 5. Prof. Ulrike Tillmann, University of Oxford, UK

Title: The shape of data

Abstract: The field of topological data analysis is still quite young. We will give an introduction, look at some applications, and discuss theoretical challenges. Beyond some basic linear algebra, little previous knowledge will be assumed.

• 6. Prof. Zhu Chengbo (National University of Singapore)

Title: Orbit Method: From Matrices to Unitary Representations

Abstract: The talk is intended as a leisurely introduction to one of the fundamental tasks of representation theory: the construction of irreducible unitary representations. I will first discuss two major sources of unitary representations of Lie groups, one from Symplectic Geometry (Kirillov philosophy) and another from Number Theory (Arthur’s conjecture). I will then introduce a constructive method called theta lifting which has been fruitful for representations of classical groups and discuss some recent applications of this method to unitary representation theory.