Abstract

An error-correcting code is locally testable (LTC) if a random tester 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). Based on joint work with I. Dinur, S. Evra, R. Livne, and S. Mozes. Fedor Pavutnitskiy Title: LLMs for the Working Mathematician Abstract: In this introductory talk I will survey foundational and recent works in the emerging area of using large language models (LLMs) in contemporary mathematical research. I will also discuss key challenges we have faced and share progress from our own ongoing projects.