Ramanujan Lecture on National Mathematics Day

This lecture is open to the public and is intended for a broad scientific audience. Join us as we celebrate India's mathematical heritage with a talk from one of the great mathematicians of our time.

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 hope also to explain how the seminal work of Howard Garland (1972) on the cohomology of quotients of the Bruhat-Tits buildings of p-adic Liegroup 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.

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