MenuDepartmental Colloquia
Spring 2024
| Date: | February 29, 2024 |
| Time: | 4:00pm |
| Location: | Bloc 117 |
| Speaker: | Alex Lubotzky, Weizmann Institute of Science |
| Title: | Good locally testable codes |
| Abstract: | 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/RightCayley 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 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. |
| Date: | March 26, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 117 |
| Speaker: | Persi Diaconis, Stanford University |
| Title: | Adding numbers and shuffling cards |
| Abstract: | When numbers are added in the usual way, 'carries' occur. Carries make a mess and it's natural to ask 'how do the carries go?' How many carries are typical and, if you just had a carry, is it more or less likely that there is a following carry? Surprisingly, the carries form a Markov chain with an 'amazing' transition matrix (are any matrices amazing?). This same matrix occurs in the analysis of the usual way of riffle shuffling cards. I will explain the 'seven shuffles theorem' and the connection. The same matrix occurs in taking sections of generating functions and in understanding the Veronese embedding. I'll try to explain all of this 'in English'. |
| Date: | March 28, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 117 |
| Speaker: | Persi Diaconis, Stanford University |
| Title: | Hyperplane walks |
| Abstract: | Picture a collection of hyperplanes in d-dimensional Euclidean space. These divided space into chambers (points not on any of the hyperplanes) and faces (points on some hyperplanes). The geometry and combinatorics of such arrangements is a world of its own, with applications in topology,algebraic geometry and every kind of algebra. I'll supplement this by introducing a simple family of random walks on the chambers. These include classical walks (Ehrenfest urn, card shuffling, dynamic storage allocation) but also lots of fresh examples(walks on parking functions!). Strangely, in more or less complete generality, there is a complete theory (all eigenvalues of the associated transition operators and sharp rates of convergence to stationarity--known). Naturally, there are open problems-- Understanding the stationary distribution of these walks involves the classical problem of sampling from an urn without replacement in various guises and there is a lot we don't know. I'll try to explain all this to a non-specialist audience. |
| Date: | April 18, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Henri Moscovici |
| Title: | Can one hear the zeros of zeta? |
| Abstract: | Preceding by a half-century the well-known challenge “Can one hear the shape of a drum?” a similar dare was raised relative to the Riemann Hypothesis, which in contemporary parlance goes by the name of “Hilbert-Polya operator”. Very recently Alain Connes worked out the heat expansion for such an operator, assuming its existence. After presenting his results I will discuss the connection with our joint work on the square-root of the prolate spheroidal wave operator, whose spectrum simulates the zeros of Zeta. |
