Analysis Seminar

Speaker: Blake Boudreaux (Western)

"Convexity in Several Complex Variables"

Time: 15:30

Room: MC 107

Abstract: In 1906, F. Hartogs discovered the existence of domains in $\mathbb{C}^n$ for which every holomorphic function can be extended to a larger domain. Domains that do not admit this extension phenomenon satisfy a complex type of convexity, known as pseudoconvexity. This type of convexity can be viewed as convexity "with respect to holomorphic functions", as opposed to classical convexity which is convexity "with respect to linear functions". $$ $$ In this talk, we will motivate and define pseudoconvexity. We will also compare and contrast its many equivalent formulations with that of classical convexity. We will also introduce a class of "pseudoconvex" manifolds known as Stein manifolds and discuss their many properties. Notions of convexity with respect to other classes of functions will also be discussed.

Transformation Groups Seminar

Speaker: Sayantan Roy Chowdhury (Western)

"Loop spaces of Davis-Januszkiewicz spaces"

Time: 10:30

Room: MC 107

Talk 6 of the learning seminar on Pontryagin rings in toric topology.

Analysis Seminar

Speaker: Purvi Gupta (Indian Institute of Science, Bangalore)

"On the dimension of bundle-valued Bergman spaces on compact Riemann surfaces"

Time: 15:30

Room: MC 107

Given a holomorphic vector bundle over a compact Riemann surface, we characterize when the Bergman space of holomorphic sections of the restriction of the bundle to an open set is infinite dimensional. This answers a problem raised by Sz{\H o}ke (2020), which was motivated by the results of Carleson and Wiegerinck for planar domains. This is joint work with Anne-Katrin Gallagher and Liz Vivas

Colloquium

Speaker: Alex Lubotzky (Weizmann Institute of Science)

"Good locally testable codes"

Time: 16:40

Room: MC 107 and zoom

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 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.