• Speaker: Martin Hairer, Imperial College London
• Title: Taming infinities
• Date: 09/15/2020
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting)

Some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! What's worse, this doesn't just happen for some exotic theories, but in the standard theories describing some of the most fundamental aspects of nature. Various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. We will dip our toes into some of the conceptual and mathematical aspects of these techniques and we will see how they have recently been used in probability theory to study equations whose meaning was not even clear until now.

• Speaker: Ruixiang Zhang, IAS
• Title: Kakeya type problems and analysis
• Date: 09/29/2020
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting)

Informally, Kakeya type problems ask whether tubes with different positions and directions can overlap a lot. One usually expects the answer to be no in an appropriate sense. Thanks to the uncertainty principle, such a quantified non-overlapping theorem would often see powerful applications in analysis problems that have Fourier aspects. Perhaps the most well-known Kakeya type problem is the Kakeya conjecture. It remains widely open in $\Bbb{R}^n (n>2)$ as of today. Nevertheless, in the recent few decades people have been able to prove new Kakeya type theorems that led to improvements or complete solutions to analysis problems that appeared out of reach before. I will give an introduction to Kakeya type problems/theorems and analysis problems that see their applications. Potentially reporting some recent progress joint with Du, Guo, Guth, Hickman, Iosevich, Ou, Rogers, Wang and Wilson.

• Speaker: Alexander Volberg, Michigan State University
• Title: Metric properties of Banach spaces, Enflo's problem, Pisier's inequality and quantum random variables
• Date: 10/20/2020
• Time: 4:00 PM - 5:00 PM
• Place: Online (virtual meeting)
• Contact: Aaron D Levin (levina@msu.edu)

A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. In the joint paper with Paata Ivanisvili and Ramon Van Handel we prove that Rademacher type and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube, which, in its turn, is based on a certain formula that we used before in improving the constants in the scalar Poincaré inequality on the Hamming cube. I will also show several extensions of Pisier's inequality with ultimate assumptions on a Banach space structure. Some of our results use approach via quantum random variables.