# Eleventh Math Colloquium BCAM - UPV/EHU

Date: Wed, Nov 24 2021

Hour: 16:00

Location: Online

Speakers: Alexander Sasha L. Volberg, is University Distinguished Professor of mathematics at Michigan State University. He obtained his PhD at Steklov Math. Institute in 1989 and in 1990 he moved to the US. Among his many awards and honors include a Salem Prize, Lars Onsager Medal for the Norwegian University of Science and Technology, and a Humboldt Foundation Professorship. He was an invited speaker at ICM-90 in Kyoto, Japan. He has covered many areas of analysis: harmonic analysis, spectral theory of functions and operators, one-dimensional complex analysis, potential analysis etc where Volberg made substantial and recognized contributions over the past 25 years.

We are glad to announce that the 11th Math Colloquium BCAM-UPV/EHU will take place on Wednesday, **November 24, at 16:00 (CET)**. Due to the COVID-19 outbreak the talk will be streamed **online** and users will be welcome to join using the video conferencing tool Webex. Registration is necessary.

*Please note that the time of the colloquium has been delayed by 2 hours.**Title**

The solution of Enflo�s problem**Abstract**

How to formulate local notions of Banach space metrically is important by the following reasons: (1) it makes it possible to extend these notions to more general metric spaces, (2) it makes it possible to study embeddings in Banach spaces of the general metric spaces, which is one of the main directions in theoretical computer science.

Purely metric properties of Banach spaces were popularized in works of Bourgain, Naor, Schechtman and many others on the so-called Ribe program.

Bourgain initiated a program to find explicit metric descriptions of local properties of Banach spaces. In the case of classical property called ``Rademacher type´´ the natural conjecture was that Enflo´s notion of metric type from 1970 is the right one. 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 posed by Enflo in the 70´s.

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 scalar Poincaré inequality on Hamming cube.

In the second part of the presentation, if time permits, I will talk about singular integrals on Hamming cube and about two approaches: Francoise Lust-Piquard´s approach via quantum random variable and another approach using Bellman function technique.

## Organizers:

Michigan University State

## Confirmed speakers:

Alexander Sasha L. Volberg

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