Séminaires : Séminaire CAGe: Combinatorics, Arithmetic and Geometry

Les sujets sont ceux décrits par le titre :). Ils doivent être compris dans un sens large.

Notre objectif est de nous réunir avec une périodicité mensuelle.

https://sites.google.com/view/combarithmgeo/home?authuser=0

14:00-15:00:  Anya Nordskova (University of Tokyo, Japan)

Title: Banach's problem in dimension 4 

Abstract: 90 years ago S. Banach asked the following question. Let V be a normed (real or complex) vector space and assume that all its subspaces of a fixed finite dimension k, where 1 < k < n = dim V, are isometric to each other. Is B necessarily Euclidean (that is, the norm is induced by an inner product)?  Translating the question into the language of convex sets: Let B be a convex centrally symmetric body in an n-dimensional normed vector space and assume that all its cross-sections by k-dimensional vector subspaces are linearly equivalent to each other. Is B necessarily an ellipsoid? 

In general, the question remains open, but affirmative answers were given in many special cases by Auerbach, Mazur and Ulam (1935), Dvoretzky (1959), Gromov (1967), Milman (1971), Bor, Hernandez-Lamoneda, Jimenez-Desantiago (2019). Almost all of these works are based on methods of algebraic topology.

Together with S. Ivanov and D. Mamaev (Invent. Math, 2023) we managed to solve Banach's problem in the smallest previously unknown case, namely, for k+1=n= 4. Due to the parallelizability of the three-dimensional sphere, topological arguments used in previous works do not provide any information in our case. Hence, we develop a different, differential geometric approach. 

This talk continues from our February 24th session, providing further details on the proof.

15:30-16:30:  Shai Haran (Technion, Israel)

Title: Non additive geometry 

Abstract: The usual dictionary between geometry and commutative algebra is not appropriate for Arithmetic geometry because addition is a singular operation at the "Real prime". We replace Rings, with addition and multiplication, by Props (=strict symmetric monoidal category generated by one object), or by Bioperad (=two closed symmetric operads acting on each other): to a ring we associate the prop of all matrices over it, with matrix multiplication and block direct sums as the basic operations, or the bioperad consisting of all raw and column vectors over it. We define the "commutative" props and bioperads, and using them we develop a generalized algebraic geometry, following Grothendieck footsteps closely. This new geometry is appropriate for Arithmetic (and potentially also for Physics).

This this continues from our February 24th session.