On k-regular maps

Date

Jan 20, 2016

Time

5:00 PM - 6:00 PM

Speaker

Prof. Dr. Tadeusz Januszkiewicz

Affiliation

IMPAN, Warschau

Series

TUD Dresdner Mathematisches Seminar

Language

en

Main Topic

Mathematik

Other Topics

Mathematik

Host

Prof. Dr. A. Thom

Description

A continuous map f:X\to R^n is called k-regular if for any k-tuple (x_1,...,x_k)of distinct points in X their images f(x_i) are affinely independent (i.e. f(x_i)-f(x_1) are linearly indepedent). When k=2 this means that f is an embedding, and the similarity with embedding theory was the reason topologists, starting with Karol Borsuk, were interested in such maps. On the other hand questions in approximation theory going back to Pafnuty Chebyshev, and studied among others by Andrei Kolmogorov, yield essentially the same class of maps. One of the first challenges isto construct such maps, and do it in an efficient way. Another challenge is to prove nonexistence results. For embeddings of R^d in R^n, this is not very interesting, but for bigger k even this case presents a challenge. Recently lower bounds on n(d,k) the minimum dimension of the euclidean space receiving a k-regular map from R^d, significantly improving previously known ones, were found by Blagoevic, Cohen, Lueck and Ziegler, using algebraic topology, while upper bounds were found bu Buczynski, Januszkiewicz, Jelisiejew and Michalek, using algebraic geometry. In some cases they meet, and provide the final answer. I will tell the story of these developments, highlighting the analogy with embeddings and immersions, and avoiding technicalities.

Links

• Dresdner Mathematisches Seminar

Last modified: Dec 3, 2015, 6:29:52 PM