On k-regular maps

Datum

20.01.2016

Zeit

17:00 - 18:00

Sprecher

Prof. Dr. Tadeusz Januszkiewicz

Zugehörigkeit

IMPAN, Warschau

Serie

TUD Dresdner Mathematisches Seminar

Sprache

en

Hauptthema

Mathematik

Andere Themen

Mathematik

Host

Prof. Dr. A. Thom

Beschreibung

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

Letztmalig verändert: 03.12.2015, 18:29:52