CENTER FOR APPLIED MATHEMATICS AND THEORETICAL PHYSICS
UNIVERZA V MARIBORU $\bullet$ UNIVERSITY OF MARIBOR
KREKOVA 2 $\bullet$ SI-2000 MARIBOR $\bullet$ SLOVENIA
Phone +(386) (2) 2355 350 and 2355 351 $\bullet$ Fax +(386) (2) 2355 360
Robnik@uni-mb.si $\bullet$ www.camtp.uni-mb.si
PROF.DR. MARKO ROBNIK, DIRECTOR

Seminarsko predavanje
Centra za uporabno matematiko in teoretično fiziko

Geometric topology of wild Cantor sets

Profesor Dr. Dušan Repovš
IMFM, Jadranska 19, Univerza v Ljubljani

Abstract: The first part will be a historical survey. We shall start with the classical ternary Cantor set on the line $R$ and its analogues in higher dimensions $R^n$. These Cantor sets are said to be tamely embedded. It was a long standing problem if all Cantor sets in $R^n$ are tame. We shall show that the answer turned out to be negative, i.e. that there exist so-called wildly embedded Cantor sets in $R^{n \ge 3}$ (the first such set was constructed in $R^3$ by Antoine in the 1920's and in $R^4$ by Blankinship in the late 1940's). We shall demonstrate how such pathological Cantor sets can give rise to wild embeddings of the standard 2-sphere in $R^3$ (the first such example dates back to the 1930's and is due to Alexander). In the second part we discuss the Bing-Borsuk Conjecture from 1960 (which in dimension 3 implies the famous Poincaré Conjecture) and the classical Hilbert-Smith Conjecture from the 1930's. They motivated our quest for nonmanifold Lipschitz homogeneous compacta: we shall present a new general technique for constructing wild Cantor sets in $R^{3}$ which are nevertheless Lipschitz homogeneously embedded into $R^3$. We present construction of rigid wild Cantor sets in $R^3$ with simply connected complement. We plan to state some open problems and conjectures.

Seminarsko predavanje (v angleškem jeziku) bo v sredo 6. septembra 2006 ob 15:15 uri v seminarski sobi CAMTP na Krekovi 2, pritličje desno. Vljudno vabljeni vsi zainteresirani, tudi študentje.

Prof.Dr. Marko Robnik
-- Direktor CAMTP --