CENTER FOR APPLIED MATHEMATICS AND THEORETICAL PHYSICS

UNIVERZA V MARIBORU UNIVERSITY OF MARIBOR

KREKOVA 2 SI-2000 MARIBOR SLOVENIA

Phone +(386) (2) 2355 350 and 2355 351 Fax +(386) (2) 2355 360

Robnik@uni-mb.si www.camtp.uni-mb.si

PROF.DR. MARKO ROBNIK, DIRECTOR

**Abstract**:
**The first part** will be a historical survey.
We shall start with the classical ternary Cantor set on the
line and its analogues in higher dimensions . These
Cantor sets are said to be *tamely embedded*. It was a long
standing problem if all Cantor sets in 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
(the first such
set was constructed in by Antoine in the 1920's and in
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
(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 which are nevertheless Lipschitz
homogeneously embedded into . We present
construction of rigid wild Cantor sets in 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** --

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