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.