Sreda, 22.06.2011. u 14h, sala 2, MI SANU, BGD

Sergej Melihov, Steklov institut RAN
COMBINATORIAL METHODS IN EMBEDDING THEORY

Abstract: We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes. The talk is based on the preprint https://arxiv.org/abs/1103.5457

Petak, 24.06.2011. u 14h, sala 2, SANU, BGD

Shahar Nevo, Department of Mathematics and Computer Science, Bar-Ilan University, Israel
Q_\alpha-NORMALITY AND ENTIRE FUNCTIONS

Abstract: We introduce the notion of Q_\alpha-normality, which is an extension of the notion of Q_1-normality (Quasi-normality) to higher orders \alpha, where alpha is an arbitrary ordinal number. Then we explain why for countable \alpha there is an entire function f=f_\alpha, such that f(n\ast z), n=1,2,3.... is "exact" Q_\alpha -normal family. This is a common work with Shai Gul.