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Abstract: Let X be a finite set in the d-dimensional Euclidean space. A set N is called weak epsilon-net for X (with respect to convex sets) if every convex set containing at least epsilon.|X| points of X intersect N. Existence results and applications will be discussed, as well as some related open problems.
Colloquium - 16:00
Abstract: It was shown by A.A. Markov in 1958 that the homeomorphism problem for manifolds is unsolvable in dimensions d \geq 4, i.e., there is no algorithm to decide whether two given manifolds M^d and N^d are homeomorphic or not. Still worse, S.P. Novikov proved that even the standard sphere S^d is not recognizable algorithmically when d \geq 5.
Despite of these results, there is a number of procedures and heuristics which, at least in special cases and situations, allow to recognize the topological type of particular manifolds. In this talk, we will survey such methods from a "practical point of view".