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Organizers Michael Brannan Matthew Kennedy Nico Spronk Kateryna Tatarko Andy Zucker |
| Date | Speaker, Title and Abstract |
|---|---|
| August 29 | Mathias Sonnleitner, University of Passau |
June 3 | Manuel Fernandez, Georgia Tech On the $\ell_0$-Isoperimetry of Measurable Sets Gibbs-sampling, also known as coordinate hit-and-run (CHAR), is a random walk used to sample points uniformly from convex bodies. Its transition rule is simple: Given the current point p, pick a random coordinate i and resample the i'th coordinate of p according to the distribution induced by fixing all other coordinates. Despite its use in practice, strong theoretical guarantees regarding the mixing time of CHAR for sampling from convex bodies were only recently shown in works of Laddha and Vempala, Narayanan and Srivastava, and Narayanam, Rajaraman and Srivastava. In the work of Laddha and Vempala, as part of their proof strategy, the authors introduced the notion of the $\ell_0$ isoperimetric coefficient of a measurable set and provided a lower bound for the quantity in the case of axis-aligned cubes. In this talk we will present some new results regarding the $\ell_0$ isoperimetric coefficient of measurable sets. In particular we pin down the exact order of magnitude of the $\ell_0$ isoperimetric coefficient of axis-aligned cubes and present a general upper bound of the $\ell_0$ isoperimetric coefficient for any measurable set. As an application, we will mention how the results give a moderate improvement in the mixing time of CHAR. |
May 9 ***4:30PM-5:20PM in MC 5417 | Sascha Troscheit, University of Oulu Dynamical Self-similar Covering Sets A classical problem in dynamical systems is known as the shrinking target problem: given a sequence of 'target' subsets A_n \subset X and a dynamic T: X \to X we ask how 'large' the set of all points R \subset X is whose n-th iterate hits the target, T^n (x) \in A_n, infinitely often. Much progress has been made on understanding this type of 'recurrent' set and I will highlight some recent results on this and the related 'dynamical covering problem' which is a dynamical generalisation of the Dvoretzky covering problem. The talk is based on joint results with Balázs Bárány, and Henna Koivusalo and Balázs Bárány. |
May 9 ***3:30PM-4:20PM in MC 5417 | Alex Chirvasitu, University at Buffalo Strange manifolds, small cohomotopy and Baire classes Pr¨ufer surfaces are non-metrizable separable 2-manifolds originally defined by Calabi and Rosenlicht by doubling the upper half-plane along a continuum’s worth of real-line boundary components. The construction and variations on it have since been studied by Gabard, Baillif and many others for the purpose of probing the pathologies of non-paracompact manifolds. The fundamental groups of such surfaces and higher-dimensional cousins are known to be (essentially) free on the sets S of connected boundary components, so their first cohomotopy groups (i.e. sets of homotopy classes of continuous maps to rather than from the circle) are identifiable with maps from S to the integers. Which functions S → Z arise in this manner is a natural question, with (perhaps) a surprising answer. The goal will be to discuss that problem, but the manifolds themselves might provide some entertainment value on their own. |