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Topology/Geometry Seminar

A Few Math Fairy Tales

Prof. D. Burago: Penn State

Location:  Hill 005
Date & time: Tuesday, 07 February 2017 at 3:30PM - 3:31PM

The format of this talk is rather non-standard. It is actually a combination of several mini-talks. They would include only motivations, formulations, basic ideas of proof if feasible, and open problems. No technicalities. Each topic would be enough for 2+ lectures. However I know the hard way that in diverse audience, after 1/3 of allocated time 2/3 of people fall asleep or start playing with their tablets. I hope to switch to new topics at approximately right times. I include more topics that I plan to cover for I would be happy to discuss others after the talk or by email/skype. I may make short announcements on these extra topics. The topics will probably be chosen from the list below. I sure will not talk on topics I have spoken already at your university. “A survival guide for feeble fishâ€ù. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water flow. This is related to homogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov. How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably “niceâ€ù mm–spaces. A notion of ho-Laplacian and its stability. Joint with S. Ivanov and Kurylev. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori has been remaining a great mystery. The main quantative invariants so far are entropies. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We were able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. These modified examples do generate positive positive metric entropy is generated in arbitrarily small tubular neighborhood of one trajectory. The technology is based on a metric theory of “dual lens mapsâ€ù developed by Ivanov and myself, which grew from the “what is insideâ€ù topic. Quite a few stories are left in my left pocket. Possibly: On making decisions under uncertain information, both because we do not know the result of our decisions and we have only probabilistic data.

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Department of Mathematics

Department of Mathematics
Rutgers University
Hill Center - Busch Campus
110 Frelinghuysen Road
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