The format of this talk is rather non-standard. It is actually a combina tion of several mini-talks. They would include only motivations, formulatio ns, 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 th at in diverse audience, after 1/3 of allocated time 2/3 of people fall asle ep or start playing with their tablets. I hope to switch to new topics at a pproximately 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 ma ke short announcements on these extra topics. The topics will probably be c hosen from the list below. I sure will not talk on topics I have spoken alr eady at your university. “A survival guide for feeble fish”. How fish can g et from A to B in turbulent waters which maybe much fasted than the locomot ive speed of the fish provided that there is no large-scale drift of the wa ter 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 e lements, triangulations and such? On manifolds and even reasonably “nice” m m–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 c ompletely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori has been r emaining a great mystery. The main quantative invariants so far are entropi es. It is easy, by modern standards, to show that topological entropy can b e 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^{inf ty} perturbations. Furthermore, a slightly modified construction resolves a nother long–standing problem of the existence of entropy non-expansive syst ems. These modified examples do generate positive positive metric entropy i s generated in arbitrarily small tubular neighborhood of one trajectory. Th e technology is based on a metric theory of “dual lens maps” developed by I vanov and myself, which grew from the “what is inside” topic. Quite a few s tories are left in my left pocket. Possibly: On making decisions under unce rtain information, both because we do not know the result of our decisions and we have only probabilistic data.

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