PDE and Applied Mathematics Seminar

Spring 2017

Organized by Peter Gordon

March 23, (CAS 220D, 3:30pm-4:30pm) Dmitri Burago (The Pennsylvania State University) will tell us "A Few Fairy Math Tales."

Abstract: The format of this talk is rather non-standard. It is actually a combination of two-three 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 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 fish". 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. The situation becomes especially interesting and complicated when the flow id time-dependent. 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.
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 quantitative 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 maps" developed by Ivanov and myself, which grew from the "what is inside" topic. Also Joint with D. Chen.
How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably "nice" mm-spaces. A notion of $\rho$-Laplacian and its stability maybe discussed. Joint with S. Ivanov and Kurylev.
How well can we approximate an (unbounded) space by a metric graph whose parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers (the most difficult one is for $\mathbb{R}^2$) are given using Dynamics and Fourier series.
Are flats in normed spaces minimize the Busemann-Hausdorff surface area? The problem goes back to Busemann. Finally, with S. Ivanov, we were able to prove this for two-dim surfaces.
"What is inside?" Imagine a body with some intrinsic structure, which, as usual, can be thought of as a metric. One knows distances between boundary points (say, by sending waves and measuring how long it takes them to reach specific points on the boundary). One may think of medical imaging or geophysics. This topic is related to the one on minimal fillings, the next one. Joint work with S. Ivanov.
Ellipticity of surface area in normed space. An array of problems which go back to Busemann. They include minimality of linear subspaces in normed spaces and constructing surfaces prescribed weighted image under the Gauss map. They include minimality of linear subspaces in normed spaces and constructing surfaces with prescribed weighted image under the Gauss map. Joint with S. Ivanov
More stories are left in my left pocket.

Previous seminars:
Fall 2016
Spring 2016
Fall 2015
Spring 2015
Fall 2014
Spring 2014
Fall 2013
Spring 2013
Fall 2012