Geometry/topology Seminar
Wednesday, April 12, 2017, 3:15pm, 119 Physics
Dmitri Burago (Pennsylvania State University)
Math Mozaic
Abstract:- The lecture includes the main part (to be chosen on the spot) and a few mini-talks with
just definitions, motivations, some ideas of proofs, and open problems. I will discuss
some (hardly all) of the following topics.
1. 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. 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.
2. 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 quantitate
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 are now able
to show that metric entropy can become infinite too, under arbitrarily small C^{infty}
perturbations, answering an old-standing problem of Kolmogorov.. Furthermore, a
slightly modified construction resolves another longstanding problem of the existence of
entropy non-expansive systems. In these modified examples positive positive metric
entropy is generated in arbitrarily small tubular neighborhood of one trajectory. Join
with S. Ivanov and Dong. Chen.
3. 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 minimal fillings and surfaces in normed spaces. Joint work with S. Ivanov.
4. 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 $\R^2$) are given using dynamics and Fourier series. Joint with Ivanov.
5.How can one discretize elliptic PDEs without using finite elements, triangulations
and such? On manifolds and even reasonably nice mmspaces. A notion of
\rho-Laplacian and its stability. Joint with S. Ivanov and Kurylev.
6. A solution of Busemanns problem on minimality of surface area in normed spaces
for 2-D surfaces (including a new formula for the area of a convex polygon). Joint with
S. Ivanov. [video]
Generated at 11:42pm Friday, April 19, 2024 by Mcal. Top
* Reload
* Login