# A few fairy math tales

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Dmitri Burago , Penn State University
The lecture consists of several mini-talks with just definitions, motivations, some ideas of proofs, and open problems. I will discuss some or 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. 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.3.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. Joint with S. Ivanov and Kurylev.4. “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.5. A solution of Busemann’s 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.