2024 Hayden-Howard Lecture

Felix Otto, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

A Variational Regularity Theory for Optimal Transportation and Its Applications to Matching

Optimal transportation, which identifies an optimal coupling between two probability measures, is a simple to state variational problem with surprisingly diverse connections.  A couple of years ago, with M. Goldman, we devised a new approach to the regularity theory for the coupling. It mimics De Giorgi's approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center: Under a non-dimensional smallness condition, the displacement is close to the gradient of a harmonic function. The main advantage of this variational regularity theory over the one based on maximum principle for the Monge-Ampere equation is that it does not require any regularity of the involved measures. Hence it can be applied to the popular matching problem, where it provides regularity on large scales.

About the Speaker:

Felix Otto has been the director at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany, since 2010. His main expertise lies in the applied analysis of partial differential equations and in the calculus of variations, with a recent focus on randomness. He is well known for his work on gradient flow structures of dissipative evolution equations, pattern formation in ferromagnets, and stochastic homogenization. His scientific honors include the A.P. Sloan Research Fellowship, the Max Planck Research Prize, the Leibniz Prize from the German Science Foundation. He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing, China.

Talks take place at 11:00-12:00 noon, Tuesday in POT 745, unless otherwise noted. For updates and more information, please visit: http://ms.uky.edu/~mhto224/seminar/PDEseminarS2018.html

If you have any questions about a particular seminar, please contact the speaker's host. For general questions about the seminar, please contact Mihai Tohaneanu using the email address mihai.tohaneanu@uky.edu. 

Talks take place at 11:00-12:00 noon, Tuesday in POT 745, unless otherwise noted. For updates and more information, please visit: http://ms.uky.edu/~mhto224/seminar/PDEseminarS2018.html

If you have any questions about a particular seminar, please contact the speaker's host. For general questions about the seminar, please contact Mihai Tohaneanu using the email address mihai.tohaneanu@uky.edu. 

Talks take place at 11:00-12:00 noon, Tuesday in POT 745, unless otherwise noted. For updates and more information, please visit: http://ms.uky.edu/~mhto224/seminar/PDEseminarS2018.html

If you have any questions about a particular seminar, please contact the speaker's host. For general questions about the seminar, please contact Mihai Tohaneanu using the email address mihai.tohaneanu@uky.edu. 

Talks take place at 11:00-12:00 noon, Tuesday in POT 745, unless otherwise noted. For updates and more information, please visit: http://ms.uky.edu/~mhto224/seminar/PDEseminarS2018.html

If you have any questions about a particular seminar, please contact the speaker's host. For general questions about the seminar, please contact Mihai Tohaneanu using the email address mihai.tohaneanu@uky.edu. 

Talks take place at 11:00-12:00 noon, Tuesday in POT 745, unless otherwise noted. For updates and more information, please visit: http://ms.uky.edu/~mhto224/seminar/PDEseminarS2018.html

If you have any questions about a particular seminar, please contact the speaker's host. For general questions about the seminar, please contact Mihai Tohaneanu using the email address mihai.tohaneanu@uky.edu. 

TITLE: Localization and spectral statistics for Schr\"odinger operators with random point interactions

Talks take place at 11:00-12:00 noon, Tuesday in POT 745, unless otherwise noted. For updates and more information, please visit: http://ms.uky.edu/~mhto224/seminar/PDEseminarS2018.html

If you have any questions about a particular seminar, please contact the speaker's host. For general questions about the seminar, please contact Mihai Tohaneanu using the email address mihai.tohaneanu@uky.edu. 

The Bernstein Problem in Two Dimensions

I will outline a complete solution to the weighted approximation problem for polynomials

on an arbitrary bounded simply connected domain  in the complex plane. In that setting

the problem was first studied extensively by Keldysh prior to 1941 in the context of L2-

approximation, and more than four decades later by Beurling where, in the latter instance,

the emphasis was on uniform approximation. Here, Beurling obtains the sharper result

with respect to the weight w, but at the expense of limiting the type of region to which

his argument applies. Ironically, however, the two problems turned out to be essentially

equivalent, but neither Beurling nor Keldysh obtained what might be considered a definitive

solution. My presentation will focus on the L2-case, where the theory of Sobolev spaces and

its associated potential theory is available. It is a simple matter to pass from there to a

solution in the case of uniform approximation.