Title: Optimal regularity for geometric flows
Speaker: William Minicozzi (MIT)
Abstract: Many physical phenomema lead to tracking moving fronts whose speed depends on the curvature. The level set method has been tremendously succesful for this, but the solutions are typically only continuous. We will discuss results that show that the level set flow has twice differentiable solutions. This is optimal.
These analytical questions crucially rely on understanding the underlying geometry. The proofs draws inspiration from real algebraic geometry and the theory of analytic functions. Developing these geometric techniques gives solutions to other analytic questions, such as Rene Thom's gradient conjecture, for degenerate equations. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even when they are not. This would explain various conjectured phenomena. This is joint work with Toby Colding.
Biosketch: William Minicozzi works in geometry and analysis and is currently the Singer Professor of Mathematics at MIT. He received his Ph.D. from Stanford in 1994 under the supervision of Rick Schoen, spent a year at Courant and then joined the faculty at Johns Hopkins where he held the Krieger-Eisenhower Chair in Mathematics starting in 2007 and was department chair in 2011-12. He was awarded a Sloan fellowship in 1998, gave an invited address at the 2006 ICM in Madrid, and received the Oswald Veblen Prize from the AMS in 2010 together with Tobias Colding for their work on minimal surfaces. He was named a Fellow of the American Mathematical Society in 2012 and elected to the American Academy of Art & Sciences in 2015.
Colloquium URL: https://web.math.osu.edu/colloquium/
