Terng Lecture Series
Tatiana Toro, University of Washington, "Uniform Rectifiability via Perimeter Minimization"
Abstract: Quantitative geometric measure theory has played a fundamental role in the development of harmonic analysis, potential theory and partial differential equations on non-smooth domains. In general the tools used in this area differ greatly from those used in geometric measure theory as it appears in the context of geometric analysis. In this course we will discuss how ideas arising when studying perimeter minimization questions yield interesting and powerful results concerning uniform rectifiability of sets. The course will be mostly self-contained.
Uhlenbeck Lecture Series
Panagiota Daskalopoulos, Columbia University, "Ancient Solutions to Geometric Flows"
Abstract: Some of the most important problems in geometric evolution partial differential equations are related to the understanding of singularities. This usually happens through a blow up procedure near the singularity which uses the scaling properties of the equation. In the case of a parabolic equation th eblow up analysis often leads to special solutions which are defined for all time $-\infty < t \leq T$, for some $T \leq + \infty$. We refer to them as ancient solutions. The classification of such solutions often sheds new insight upon the singularity analysis.
In this lecture series we will discuss Uniqueness Theorems for ancient solutions to parabolic partial differential equations, starting from the Heat equation and extending to the Semi-linear heat equation, the Mean curvature flow, the Ricci flow and the Yamabe flow. We will also discuss the construction of new solutions from the gluing of two more solitons.
Prerequisites:(for both courses)
Multivariable calculus and an undergraduate course in analysis.