**SAMUEL EILENBERG LECTURES **

SPRING SEMESTER 2015

**Prof. Luis A. Caffarelli **_{ }**(University of Texas) **_{ –}

** “Future Directions in Nonlinear Partial Differential Equations”**

** “**The first three lectures will consists of a description of free boundary problems. Typical examples of free boundary problems are a solid liquid interphase, the edge of a flame, the shape of a drop sitting on a surface, a discontinuous change if strategy in a game. Mathematically their structure consists of a domain where some phenomena takes place (cooling of a substance, value of a portfolio, wetting of a region) describe by some variable, like temperature, height of the drop, etc. and the behavior of the phenomena, changes discontinuously when this variable goes to some treshold value (solid to liquid, unburnt to burnt, the edge of the drop, etc.) Mathematically this means that the transition surface is part of the problem itself, and a range of issues arises like existence of solutions, regularity of the solution and of the transition surface etc. We will describe some of these problems, and how to attack these regularity issues. For the rest of the course , one possibility will be to study a family of non-linear parabolic equations of local and non-local nature that for their character are naturally treated by energy method , sort on the style of De Giorgi Nash Moser way. These include local and non-local equations like porous media, non-linear diffusion with memory, but I will be glad to discuss other topics.”

** FALL SEMESTER 2014**

**Simon Brendle ( Stanford University) **

**“Partial Differential Equations in Geometry”**

“A central theme in geometry is the study of manifolds and their curvature. In this lecture series, we will discuss how techniques involving partial differential equations have shed light on several longstanding problems in global differential geometry. In the first part of the course, we will focus on the geometry of hypersurfaces, and discuss our proof of Lawson’s conjecture concerning minimal tori in S^3, as well as new results on mean curvature flow with surgery. In the second part, we will focus on the Ricci flow, including the Differentiable Sphere Theorem and Perelman’s question concerning the uniqueness of the Bryant soliton.”

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