# Mathematics Colloquium Spring 2011

Jack Baskin Engineering Room 301A

Refreshments served at 3:40

For further information please contact the Mathematics Department at 459-2969

**April 19, 2011**

**Rigidity of solutions of elliptic local and non local equations on Riemannian manifolds.**

**Rigidity of solutions of elliptic local and non local equations on Riemannian manifolds.**

**Yannick Sire, Department of Mathematics, Marseille, France **

Thirty years ago, De Giorgi formulated a conjecture about the symmetry of bounded monotone solutions of the Allen-Cahn equation. In the spirit of this conjecture, I will describe several recent results about the symmetry of solutions of elliptic equations in the Euclidean space and on riemannian manifolds. More precisely, I will stress on the symmetry properties of local Allen-Cahn type equations and on nonlocal elliptic equations. As far as nonlocal equations are concerned, I will introduce fractional powers of the Laplace-Beltrami operator and study several regularity and symmetry results of these equations. Depending on the geometry of the manifold, I will introduce Liouville type results and one-dimensionnal symmetry of stable solutions.

**April 26, 2011**

**Linearization methods for the infinity Laplacian PDE**

**Linearization methods for the infinity Laplacian PDE**

**Lawrence C Evans, Department of Mathematics, UC Berkeley**

The "infinity Laplacian" equation is an extremely degenerate nonlinear second-order elliptic PDE that naturally appears in certain sup-norm variational problems.

I will discuss the many mysteries of this PDE, and describe recent work (with C Smart) showing how using Green's function for the linearization leads to new estimates and in particular a proof of the everywhere differentiability of solutions.

**May 3, 2011**

**Why do we care about the Riemann Hypothesis?**

**Why do we care about the Riemann Hypothesis?**

**Keith Conrad, University of Connecticut **

All mathematicians have heard that the Riemann Hypothesis is a significant open problem, but why is it such a big deal?

One can name plenty of older unsolved problems in number theory that get far less publicity. We care about the Riemann Hypothesis because it is connected to a large number of significant questions. I will discuss the history, scope, and range of consequences of the Riemann Hypothesis, with the aim of convincing mathematicians outside number theory that it deserves to be counted as one of the most important unsolved problems in mathematics.