# Seminars & Events for Differential Geometry & Geometric Analysis Seminar

##### A prime orbit theorem, trace formulae, and interactions between quantum and classical mechanics on asymptotically hyperbolic manifolds

Asymptotically hyperbolic manifolds are a natural generalization of infinite volume hyperbolic manifolds and enjoy similar features. They are of particular interest in physics because all Poincar\'e-Einstein manifolds, which arise in adS-CFT correspondence, are asymptotically hyperbolic. In this talk, we'll recall the definition of these spaces and see some examples. After a brief discussion of their spectral theory and dynamics, I will present a prime orbit theorem and two ``dynamical wave trace formulae.'' Based on the prime orbit theorem and the trace formulae, we will determine a relationship between the existence of pure point spectrum and the topological entropy of the geodesic flow. We can interpret this physically as an interaction between the quantum and classical mechanics.

##### Regularity of solutions to the complex Monge-Ampere equation

In geometric analysis many arguments rely on a suitable regularity theory for the analyzed differential equations. Similarly to the solution of the Calabi conjecture often deriving suitable a priori estimates is in fact the heart of the matter. In the talk a regularity result for the complex Monge-Ampere equation will be presented. We will prove that any $C1,1$ smooth plurisubharmonic solution u to the problem $det(uij) = f$ with $f$ strictly positive and $H¨$ older continuous has in fact $H¨$ older continuous second derivatives. For smoother $f$ this follows form the classical Evans-Krylov theory yet in our case it cannot be applied directly. Instead we shall follow closely an idea of Xu-Jia Wang.

##### Liouville-type theorem for nonnegative Bakry-Emery Ricci tensor

##### Complex Monge-Ampere equations on symplectic and hermitian manifolds

We will discuss a program to generalize the complex Monge-Ampere equation to symplectic and hermitian manifolds. We will explain to which extent the classical theory on Kahler manifolds extends to these two cases, and give some applications. This is joint work with B. Weinkove and partly with S.-T. Yau.

##### Front propagation and phase transitions for fractional diffusion equations

Long range or *anomalous* diffusions, such as diffusions given by the fractional powers $(-\Delta)^s$ of the Laplacian, attract lately interest in Physics, Biology, and Finance. From the mathematical point of view, nonlinear analysis for fractional diffusions is being developed actively in the last years. In this talk, I will describe recent results concerning front propagation for the nonlinear fractional KPP heat equation, $\partial_t tu+(-\Delta)^su = u(1-u) in (0;1) Rn, 0 u 1$, with $s 2 (0; 1)$. In collaboration with J. M. Roquejore, we establish that fronts propagate at exponential speed |in contrast with the classical case $s=1$ for which there is propagation at a constant KPP speed. I will also describe works in collaboration with Y. Sire and E. Cinti.