Rapidly oscillating functions associated with Lagrangian submanifolds play a fundamental role in semi-classical analysis. In this talk I will describe how to associate spaces of semi-classical oscillatory functions to isotropic submanifolds of phase space, and sketch their symbol calculus. As a special case we obtain the semi-classical version of the Hermite distributions of Boutet the Monvel and Guillemin. I will also discuss a couple applications of the theory. This is based on joint works with Victor Guillemin and Alejandro Uribe.
 

A complete Kahler metric g on a Kahler manifold M is a "gradient Kahler-Ricci soliton" if there exists a smooth real-valued function f:M-->R  with \nabla f holomorphic such that Ric(g)-Hess(f)+\lambda g=0 for \lambda a real number. I will present some classification results for such manifolds. This is joint work with Alix Deruelle (Université Paris-Sud) and Song Sun (UC Berkeley).

We'll discuss a new technique for relating scalar curvature bounds to the global structure of 3-dimensional manifolds, exploiting a relationship between the scalar curvature and the topology of level sets of harmonic functions. We will describe several geometric applications in both the compact and asymptotically flat settings, including a simple and effective new proof (joint with Bray, Kazaras, and Khuri) of the three-dimensional Riemannian positive mass theorem.

In joint work with F. Bonsante and A. Seppi, we solve a
Dirichlet-type problem for entire constant mean curvature hypersurfaces in
Minkowski n+1-space, proving that such surfaces are essentially in bijection
with lower semicontinuous functions on the n-1-sphere. This builds off of
existence theorems by Treibergs and Choi-Treibergs, which themselves rely on
the foundational work of Cheng and Yau. I'll present their maximum principle
argument as well the extra tool that leads to our complete existence and
uniqueness theorem. Time permitting, I'll compare with the analogous problem
of constant Gaussian curvature and present a new result on their intrinsic
geometry.

Conference website: https://www.math.uci.edu/scgas

Please register at the conference website if planning to attend.