Semi-classical isotropic functions and applications

Speaker: 

Zuoqin Wang

Institution: 

USTC and MIT

Time: 

Thursday, February 20, 2020 - 2:00pm to 3:00pm

Host: 

Location: 

340P Rowland Hall

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.
 

Semi-classical isotropic functions and applications

Speaker: 

Zuoqin Wang

Institution: 

USTC and MIT

Time: 

Thursday, February 20, 2020 - 2:00pm to 3:00pm

Host: 

Location: 

RH 340P

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.

SEMINAR CANCELLED - Classification Results for Expanding and Shrinking gradient Kahler-Ricci solitons.

Speaker: 

Ronan Conlon

Institution: 

Florida International University

Time: 

Friday, March 13, 2020 - 3:00pm to 3:50pm

Host: 

Location: 

RH 306

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).

Isoperimetry and volume preserving stability in real projective spaces

Speaker: 

Celso Viana

Institution: 

UC Irvine

Time: 

Tuesday, February 25, 2020 - 4:00pm

Location: 

RH 306

In this talk I will address the problem of classifying volume
preserving stable constant mean curvature hypersurfaces in Riemannian
manifolds. I will present recent classification in the real projective space
of any dimension and, consequently, the solution of the isoperimetric
problem.

Scalar curvature and harmonic functions

Speaker: 

Daniel Stern

Institution: 

University of Toronto

Time: 

Tuesday, February 18, 2020 - 4:00pm

Host: 

Location: 

RH 306

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.

 

Epsilon-regularity for minimal surfaces near quadratic cones

Speaker: 

Luca Spolaor

Institution: 

UCSD

Time: 

Tuesday, January 21, 2020 - 4:00pm to 5:00pm

Location: 

RH 306

Every area-minimizing hypercone having only an isolated singularity fits into a foliation by smooth, area-minimizing hypersurfaces asymptotic to the cone itself. In this talk I will present the following epsilon-regularity result: every minimal surfaces lying sufficiently close to a minimizing quadratic cone (for example, the Simons' cone), is a perturbation of either the cone itself, or some leaf of its associated foliation. This result also implies the Bernstein-type result of Simon-Solomon, which characterizes area-minimizing hypersurfaces asymptotic to a quadratic cone as either the cone itself, or some leaf of the foliation, and it also allows to study convergence to singular minimal hyper surfaces. This is a joint result with N. Edelen

CMC surfaces in Minkowski space

Speaker: 

Peter Smillie

Institution: 

Caltech

Time: 

Tuesday, January 14, 2020 - 4:00pm

Location: 

RH 306

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.

Some study of biharmonic maps and submanifolds since 2000 (Cancelled)

Speaker: 

Yelin Ou

Institution: 

Texas A&M Commence

Time: 

Tuesday, March 10, 2020 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Biharmonic maps are maps between Riemannian manifolds which are critical points of the bi-energy. They are solutions of a system of 4thorder PDEs and they include harmonic maps and biharmonic functions as special cases. Biharmonic submanifolds (which include minimal submanifolds as special cases) are the images of biharmonic isometric immersions. The talk will review some problems, including classification of biharmonic submanifolds in space forms, biharmonic maps into spheres, biharmonic conformal maps, and unique continuation theorems, studied in this field and their progress since 2000. The talk also presents some recent work on equivariant biharmonic maps and the stability and index of biharmonic hypersurfaces in space forms.

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