# Distinguished Lecture

### South Hall 6635

### 3:30-4:30pm

## by Gunther Uhlmann

### University of Washington

Please note special time and room

Here is a poster created by Hanming Zhou

### 2/14/18 3-4pm, Flying A Studios Room, UCen

### Lecture 1 "Harry Potter's Cloak via Transformation Optics"

fascination for millennia in Greek mythology, movies, science fiction,

etc. including the legend of Perseus versus Medusa and the more recent

Star Trek and Harry Potter. In the last decade or so there have been

several scientific proposals to achieve invisibility. We will introduce

some of these in a non-technical fashion concentrating on the so-called

"transformation optics" that has received the most attention in the

scientific literature.

### 2/15/18 3-4pm, Flying A Studios Room, UCen

### Lecture 2
"Journey to the Center of
the Earth"

Abstract: We will consider the inverse problem of determining the sound

speed or index of refraction of a medium by measuring the travel times of

waves going through the medium. This problem arises in global seismology

in an attempt to determine the inner structure of the Earth by measuring

travel times of earthquakes. It has also several applications in optics

and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the

Riemannian metric of a Riemannian manifold with boundary by measuring

the distance function between boundary points? This is the boundary

rigidity problem. We will also consider the problem of determining

the metric from the scattering relation, the so-called lens rigidity

problem. The linearization of these problems involve the integration

of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, join with Plamen Stefanov

and Andras Vasy, on the partial data case, where you are making

measurements on a subset of the boundary. No previous knowledge of

Riemannian geometry will be assumed.

### 2/16/18 3:00-4:00pm, Broida Hall 1640

### Lecture 3 "Seeing Through Space Time"

### Abstract: We consider inverse problems for the Einstein equation with a

time-depending metric on a 4-dimensional globally hyperbolic Lorentzian

manifold. We formulate the concept of active measurements for relativistic

models. We do this by coupling Einstein equations with equations for

scalar fields.

The inverse problem we study is the question of whether the observations

of the solutions of the coupled system in an open subset of the space-time

with the sources supported in this set determines the properties of the

metric in a larger domain. To study this problem we define the concept of

light observation sets and show that knowledge of these sets determine the

conformal class of the metric. This corresponds to passive observations

from a distant area of space which is filled by light sources.

We will start by considering inverse problems for scalar non-linear

hyperbolic equations to explain our method. No previous knowledge of

Lorentzian geometry or general relativity will be assumed.

This is joint work with P. Hinz, Y. Kurylev, M. Lasss and Y. Wang.

## by Rick Schoen, UC Irvine

### 10/27/16
"Geometries that optimize
eigenvalues: closed surfaces"

### 10/28/16
"Geometries that optimize
eigenvalues: surfaces with boundary"

Abstract:When we choose a metric on a manifold we determine th spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. In these talks we will survey two cases in which progress has been made, focusing in the first lecture on closed surfaces and in the second on surfaces with boundary. We will describe the geometric structure of the critical metrics which turn out to be the induced metrics on certain special classes

of minimal (mean curvature zero) surfaces in spheres and euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. We will also discuss some of these.

## by Alice Chang, Princeton University

### 5/12/14
"**On a class of conformal covariant
operators and conformal invariants**"

Abstract: In 2005, Graham-Zworski introduced a continuous family of conformal covariant operators of high orders via scattering theory on conformal compact Einstein manifolds. This class of operators P° and their associated curvature Q° has played important roles in problems in conformal geometry and in the study of some geometric invariants in the Ads/CFT setting. In the talk, I will survey some of the recent progress in this field, and also report some recent joint work with Jeffrey Case about positivity property of this class of operators under curvature assumptions.

## by Paul Yang, Princeton University

### 5/14/14
"**CR
geometry in 3-D**"

Abstract: In this talk, I report on the Embedding problem for 3-D CR structures. As a consequence of the embedding criteria, we obtain a positive mass theorem. Further application is the analysis of the new operator on the pluriharmonic functionsand the associated Q'curvature. The work were joint with Case, Chanillo, Cheng, Chiu, and Malchiodi.

### 5/15/14
"**A
fourth order operator in conformal geometry**"

Abstract: In this talk, I report on the Paneitz operator and the Q-curvature equation. We obtain criteria for the sign of the Green's function for this operator, and hence the solvability of the Q-curvature equation in all dimensions. This is a joint work with Fengbo Hang.

## by Gang Tian, Princeton and Beijing Univ.

### 10/28/13
"Kahler-Einstein metrics with positive scalar curvature"

Abstract: It has been a long-standing problem to studying the existence of Kahler-Einstein metrics on compact Kahler manifolds with positive first Chern class. There are obstructions to the existence, by Matsushima in late 50s, by A. Futaki in early 80s and then K-stability in 90s. In this general talk, I will give a brief tour on the study of Kahler-Einstein metrics on Fano manifolds in last two decades and then discuss some recent progresses on this existence problem.

### 10/29/13 "K-stability and GIT"

Abstract: In this talk, I will discuss the K-stability, its original definition as well as new formulations. I will show how it is related to the Geometric Invariant Theory.

## by Richard Melrose, MIT

## 11/26/12 3:30-4:30pm "Towards analysis on loop manifolds"## 11/28/12 3:30-4:30pm "Toeplitz operators and Bott-Virasoro"## 11/30/12 3-4pm, "Spin representation and Dirac-Ramond operator" |