# Differential Geometry Seminar Schedule for Spring 98

# Fridays 3:30-5pm, SH 6623

This quarter we are having Reading Seimnar in Geometry
(A Part of the Geometry Seminar)
Graduate Students are strongly encouraged
to participate.

We are going to cover two very exciting topics
of Riemannian Geometry and Geometrical Analysis,
with applications to physics in mind.

Topic 1: Gromov's macroscopic picture of scalar
curvature

Scalar curvature is the trace part of the curvature
tensor, and is important for understanding conformal
geometry, Einstein metrics, general relativity and
many other subjects. According to Gromov, manifolds have
"macroscopic dimension", which is the dimension you see
when you view a manifold from a far distance. Small
directions of the manifold are compressed in such view,
so you get a macroscopic picture. (So we humans are
0-dimensional in this view!)
The theme of Gromov is this: manifolds of positive
scalar curvature have macroscopic dimension n-2, i.e.
2 dimensions are compressed when viewed from afar.
Many interesting topics such as spectrum pop up in the
course of analysing this picture.

Topic 2: Penrose Conjecture in General Relativty and
Spin Geometry

Do you know what is the most important global physical
quantity associated with an isolated space-time (such
as our solar system)? It's the so-called ADM mass.
It is a rather mysterious, yet fundamental physical
entity. A fundamental conjecture of Penrose gives a lower
bound for the ADM mass of a universe in terms of its
boundary area (the area of the horrible event
horizon). Of course, this can be formulated in terms
of geometry (indeed, Riemannian geometry). What have
geometers done about this conjecture? They proved it.
We are going to present the details of the proof.
The tool is spin geometry and PDE.
(A secret message: the story of Penrose conjecture is
not yet finished. Using the methods, one may do more.)

### 4/10 Guofang Wei ``Introduction to Scalar Curvature"

### 4/17 Chad Sprouse, UCLA ``Integral curvature bounds and the diameter of Riemannian manifolds"

### 4/24 Rugang Ye ``The Penrose Conjecture in General Relativty I"

### 4/30 Rugang Ye ``The Penrose Conjecture in General Relativty II"

### 5/8 Rugang Ye ``The Penrose Conjecture in General Relativty III"

### 5/15 Xianzhe Dai ``The Penrose Conjecture: analysis background"

Return to Seminars and Colloquium page

Return to Guofang Wei's home page

# Schedule of Winter 98

### 01/9 no meeting

### 1/16 R. Bryant, Duke University ``Finsler Manifolds with Constant Flag Curvature"

### 1/22, 2pm, Y. Ruan, University of Wisconsin ``Symlectic Surgery and Gromov-Witten Invariants of
Calabi-Yau 3-Folds"

### 1/22, 3:45pm, R. Bott, Harvard University ``Configuration Space Invariants for 3-Manifolds"

### 1/23 E. Witten, Institute for Advanced Study ``Integration over the u-Plane in Donaldson Theory"

### 1/30 S. Wu, ``On the instanton complex of holomorphic Morse theory"

### 2/11 Fanghua Lin, CIMS, NYU, ``Sobolev mappings,Fundamental groups and defect measures"

### 2/27 Xianzhe Dai, ``Torsions for mnaifolds with boundary"

### 3/6 Xianzhe Dai, ``Torsions for mnaifolds with boundary (continued)"

### 3/13 Xiaochun Rong, Rutgers University ``Positive curvature, local symmetry and fundamental groups"

### 3/20 Paul Yang, USC ``Regularity of biharmonic maps"