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"