主 题: 鐗瑰埆鏁板?璁插骇绗?4鏈
时 间: 2011-06-13 08:00 - 2011-07-13 18:00
地 点: 鍖椾含鍥介檯鏁板?鐮旂┒涓?績 璧勬簮澶у帵1218鏁欏?
鏃堕棿锛 2011骞?鏈?3鏃 - 2011骞?鏈?3鏃
1. Current Progress on Stability of Black Holes
Speaker: Pieter Blue
Abstract: General relativity is a geometric theory of gravity, and a theory that has been tested to an incredible accuracy. In general relativity, the universe is described by a four dimensional manifold of space-time points with a Lorentz metric. The metric satisfies a hyperbolic partial differential equation, known as Einstein's equation. A class of particular important solutions is the family of Kerr solutions, which includes the Schwarzschild subfamily. It is generally accepted by physicists that all stationary, asymptotically flat solutions that contain a single black hole belong to the Kerr family of solutions, and that all black holes will asymptotically approach a Kerr solution under the evolution given by the Einstein equation. In particular, it is expected that the Kerr black holes are asymptotically stable. In these lectures, we will discuss the background for, results related to, and current progress on this problem.
2. SOLUTIONS OF THE EINSTEIN INITIAL VALUE CONSTRAINT EQUATIONS
An initial data set for Einstein's gravitational field equations cannot be chosen freely; it must satisfy the Einstein constraint equations. In this series of lectures we discuss what we know about the solutions of these constraint equations. We focus in particular on what the use of the conformal method and the use of gluing techniques has taught us about the parametrization and the construction of initial data sets satisfying the constraints. We review the well-known results regarding constant mean curvature and near constant mean curvature solutions, and we report on recent progress that has been made in studying solutions which fit in neither of these categories.
3. The Penrose Inequality, Quasi-Local Mass, and the Hoop Conjecture
Speaker: Marcus Khuri
Abstract: We will begin by reviewing the basic mathematical formulation of General Relativity, with the aim of understanding the consequences and proofs of the positive mass theorem. Next we will study a refinement of the positive mass theorem to the case in which spacetime contains black holes. This is known as the Penrose Inequality, and provides a lower bound for the mass in terms of the area of the black hole horizons. It has been proven in the time symmetric case, but remains open in general. We will study the known proofs and also the proposals for establishing the general theorem. The second part of the course will focus on the existence problem for black holes, known as the hoop conjecture. The goal is to provide a rigorous mathematical theorem for the heuristic physical intuition that black holes form when too much matter/energy is enclosed in a sufficiently small region. Directly related to this problem is the question of quasi-local mass, which asks for a geometric quantity describing the total mass (gravitation plus matter) content of a bounded region. Various proposals for such a definition will be surveyed.
4. Topics in Geometric Analysis
Speaker: Xiaodong Wang
Abstract: We plan to focus on the scalar curvature in Riemannian geometry. In the first part we will discuss some conformal geometry involving the scalar curvature. In particular we will study the Yamabe problem and its solution. In the second part, we will discuss some topological results involving scalar curvature. There are two approaches: one using spinors and Dirac operators and the other using minimal hypersurfaces and the second variation formula. We will discuss both approaches. In the third part we will discuss the positive mass theorem and its geometric applications.
5. The Yamabe Problem: existence and compactness questions
Speaker: Fernando Coda Marques
Abstract: The Yamabe