Program
 Spinor Analysis on Riemannian 3-manifolds Student No.： 50 Time： Tue/Thu15:30-17:00, 2013-02-25~ 2013-06-05(except for public holidays) Instructor： Abul Masood-ul-Alam  [Tsinghua University] Place： Conference Room 3, Floor 2, Jin Chun Yuan West Building Starting Date： 2014-2-25 Ending Date： 2014-6-5

Description:

Spinors have important applications in general relativity and differential geometry. Unlike the case of tensor analysis much of spinoranalysis is masked by representation theory jargons, which vary considerably with dimension and metric signature, thus becoming difficult and time consuming for analysts. Although tensors are also elements of representation spaces, and we decompose tensors belonging to invariant subspaces, with tensors we do much computation without going deep into representation theory. Aim of this course is to learn performing similar applications with spinors. We start with SU(2) spinors in 3-dimension from very basics, learn to do calculation in local coordinates, study Dirac equation and positive mass theorem. Finally we study SL(2,C) spinors in 3+1 space-times. The course is suitable for both graduate and senior level undergraduate students.

Prerequisite:

Geometry: manifolds, knowledge of Christoffel symbols and curvature, or concurrent enrollment in a Riemannian Geometry course.

Reference:

There is no single reference. Following books and articles are useful.
1. Spinors in physics by Jean Hladik ;.
2. The Theory of Spinors by ElieCartan.
3. Spinors and space-time, Roger Penrose and Wolfgang Rindler.

4. Elliptic operators, topology and asymptotic methods, John Roe.
5. “Positive scalar curvature and the Dirac operator on complete Riemannian manifold,” M. Gromov and H. B. Lawson, Publ. Math. IHES 58 (1983).
6. “Essential self-adjointness for the Dirac operator and its square,” J. Wolf, Indiana Univ. Math. J. 22 (1972/73) 611-640.
7. “The Mass of an Asymptotically Flat Manifold,” R. Bartnik, Commun. Pure
Appl. Math. XXXIX (1986) 661-693.

8. “Eigenvalue Boundary Problems for the Dirac Operator,” O. Hijazi, S. Montiel, A. Roldan, Commun. Math. Phys. 231 (2002) 375-390.

Multivariable Calculus, Real Analysis and Linear Algebra.