Spinor Analysis on 3- and 4-manifolds
Student No.:50
Time:Mon/Wed 15:10-17:00(except for public holidays and the 8th week of the calendar)
Instructor:Abul Masood-ul-Alam  [Tsinghua University]
Place:Conference Room 1, Floor 1, Jin Chun Yuan West Building
Starting Date:2015-9-21
Ending Date:2015-12-30



Spinors have important applications in differential geometry and general relativity. Unlike the case of tensor analysis much of spinor analysis 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, with tensors we do much computation without going deep into representation theory. Aim of this course is to learn performing similar applications with spinors. Starting with SU(2) spinors on 3-dimensional Riemannian manifolds from very basics, we learn to do calculation in local coordinates, study positive mass theorem, and spinor representation of surfaces in Euclidean 3-space. We study spinors on Riemannian 4-manifolds and spend few hours on CP2 as an example of non-spin manifold. Finally we study SL(2,C) spinors in 3+1 space-times. The course is suitable for both graduate and senior level undergraduate students.





Multivariable Calculus, Real Analysis and Linear Algebra.
Geometry: manifolds, knowledge of covariant derivative and curvature, or concurrent enrollment in a Riemannian Geometry course.





There is no single reference. Following books are useful.
1. “Spinors in physics” by Jean Hladik.
2. “The Theory of Spinors” by Elie Cartan.
3. “Spinors and space-time” by Roger Penrose and Wolfgang Rindler.
4. “Elliptic operators, topology and asymptotic methods” by John Roe.
5. “Dirac operators in Riemannian Geometry” by T. Friedrich.