Modern Geometry: Mathematics GR6402 (Fall 2017)


Tuesday and Thursday  10:10-11:25
307 Mathematics

This is the first part of a full-year course on differential geometry, aimed at first-year graduate students in mathematics, while also being of use to physicists and others.  The spring semester second part of the course will be taught by Simon Brendle, and concentrate on topics in Riemannian geometry.


Prerequisites

An undergraduate course of some kind dealing with differentiable manifolds, such as our GU4081, Introduction to Differentiable Manifolds, will be assumed. Much of the material of such a course will be reviewed quickly during the early part of this course.


Tentative Syllabus

1. Review of the definition of a differentiable manifold, tangent spaces, vector fields.
2. Tensor algebra, tensor fields, differential forms.
3. de Rham cohomology.
4. Lie groups and Lie algebras.
5. Principal bundles, vector bundles.
6. Connections and curvature on a principal bundle. Connections and curvature for vector bundles.
7. Chern-Weil theory.
8. Frame bundles, torsion, metrics.
9. Laplacians, Hodge theory.
10. Gauge theory, Maxwell's equations, Yang-Mills equations.
11. Clifford algebras, spinors, the Dirac equation.


Actual Syllabus

Tuesday,  September 5:  Introduction to course. Definition of a differentiable manifold.

Thursday, September 7:  Examples of differentiable manifolds.  The tangent space.

Tuesday, September 12:  More on the tangent space. The differential of a smooth map. Vector fields. The Lie bracket. Definition of a Lie algebra.

Thursday, September 14:  The tangent bundle. Vector bundles and sections. The cotangent bundle. Differential 1-forms.

Tuesday, September 19:  Tensor products.  Tensor fields.

Thursday, September 21:   Tensors as multi-linear functions. Symmetric and exterior algebras.  Differential forms.

Tuesday, September 26:  The exterior derivative and its properties.

Thursday, September 28:  Some symplectic geometry and Hamiltonian mechanics.  The Lie derivative.

Tuesday, October 3:  Lie derivative of vector fields.  de Rham cohomology.  Poincaré lemma.

Thursday, October 5:  Some homological algebra. Mayer-Vietoris sequence, computation of cohomology of spheres. Comments on de Rham theorem.

Tuesday, October 10:  Orientations, integration of differential forms, Stokes theorem.

Thursday, October 12:  Lie groups: definitions and examples.

Tuesday, October 17:  Submanifolds. Actions of Lie groups on manifolds.

Thursday, October 19:  Lie algebras: left-invariant vector fields, the exponential map.

Tuesday, October 24:  The adjoint representation. Baker-Campbell-Hausdorff formula. Left-invariant differential forms on Lie groups.

Thursday, October 26:  The Maurer-Cartan equation. Lie algebra cohomology.

Tuesday, October 31:  Principal bundles, the gauge group.

Thursday, November 2:  Connections on principal bundles. Curvature. Bianchi identities.

Thursday, November 9:  Frame bundles, torsion.  Cartan connections.

Tuesday, November 14: Riemannian geometry, the Levi-Civita connection. Formulas in local coordinates.

Thursday, November 16:  More about local coordinates.  Topology of bundles, examples.

Tuesday, November 21:  Chern-Weil theory.

Tuesday, November 28:  More Chern-Weil theory, Maxwell's equations.

Thursday, November 30:  Yang-Mills equations, General Relativity.

Tuesday, December 5:  Covariant derivatives and connections on vector bundles.

Thursday, December 7:  Clifford algebras, spinors and the Dirac equation.

Thursday, December 21: Final Exam 9-12 am Math 307


Homework Assignments

Assignment 1 (due Thursday, September 14)
Assignment 2 (due Thursday, September 21)
Assignment 3 (due Thursday, September 28)
Assignment 4 (due Thursday, October 5)
Assignment 5 (due Thursday, October 12)
Assignment 6 (due Thursday, October 19)
Assignment 7 (due Thursday, October 26)
Assignment 8 (due Thursday, November 2)
Assignment 9 (due Thursday, November 16)
Assignment 10 (due Thursday, November 30)
Assignment 11 (due Thursday, December 7)
Assignment 12  (due Thursday, December 21)


Final Exam

There will be an in-class final exam 9-12 am Thursday December 21.

Office Hours

I should be available after class in my office (Math 421).  Feel free to come by Math 421 at any time and I will likely have some time to talk, or make an appointment by emailing me.

Teaching Assistant

The teaching assistant for the course is Pei-ken Hung (pkhung@math.columbia.edu). He will be available for help with the material of the course Fridays 9-12.


Books and Online Resources

Some recommended textbooks:

Morita, Geometry of Differential Forms.
Kobayashi and Nomizu, Foundations of Differential Geometry, Volume 1.
Spivak, A Comprehensive Introduction to Differential Geometry, Volumes 1 and 2.
Warner, Foundations of Differentiable Manifolds and Lie Groups.
Tu, An Introduction to Manifolds  (this is an undergraduate level book, we will assume much of this material).
Tu, Differential Geometry: Connections, Curvature and Characteristic Classes.
Lee, John,  Introduction to Smooth Manifolds.
Lee, Jeffrey,  Manifolds and Differential Geometry.
Bleecker, David, Gauge Theory and Variational Principles.
Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program.
Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature.
Baum, Helga, Eichfeldtheorie. (only available in German).
Marsh, Adam, Mathematics for Physics: An Illustrated Handbook.

An excellent review article outlining the subject and the relation to physics:
Eguchi, Gilkey and Hanson, Gravitation, Gauge Theories and Differential Geometry.

An excellent series of lecture notes on connections on bundles and gauge theory:
Figueroa-O'Farrill, Gauge Theory