Math 7590, Fall 2001

Ambar Sengupta, Office: Lockett 324

e-mail: sengupta@math.lsu.edu

Home: http://www.math.lsu.edu/~sengupta

Class will meet MWF 1:40pm-2:30pm, Lockett 111

Office hours: 10am-12 noon, Mondays and Wednesdays, and by appointment.

The following notes in PDF format:

1. Superalgebra
2. Abstract Differential Calculus
3. Extracting/Encoding Geometry in Algebra
4. Tensor Products
5. Vector-valued Differential Forms, Representations
6. Bundles, Connections, Covariant Derivatives
7. Differential Forms II
8. Homework 1 on superalgebra.
9. Homework 2 on sections of bundles and Lie derivatives of sections.

This is an introduction to calculus on bundles. The course plan looks roughly as follows:

1. Background from Algebra
2. Differential Calculus on Manifolds
3. Review of Lie groups and Lie algebras
4. Principal bundles
5. Connections
6. Parallel transport
7. Curvature
8. Associated bundles

We shall also pay attention to the symbiotic relationship these topics have with parts of physics (gauge theory). No single text will be used but notes, as well as a guide to the literature, will be provided.

Knowledge of manifolds and basic algebraic notions (rings, modules, algebras) will be necessary for this course.

Useful books include:

• Foundations of Differential Geometry volume I, by Shoshichi Kobayashi and Katsui Nomizu, published by John Wiley and Sons
• Gauge Theory and Variational Principles by David Bleecker, published by Addison-Wesley

Problems will be assigned as we progress through the topics and grades will be decided on the basis of performance on these problems. Satisfactory work on eighty percent or more of the problems will correspond to a grade of A. Sixty to eighty percent will correspond to B. Forty to sixty percent will be needed to get a C.

The following tentative plan will be followed in a flexible way.

Week 1. Algebra

1. Rings, Modules, Algebras, Morphisms
2. Free modules, tensor and exterior products
3. Exterior algebras, derivations

Week 2. Superalgebra and abstract calculus

1. Superalgebra
2. Superalgebra
3. Abstract differential calculus

Week 3.

1. Abstract differential calculus
2. Review of multivariable calculus
3. Manifolds

Week 4.

1. Manifolds, Lie groups
2. Principal bundles
3. Principal bundles

Week 5. Associated Bundles

1. Principal bundles, frame bundle
2. Associated bundles
3. Associated bundles

Week 6.

1. Tangent bundle, cotangent bundle
2. Tensor bundles
3. Sections of bundles, vector fields

Week 7.

1. Vector fields
2. Differential Forms
3. Lie algebra

Week 8. Connections

1. Connections
2. Connections
3. Parallel-transport
4. Connections, holonomy

Week 9. Curvature

1. Curvature
2. Curvature
3. Curvature