Introduction to Differential Geometry and Topology

Description: This course is intended as an introduction to the modern differential geometry and topology.

In the last fifty years the ideas and methods of differential geometry and topology become very important tool in many different areas of physics, e.g., physics of condenced matter, quantum field theory, non-linear optics, supersymmetry etc. Surprisingly, many interesting mathematical structures which appear in differential geometry and topology, found a direct realization in various physical systems. From a pragmatic point of view the language and methods of differential geometry and topology provide a substantial extension of the 'tool kit' of a physicist. They also serve as valuable source for intuition about the pattern of evolution of physical systems.

This course aims at giving a self-contained introduction in the differential geometry and topology, it provides a set of simple examples of the physical concepts and mathematical ideas which where developed in that context.

Syllabus:

1. Basic ideas and concepts of topology. Eiler index and topological classification.
2. Projective spaces.
3. Homotopy, groups of homotopy.
4. Simplicial complex. Homology and groups of homology. Betti numbers.
5. Differential forms. Wedge product and other basic operations on the forms
6. Hodge duality.
7. Stokes theorem, Maxwell electrodynamics in the formalism of the differential forms.
8. Vector fields on the manifolds. Hopf-Poincare theorem.
9. Lee derivative of the vector fields and differential forms.
10. Fiber bundes, connection and curvature.
11. Hopf bundle.

Lectures

• Lectures 1-3
• Lectures 4-6
• Lectures 7-9
• Lectures 10-12