Mathematics

In the course we will cover the main ideas from Riemannian geometry (distances, volumes, tubes, ...) with a point of view on metric and topological properties of real algebraic varieties.  The list of topics to be covered includes:

• Basics of Riemannian geometry
• Definition of volume on Riemannian manifolds and examples
• Gaussian measures, volumes, probabilities
• Random linear spaces, Random matrices (GOE and more...)
• Invariant measures on space of polynomials (representation theory etc.)
• Normal bundles, exponential map
• Weyl's tube formula
• Distance function to submanifolds, properties
• Eckart-Young Theorem and generalizations
• Integral Geometry
• Poincarè formulas in Riemannian homogeneous spaces
• Degree and volume
• Morse theory of distance functions

 References

• Do Carmo, Riemannian Geometry
• Howard, The kinematic formula in Riemannian homogeneous spaces
• Lecture notes (to be prepared)