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Differential Geometry

Lecturer: 
Course Type: 
PhD Course
Master Course
Anno (LM): 
Second Year
Academic Year: 
2023-2024
Period: 
March-May
Duration: 
40 h
Description: 

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)
Next Lectures: 
Tuesday, March 5, 2024 - 16:00 to 18:00
Wednesday, March 6, 2024 - 14:00 to 16:00
Tuesday, March 12, 2024 - 16:00 to 18:00
Wednesday, March 13, 2024 - 14:00 to 16:00
Tuesday, March 19, 2024 - 16:00 to 18:00
Wednesday, March 20, 2024 - 14:00 to 16:00
Tuesday, March 26, 2024 - 16:00 to 18:00
Wednesday, March 27, 2024 - 14:00 to 16:00

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