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Real Geometry

Course Type: 
PhD Course
Master Course
Academic Year: 
60 h

Real Geometry
(Differential Geometry for LM)

The lectures will also be streamed on zoom, access through this link:  

TThere is a whatsapp channel for the course, we will use it as a diary for the class and for practical communications, it can be joined through this link:

1. Semialgebraic geometry

  • Semialgebraic sets and functions
  • Semialgebraic triviality and applications
  • Triangulation of semialgebraic functions
  • Regularization of semialgebraic sets

2. Complex and real discriminants

  • Dimension and stratifications
  • Thom's isotopy lemma
  • Generic versus random

3. Volume and measures

  • 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.)

4. Tubes

  • Normal bundles, exponential map
  • Weyl's tube formula
  • Distance function to submanifolds, properties
  • Eckart-Young Theorem and generalizations, random matrices and applications
  • Geometry of the Veronese variety

5. Integral Geometry

  • Poincarè formulas in Riemannian homogeneous spaces
  • Degree and volume
  • How many zeroes of a random polynomial are real?
  • Probabilistic Schubert Calculus
  • Kac-Rice formulas

6. Topology of real algebraic varieties

  • Nash theorem
  • Morse theory and Thom-Milnor theorem
  • Random algebraic varieties
TBC(to be checked)
A-128 Monday, A-136 Wednesday
Next Lectures: 

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