Lecturer:
Course Type:
PhD Course
Master Course
Academic Year:
2021-2022
Period:
October-January
Duration:
60 h
Description:
Real Geometry
(Differential Geometry for LM)
The lectures will also be streamed on zoom, access through this link:
https://sissa-it.zoom.us/j/89166275709
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:
https://chat.whatsapp.com/GTJh8hjOzy9IY9fHLzZNhh
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
Location:
TBC(to be checked)
Location:
A-128 Monday, A-136 Wednesday