Metric algebraic geometry
Abstract and reference
In this short course we will look at real algebraic geometry (i.e. the study of zero sets of real polynomial equations) from the point of view of metric geometry and measure theory. More precisely, we will put a natural metric on the space of real polynomials and study classical notions (e.g. the notion of degree) using tools from metric measure theory. We will discuss the geometry of the discriminant, generalizing Eckart-Young Theorem, and of its complement, studying the topology of "most" real hypersurfaces.
1. Real representations of compact groups
2. Spherical harmonics and a "real" version of the Fundamental Theorem of Algebra
3. Invariant scalar products and the Bombieri-Weyl metric
4. Eckart-Young theorem and the distance to the discriminant
5. Topology of real algebraic hypersurfaces.
Preliminary lecture notes available here https://drive.google.com/file/d/1A6UzYuv1OjucRscwZOQ4mDakKSfk_c77/view