This course is a follow-up of a previous course that I taught in 2021, with the same title. In the 2021 version of this course we covered topological properties of real algebraic varieties,  using tools from differential geometry. In this course we examine metric properties of varieties, using tools from Riemannian geometry and probability theory. The main focus will be on two classical topics from Riemannian geometry: the Integral Geometry Formula in Riemannian homogeneous spaces and Weyl's tube formula. We will cover the basics of Riemannian geometry and then move to these topics and their applications to real and complex algebraic geometry. 

The exam consists in a seminar deepening some of the topics we covered in class, possibly studying one the papers listed below  (the list will be updated).
I will be happy to discuss more topics by appointment. 

The lecture notes of the course are available at this link (last version June 1, 2024)
They are still incomplete, they will be updated weekly.

Other interesting references from where I will take inspiration for the lectures are:
A. Gray, "tubes"
R. Howard, "The kinematic formula in Riemannian homogeneous spaces"

Here is a list of possible topics for the exam (contact me if you cannot find the source):

• "Complexity of Bezout's Theorem II" by Mike Shub and Steve Smale

• "How many zeroes of a random polynomial are real?" by Edelman and Kostlan

• "The probability that a numerical analysis problem is difficult" by James W. Demmel

• Chapter 11 "Condition number and the loss of precision in linear equations" from the book "Complexity and real computations", by Leonore Blum, Felipe Cucker, Mike Shub and Steve Smale (it contains the proof the general Eckart-Young theorem)

• Volume of tubes around algebraic sets.

• "On the total curvature of immersed manifolds" by Chern and Lashof

• "Volume of tubes about Khaler submanifolds" by Gray

• Grassmannian integral geometry and probabilistic Schubert Calculus

• Nonarchimedean integral geometry

• "Hermitian Integral Geometry" by Bernig and Fu

• full proof of Bernstein-Khovanskii-Kouchnirenko (with mixed volume) and the topology of monomial hyeprsurfaces (a-la Khovanskii)

• Alexandrov-Fenchel and Hodge inequality (chapter 27 from "Geometric inequalities" by Burago and Zalgaller)

• "Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory" by Kaveh and Khovanskii

• Weyl's paper on volume of tubes (look also at the book "Condition" by Burgisser and Cucker and the book "tubes" by Gray)

• The moment map in Hamiltonian geometry:
  Angular momentum, convex Polyhedra and Algebraic Geometry  by Atiyah
  "Convexity and commuting hamiltonians" by Atiyah
  Duistermaat-Heckman Theorem (section 2 from this book)

• Kozhasov's paper on critical points of spherical harmonics