This course explores the topology of random algebraic hypersurfaces. The starting point is Hilbert sixteenth Problem, which asks for the possible topologies of real algebraic hypersurfaces of degree $d$. In the case of plane curves, Harnack’s inequality provides a bound on the number of connected components, but as d grows the number of isotopy types grows super-exponentially, making a deterministic classification hopeless.
A probabilistic viewpoint offers a new perspective: by endowing the space of real algebraic hypersurfaces with a natural probability distribution, one can study their statistical topology. This leads to a probabilistic reformulation of Hilbert’s problem and reveals a striking phenomenon: in expectation, the complexity of the real world grows like the square root of its complex counterpart.
The course will introduce the techniques underlying this circle of ideas. We will start with the simplest case: the distribution of zeros of univariate random polynomials. From there, we will move to higher dimensions, discussing random Morse theory, the Nazarov–Sodin barrier method, and the low-degree approximation principle (showing that “most” degree-$d$ hypersurfaces are ambiently isotopic to a degree-$\sqrt{d}$ one — with “most” to be understood in a probabilistic sense). A central theme will be the choice of a meaningful probability measure on the space of real algebraic hypersurfaces.
Beyond real algebraic geometry, these methods connect to the study of random nodal sets of Gaussian fields.
LiteratureA non-exhaustive list of references:
- Lecture notes by the lecturer will be provided
- D. Diatta, A. Lerario: Low degree approximation of random polynomials, FoCM (2022).
- D. Gayet, J.-Y. Welschinger: Betti numbers of random real hypersurfaces and determinants of random symmetric matrices, JLMS (2014).
- D. Gayet, J.-Y. Welschinger: Expected topology of random real algebraic submanifolds, JEMS (2016).
- A. Lerario: What is… random algebraic geometry? (lecture notes).
- F. Nazarov, M. Sodin: Asymptotic laws for the number of components of zero sets of Gaussian fields, JAMS (2016).
- P. Sarnak, I. Wigman: Topologies of nodal sets of band-limited functions, CPAM (2019).