Topology of random hypersurfaces
PhD course at KTH
Duration: 20h (10 lectures)
Starting date and schedule.
First lesson on February 7 at 2pm in room 3418
The course continues with the following schedule:
Tue 2pm--4pm, room 3418
Wed 10am--12pm, room 3418
No lectures in the week February 20--25!
We resume on February 28!
Lecture notes available here.
(Last update 15/02/23)
Course description.
Hilbert's Sixteenth Problem (H16), in a modern formulation, asks for the study of the number, shape and position of the components of a smooth algebraic hypersurface of degree d in the n-dimensional real projective space. For example, in the case n=2, Harnack's inequality provides the bound (d-1)(d-2)/2+1 for the number of connected components of a smooth real algebraic curve of degree d in the plane. However, the number of possibilities in which these components can be embedded in the plane, up to isotopies, grows super-exponentially in d, and this seems to rule out a large-degree approach to H16.
In this course I will introduce an alternative point of view on this problem, by endowing the space of hypersurfaces with a natural probability distribution and studying instead the statistical behavior of these objects. I will therefore formulate a random version of H16, trying to explain in which sense real algebraic geometry, on average, behaves as the ''square root'' of complex algebraic geometry. This approach fits into the framework of the emerging field of "Random Algebraic Geometry", which aims at replacing, over the Reals, the notion of "generic" (from complex algebraic geometry) with the notion of "random". I will introduce the basic tools from this subject and show how they can be applied more generally for the study of the topology of "random objects" (for instance random simplicial complexes emerging from random point clouds).
Course literature.
Auffinger, A., Lerario, A. and Lundberg, E. Topologies of Random Geometric Complexes on Riemannian Manifolds in the Thermodynamic Limit, International Mathematics Research Notices, Volume 2021, Issue 20, October 2021, Pages 15497–15532, https://doi.org/10.1093/imrn/rnaa050
Diatta, D.N., Lerario, A. Low-Degree Approximation of Random Polynomials. Found Comput Math 22, 77–97 (2022). https://doi.org/10.1007/s10208-021-09506-y
Gayet, D. and Welschinger, J.-Y. Lower estimates for the expected Betti numbers of random real hypersurfaces. Journal of the London Mathematical Society, (2014) 90: 105-120. https://doi.org/10.1112/jlms/jdu018
Gayet, D. and Welschinger, J.-Y. Betti numbers of random real hypersurfaces and determinants of random symmetric matrices. J. Eur. Math. Soc. 18 (2016), no. 4, pp. 733–772
Kahle, M. Random Geometric Complexes. Discrete Comput Geom 45, 553–573 (2011). https://doi.org/10.1007/s00454-010-9319-3
Lerario, A. What is...random algebraic geometry? available at https://drive.google.com/file/d/1qZ2ebeV4gv7TTEyFRdebGhWZpEp8XY5V/view
Nazarov, F.; Sodin, M. Asymptotic Laws for the Spatial Distribution and the Number of Connected Components of Zero Sets of Gaussian Random Functions. J. Math. Phys. Anal. Geom. 2016, 12, 205-278.
Sarnak, P. and Wigman, I. Topologies of Nodal Sets of Random Band-Limited Functions. Comm. Pure Appl. Math., 72: 275-342. https://doi.org/10.1002/cpa.21794
Intended learning outcomes.
To understand and be able to apply probabilistic techniques for the study of the topology (e.g. Betti numbers) of random geometric objects.
Course structure.
Lectures and presentations by course participants.
Prerequisites.
No advanced prerequisites will be needed.
Examination.
The exam will consist either in a short research project, supervised by the lecturer, or in a short presentation of some research articles related to the course.