# Academic year 2023-2024 : Geometrical aspects in probability

General introduction:

This year of master is dedicated to contemporary links between probability and geometry. These links have at least three aspects: first, many natural probabilistic questions arise in geometrical contexts, such as Bernoulli percolation, which aims at understanding the large scale topological features of random subgraphs of $\mathbb{Z}^d$. Second, a probabilistic viewpoint on a deterministic geometric object can provide unexpected understanding of this object, such as random projections of convex subsets in large dimensions. Finally, some geometric questions have no single deterministic answer, so it is natural to turn the problem into a probabilistic one, such as the statistics of the topology of the vanishing locus of a real polynomial. The topics of this year's master's program cover some very active areas in geometric probability. All courses are designed to provide useful general tools in addition to explaining current theories. During the first semester, the first course is dedicated to important classical theorems and tools in differential geometry (Morse theory in geometry and topology), the second one is devoted to random structures on lattices, like percolation and Ising models (Random models on lattices) and the third one is devoted to the links between Brownian motion on manifolds and analysis, e.g through the study of the heat equation (Analysis and probability on manifolds). During the second semester two specialized courses are proposed: the first one is dedicated to the topology of smooth random hypersurfaces (Topology of random hypersurfaces), and the second one to geometrical characteristics of convex subsets in high dimensional normed spaces (Probabilistic and geometric techniques in high dimension). Students must follow two of the three courses of the first semester, and one of the two courses of the second semester.

Semester 1: introductive courses

1. Morse theory in geometry and topology (Sylvain Courte and Paolo Ghiggini)

Description:

Morse theory is the study of critical points of real-valued functions on manifolds and the dynamics of their associated gradient vector fields. It has been extensively used in differential topology as a tool to study the topology of manifolds. The closely related theory of handle decompositions is also prominent in topology and is the basis of Smale's proof of the Poincaré conjecture in high dimension (a manifold homotopy equivalent to the sphere is homeomorphic to it). A more modest but extremely fruitful approach is to use Morse theory to compute the homology of a manifold, via what is now called Morse homology. When extended to infinite dimensional manifolds such as the loop space of a manifold, Morse theory can help to study the geodesics of Riemannian metrics. In turn a thorough study of geodesics in symmetric spaces can be used to prove homotopical phenomena such as the Bott periodicity theorem concerning homotopy groups of the classical Lie groups. Morse theory has also proved useful in the study of the topology of complex algebraic
varieties, see for instance Lefschetz's hyperplane section theorem. These are many great applications of Morse theory to geometry and topology that we will try and cover during the lectures, as time permits.

Program:

The lectures will also contain more classical material from differential geometry and topology, including:

• Manifolds, differential forms, vector fields, transversality, vector bundles, submanifolds and their tubular neighborhoods.
• Riemannian metrics, curvature, geodesics.
• Homology, cohomology, homotopy groups.

Prerequisites:

• Differential calculus in $\mathbb{R}^n$, submanifolds.
• Modules over a ring.

Bibliography:

• Milnor - Morse theory
• Milnor - Lectures on the h-cobordism theorem
• Milnor - Topology from the differential viewpoint
• Laudenbach - Courants, transversalité, théorie de Morse
• Hirsch - Differential topology
• Audin, Damian - Théorie de Morse et homologie de Floer

2. Random models on lattices (Loren Coquille and Hugo Vanneuville)

Description:

Random models on lattices constitute a mathematical framework for the study of phase transitions, which are sudden changes of behaviour of a medium as some parameter crosses a critical value (think of water turning into ice as temperature goes to zero --
we are all very used to this phenomenon, but isn’t it surprising that a system can be so different at temperatures T = +0.001°C and T = −0.001°C, while the microscopic interaction of the particles has essentially not changed?).

In this course, we will study the counterpart of this phenomenon for models on lattices, by investigating:

• the percolation transition in models of random graphs (Percolation model);
• the magnetisation transition in models of random spins (Ising/Potts models).

A framework which encapsulates many features of these models is the following: Consider a random spin configuration on the lattice $\mathbb{Z}^d$, that is, a random function $\sigma : \mathbb{Z}^d \rightarrow \{-1,+1\}$ whose law depends on a "temperature parameter" $T>0$, in such a way that, as $T$ decreases, the local interaction between two neighboring vertices $v$ and $w$ increases continuously (the "interaction between $v$ and $w$" can be for instance measured by the probability that $\sigma(v)$ equals $\sigma(w)$).

We will be interested in understanding the emergence of large scale interactions, that can be thought:

• in the geometric sense: prove that two very distant sites have a non-negligible probability to be in the same connected component of +1's or −1's;
• in the probabilistic sense: prove that, even when $v$ and $w$ are far from each other, there is some non-negligible dependence between $\sigma(v)$ and $\sigma(w)$.

All in all, the question of phase transitions in the context of random models on lattices can be stated as follows:

Does there exist a critical value such that, when $T$ crosses this value, interactions at large scales suddenly appear?

We will prove that such sudden changes happen in percolation and spin models, and we will see that these changes coincide with the emergence of fascinating geometric (fractal) structures.

More precisely, the goal of this course is to:

• Give an introduction to important examples of random models on lattices;
• Prove the existence and describe the phase transitions for these models;
• Study interactions in the probabilistic or geometric sense. Key tools include "geometric representations of lattice models", that enable to unify the notions of probabilistic and geometric interactions;
• Study geometric objects emerging at large scales: fractal curves, minimal surfaces, ... , and study geometric properties of clusters and interfaces: fluctuation, connectivity, ...

Program:

• Percolation: phase transition, correlation inequalities, coincidence of various critical parameters, fractal properties of clusters at the critical parameter;
• The Ising and Potts models: phase transition, correlation inequalities, geometric representations, minimal surfaces, fluctuations of interfaces.

Prerequisites:

Master 1 probability course (conditional expectation, martingales and Markov chains).

Bibliography:

• Percolation, B. Bollobás and O. Riordan
• Lectures on the Ising and Potts models on the hypercubic lattice, H. Duminil-Copin
• Percolation, G. Grimmett
• Statistical Mechanics of Lattice Systems, S. Friedli and Y. Velenik

3. Analysis and probability on manifolds (Vincent Beffara and Baptiste Devyver)

Description:

Among the many success stories in probability theory, the probabilistic method is the use of results from probability theory to other domains of mathematics: for instance, one can show that an object with prescribed properties exists by showing that a random object of the same type has these properties with positive probability, which is often easier to do than giving an explicit example. In the case of analysis, one can use probabilistic tools to show the existence of solutions to (partial) differential equations, and to obtain estimates on such solutions that are sometimes dificult or technical to derive in other ways.

The aim of this course is to develop the general theory of diffusions and stochastic calculus, first in Euclidean space and then on manifolds, and to apply it to a few questions in analysis and geometry.

Program:

The course will be subdivided into two main parts, the first one more probabilistic and the second one with a more geometric theme.

Part 1: Brownian motion and stochastic calculus

We will  introduce the relevant theory on filtrations and stochastic processes in continuous time, Brownian motion in $\mathbb{R}^d$, Itô calculus, and diffusions. Then we will study some of their applications in analysis by looking into probabilistic proofs of classical theorems, for instance:

• The solution of the Dirichlet problem in a domain in terms of stopped Brownian motion,
• The use of coupling to show Liouville's theorem that bounded harmonic functions on $\mathbb{R}^d$ are constant,
• The Feynman-Kac representation of solutions of parabolic partial differential equations,
• The use of planar Brownian motion to show the Riemann mapping theorem in complex analysis,
• If time allows, applications of branching processes to some non-linear PDEs.

Part 2: Processes on manifolds and applications

In this second part we will exploit stochastic calculus to first define Brownian motion on a smooth manifold, and then use it to obtain analytic and geometric results such as,

• Feynman-Kac formulas for solutions of the heat equation on differential forms,
• Probabilistic techniques in harmonic analysis: Littlewood-Paley-Stein inequalities and applications to the boundedness of Riesz transform on manifolds,
• If time allows, Bismut formula for representing the gradient of solutions to the heat equations.

Prerequisites:

• M1 level probability theory (Gaussian vectors, martingales and Markov chains in discrete time)
• Differential calculus and differential equations in $\mathbb{R}^d$

Bibliography:

• Bass, Probabilistic Techniques in Analysis
• Bismut, Mécanique aléatoire
• Rogers and Williams, Diffusions, Markov Processes, and Martingales
• Oksendal, Stochastic Differential Equations
• Meyer, P. A., Démonstration probabiliste de certaines inégalités de Littlewood-Paley, Semin. Probab. X, Univ. Strasbourg 1974/75, Lect. Notes Math. 511, 125--183 (1976)
• D. Bakry, Etude des transformations de Riesz dans les variétés riemanniennes à  courbure de Ricci minorée, Sémin. probabilités XXI, {\em Lect. Notes Math.} 1247, 137-172 (1987)
• D. Elworthy, Stochastic methods and differential geometry, Semin. Bourbaki, 33e annee, Vol. 1980/81, Lect. Notes Math. 901, 95--110 (1981)
• B. Driver and A. Thalmaier, Heat equation derivative formulas for vector bundles, J. Funct. Anal. 183, No. 1, 42--108 (2001)

Semester 2: advanced courses

1. Topology of random hypersurfaces (Damien Gayet)

Description:

This course is devoted to the topology of random submanifolds in a fixed manifold. This subject arose in first in the 40's with random polynomials with questions like: if $p$ is a polynomial of degree $d$ with independent standard Gaussian coefficients, what is its average number of roots? What is the probability that this number is maximal?

In higher dimensions, roots are replaced by hypersurfaces, and the number of roots by the number of components. However in this higher dimensions new topological observables can be studied, like the other Betti numbers or the Euler characteristic.

In another direction and in $\mathbb{R}^n$ for $n\geq 2$, one can ask questions about the existence of large connected components of the random hypersurfaces, begining with the probability that there is an infinite one. This subject is linked with classical Bernoulli percolation over lattices.

Program:

• The Kac-Rice formula for computing the average number of zeros. This is a wonderful formula based on the coarea formula, which allows to express the average number of zeros of a random function $f : M^n \rightarrow \mathbb{R}^n$ under very general assumptions on the probability measure for $f$. It is extremely useful in the topic of random topology, but also for metric observables like the volume of the random hypersurface.
• Critical points, Morse theory and Betti numbers of random hypersurfaces. Thanks to the Kac-Rice formula, we can access to the average number of critical points of given index of a random function. This will be used to give upper bounds for the average of Betti numbers of the random hypersurfaces.
• Barrier method. The barrier method is a useful tool for proving that certain topological types arrive with a controlled probability as the trace of random hypersurface in a fixed open subset of the manifold. In particular, this provides a lower bound for the mean Betti numbers of the hypersurface.
• Piterbarg formula. This formula computes the dependency of two topological events in two open sets, in terms of an integral over the two subsets, the covariance of the random Gaussian function and others.
• Random nodal lines and percolation. The main question in this topic is the following: if $f: \mathbb{R}^2\rightarrow \mathbb{R}$ is random, and if $R\subset \mathbb{R}^2$ is a fixed rectangle, what is the probability that there is a connected component of $R\cap \{f=0\}$ touching the left and right sides of $R$? Is this probability uniformly bounded above by a positive constant when $R$ is larger and larger, with a fixed relative shape?  We will explain that the answer to the last question is yes for quite natural random functions.

Prerequisites:

• The first semester course on Morse theory
• Basic knowledge on probability, Gaussian vectors if possible, but not mandatory.

Bibliography:

• Adler, Robert J. and Taylor, Jonathan E.,  Random fields and geometry, 2009, Springer
• Gayet D., Topology of random algebraic and analytic hypersurfaces, Graduate course
• Welschinger J.-Y., Topology of random real hypersurfaces, CIMPA course
• Tassion, V., Crossing probabilities for Voronoi percolation

2. Probabilistic and geometric techniques in high dimension (Cécilia Lancien)

Description:

In this course we will study normed vector spaces (in particular their convex subsets) of finite but large dimension. The general objective will be to understand what are the typical properties of such spaces, which arise precisely in this high-dimensional regime. We will see how probabilistic and geometric techniques can be combined to achieve this goal. The main phenomenon that we will highlight is that of concentration of measure, which can itself be interpreted both from a probabilistic and a geometric point of view, namely: in high dimension, a function of a random vector has an extremely small probability of deviating from its average and a convex set has most of its mass concentrated in a small volume.

We will see several striking consequences of this concentration property. For instance, it can be shown that, when projecting a convex set on a lower-dimensional subspace, a threshold phenomenon happens between two regimes: one where the geometry of the original set is generically almost perfectly preserved and one where it is generically almost entirely lost. Statements of this kind have implications in numerous fields, such as computer science, information theory, etc.

Program:

• First notions and results from convex geometry: Convex bodies. Log-concavity of the volume (Brunn-Minkowski and Prékopa-Leindler inequalities). Polarity. Maximality of Mahler volume for Euclidean balls (Blaschke-Santalò inequality).
• Distance between convex bodies: Banach-Mazur distance. Ellipsoids. Distance of a convex body to a Euclidean ball (John theorem). Distances between usual convex bodies. Khintchine inequality.
• Concentration of measure phenomenon: Measure of spherical caps. Covering and packing on the sphere. Isoperimetry and concentration on the sphere. Isoperimetry and concentration on the Gaussian space. Dimension reduction (Johnson-Lindenstrauss lemma).
• Almost Euclidean sections of convex bodies: Haar measure on the orthogonal group and the Grassmann manifold. Concentration of Lipschitz functions on random subspaces. Embedding a convex body into a cube (Dvoretzky-Rogers lemma). Almost Euclidean sections of a convex body (Dvoretzky-Milman theorem).
• Almost full-dimensional sections of convex bodies: Volumetric estimates on covering numbers. Distance of a random subspace to a fixed point. Volume ratio theorem. Splitting (Kashin theorem).
• Gaussian processes: Comparison inequalities. Metric entropy. Upper bound on the supremum of a Gaussian process (Dudley inequality). Lower bound on the supremum of a Gaussian process (Sudakov and dual Sudakov inequalities). $\ell$-norm, $K$-convexity constant and $MM^*$-estimate.
• Random vectors and matrices: Subgaussian concentration. Typical norm of a random vector. Typical norm of a random matrix.

Bibliography:

• S. Artstein-Avidan, A. Giannopoulos, V.D. Milman. Asymptotic geometric analysis, Part I. American Mathematical Society, Mathematical Surveys and Monographs 202, 2015
• S. Artstein-Avidan, A. Giannopoulos, V.D. Milman. Asymptotic geometric analysis, Part II. American Mathematical Society, Mathematical Surveys and Monographs 261, 2021
• G. Aubrun, S.J. Szarek. Alice and Bob meet Banach: The interface of asymptotic geometric analysis and quantum information theory. American Mathematical Society, Mathematical Surveys and Monographs 223, 2017
• M. Ledoux. The Concentration of Measure Phenomenon. American Mathematical Society, Mathematical Surveys and Monographs 89, 2001
• G. Pisier. The volume of convex bodies and Banach space geometry. Cambridge Tracts in Mathematics 94, Cambridge University Press, 1989
• R. van Handel. Probability in high dimension. Lecture notes
• R. Vershynin. High-dimensional probability: An introduction with applications in data science. Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 2018
• R. Vershynin. Geometric functional analysis. Lecture notes

Prerequisites:

• Standard notions from topology of normed vector spaces and functional analysis.
• Basic notions from measure theory and probability.