Below is a list of Ph.D. courses currently taught during the academic year 2022-2023.

#### Combinatorics of the flag variety (starting date: April 4)

**Lecturer:** Philippe Nadeau (Université Lyon 1)

**Schedule:** The Course will start on April, 4 (Aula Seminari)

**Duration: **18 hours

In this course, I will present various combinatorial aspects of the complex Grassmannian (briefly) and flag variety (in more detail). We will first recall the classical path going from intersection problems in these varieties to algebraic and combinatorial techniques. In the Grassmannian case, integer partitions and symmetric polynomials (in particular Schur polynomials) play a fundamental role. In the case of the complete flag variety, the combinatorics is that of permutations and Schubert polynomials, which form a basis of multivariate polynomials. We will present these objects in detail, leading to current research questions about them.

#### Monotonicity formulas in non-linear potential theory and their geometric applications (starting date: April, 12)

**Lecturer:** Luca Benatti (Università di Pisa)

**Schedule:** The course will be held from April 12, 2023 –

**Duration: **12 hours

**Description:** I will describe an approach based on non-linear potential theory toward the proof of relevant geometric inequalities, holding in the framework of Riemannian geometry. This approach consists in establishing monotonicity formulas along the level sets of the solutions to well-chosen PDE’s that, combined with careful asymptotic analysis, can be employed to compare the value of a certain geometric quantity with the one on the model. Some selected examples discussed in this course will be the Minkowski inequality and the Riemannian Penrose inequality.

**Prerequisites.** Standard tools in analysis: Sobolev spaces, coarea formula, integration by parts, Holder’s inequality, etc. Basic knowledge of elliptic PDE’s and Riemannian geometry is required.

**Preliminary program:**

- Non-linear potential theory and weak inverse mean curvature flow (IMCF for short): difference and similarity, some monotonicity formulas in literature, notions of p-capacities and strictly outward minimising hulls, the convergence of p-potentials to weak IMCF (with proof, time permitting).Monotonicity formulas along the level set flow of the p-capacitary potential: Bochner’s identity, Kato-type inequality, the sketch of the proof of monotonicity formulas in the setting of Riemannian manifolds with non-negative Ricci curvature.Geometric applications: Minkowski inequality and Riemannian Penrose inequality for the isoperimetric mass.

#### Breaking Nonconvexity: Consensus-Based Optimization (starting date: May, 5)

**Lecturer:** Massimo Fornasier (Technische Universität München)

**Schedule:** The course will be held from May 5, 2023

**Duration: **18 hours

**Description:**

High-dimensional global optimization of nonconvex/nonsmooth functions can be a formidable mathematical and numerical problem with a vast number of possible applications, not last in machine learning. Consensus-based optimization (CBO) is a multi-particle metaheuristic derivative-free optimization method that can globally minimize nonconvex nonsmooth functions and is amenable to theoretical analysis. In fact, optimizing agents (particles) move on the optimization domain driven by a drift towards an instantaneous consensus point, which is computed as a convex combination of particle locations, weighted by the cost function according to Laplace’s principle, and it represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached the stochastic component vanishes. As the dynamics of the algorithm can be described as an Euler-Maruyama approximation scheme of a system of stochastic differential equations (SDE), the first part of the course is about a concise primer on stochastic calculus and stochastic differential equations. We also need to introduce the concept of mean-field limit, to show how the law of the the system of SDE converges in a suitable (weak) sense for the large particle limit to the solution of a partial differential equation (PDE) of Fokker-Planck-type. The combination of the analysis of the large time behavior of the solution of the PDE with the mean-field limit will then be the key strategy to prove the global convergence of the algorithm.

The results unveil the internal mechanisms of CBO that are responsible for the success of the method. In particular, the convergence proof will show that essentially CBO performs a convexification of a very large class of optimization problems as the number of optimizing agents goes to infinity. We further present formulations of CBO over compact hypersurfaces and the proof of convergence to global minimizers for nonconvex nonsmooth optimizations on the hypersphere. We further mention further variations of CBO to include anisotropic noise and impulsive noise exploration, to approximate other methods such as particle swarm optimization, and numerical tricks and implementations. We conclude the course with several numerical experiments, which show that CBO scales well with the dimension and is extremely versatile. To quantify the performances of such a novel approach, we show that CBO is able to perform essentially as good as ad hoc state of the art methods using higher order information in challenging problems in signal processing and machine learning, namely the phase retrieval problem, the robust subspace detection, and training of neural networks.

– Gradient descent for smooth and convex functions

– Stochastic gradient descent

– Simulated annealing

– Gradient descent for non-smooth and non-convex functions

– A primer on stochastic calculus and stochastic differential equations

– From particle swarm optimization to consensus-based optimization (CBO)

– Well-posedness of (CBO) and its mean-field limit

– Large time behavior and proof of global convergence

– Variations on the theme: anisotropic noise, gradients, memory, and CBO over compact hypersurfaces

**Links:**

https://arxiv.org/abs/2103.15130

https://arxiv.org/abs/2211.12184

https://arxiv.org/abs/2001.11994

https://arxiv.org/abs/2001.11988

#### L-functions (starting date: March)

**Lecturer:** Davide Lombardo (Università di Pisa)

**Schedule:** The course will be held from March 7, 2023 – Tuesday 2-4 p.m. Sala Riunioni and Thursday 11 a.m. -1 p.m Aula Magna.

**Duration: **30 hours

**Description:** The course aims to introduce the notion of an L-function, a tool at the boundary between algebraic and analytic number theory, and to prove some classical results in arithmetic using this language. It will consist of approximately 30 hours of lectures and aims to also be accessible to motivated master’s students. The lectures will include a review of the prerequisite notions from number theory.

**Preliminary program:**

- Classical L-functions: Riemann’s zeta function, Dirichlet’s L-functions, analytic continuation and functional equation. Arithmetic applications: the prime number theorem, Dirichlet’s theorem on arithmetic progressions, Chebotarev’s density theorem.
- Special values of zeta functions: the analytic class number formula, regular primes.
- Review of algebraic number theory, adèles, and idèles. The L-function of a Galois representation. Artin and Hecke L-functions.
- Fourier analysis on the adèles and Poisson summation. Tate’s approach to analytic continuation for Hecke L-functions.
- More general L-functions (if time permits).

#### Interdisciplinary Celestial Mechanics (starting date: March 2)

**Lecturer:** Giovanni Federico Gronchi (Università di Pisa)

**Schedule:** The Course will start on March, 2 (Sala Riunioni)

**Duration: **30 hours

**Description:** In this course, I’ll describe some results concerning the applications of techniques of calculus of variations or computational algebra to classical problems of celestial mechanics, like searching for periodic orbits of the N-body problem or computing an orbit of a solar system body using observations made from optical telescopes.

**Preliminary program:**

Preliminaries of celestial mechanics

- the N-body problem;
- collisions and regularization: methods by Levi-Civita and Kustaanheimo-Stiefel;
- non-collision singularities, Von Zeipel’s theorem.

Periodic orbits with variational methods

- different methods to search for periodic orbits;
- the “figure eight” by Chenciner and Montgomery;
- orbits with symmetry and topological constraints;
- examples of Gamma-convergence in celestial mechanics.

Algebraic problems in orbit determination (OD)

- classical methods in OD: methods by Laplace and Gauss;
- multiple solutions in preliminary OD;
- the Keplerian integrals method;
- Groebner’s bases in OD problems.

#### Mapping Class Groups and relatives: hierarchically hyperbolic spaces (starting date: March 1)

**Lecturer:** Alessandro Sisto (Heriot-Watt University)

**Schedule:** March 1, 2-4 pm, March 2, 11 am -13pm, March 3, 2-4 pm – all lectures in Sala Riunioni

**Duration: **6 hours

**Abstract:** Hyperbolic spaces and groups were introduced by Gromov and have been studied extensively over the past few decades from a huge variety of points of view, and with applications to various fields such as the study of hyperbolic 3-manifolds. Many groups of interest in geometry and topology, most notably mapping class groups, are however not hyperbolic, and it is therefore natural to try to generalise the notion of the hyperbolic group to encompass as many of these examples as possible. The notion of hierarchical hyperbolicity, inspired by the deep work of Masur and Minsky, achieves just that. Roughly speaking, a hierarchically hyperbolic structure on a metric space or group is a coordinate system where the coordinates take values in hyperbolic spaces. I will give an introduction to hierarchically hyperbolic spaces, thereby describing the (coarse) geometry of mapping class groups, and explain what this kind of structure is good for and how to show that a given space or group is hierarchically hyperbolic.

#### Monotonicity formulas in free boundary and geometric variational problems (starting date: February 27)

**Lecturers:** Bozhidar Velichkov (Università di Pisa), Roberto Ognibene (Università di Pisa), and Giorgio Tortone (Università di Pisa)

**Schedule:** The Course will start on February, 27 (n. 2 lessons per week)

**Duration: **30 hours

**Syllabus:** The course is an introduction to the regularity theory for free boundary problems and geometric variational problems. The focus is on the role of the monotonicity formulas in the analysis of the structure of nodal sets, free boundaries, and free discontinuities. In particular, on the analysis of the singularities.

** **The course will be divided into four main parts:

*Part 1. Regularity theory for the one-phase Bernoulli problem* (approx. 15 hours, Bozhidar Velichkov). We will provide a complete analysis of the free boundary for the one-phase problem. We will prove the optimal (Lipschitz) regularity of the solutions, and we will introduce the notions of blow-up sequences and blow-up limits, for which a key tool will be the Weiss monotonicity formula. We will show how to decompose the free boundary into a “regular set” and a “singular set” and we will prove that the “regular set” is a smooth manifold via an epsilon-regularity theorem. We will then introduce Federer’s dimension reduction principle in order to estimate the dimension of the “singular set”. We will then discuss the application of the same strategy in the case of minimal surfaces.

*Part 2. Almgren frequency function and unique continuation *(approx. 5 hours, Roberto Ognibene). The aim is to discuss Almgren’s frequency function for solutions to elliptic PDEs and to show how it can be used to study the structure of their nodal sets. We will show for instance how to deduce the unique continuation property (which in its classical form is the fact that the nodal set has an empty interior) for certain classes of PDEs.

*Part 3. Alt-Caffarelli-Friedman’s monotonicity formulas *(approx. 5 hours, Giorgio Tortone). We will discuss the celebrated Alt-Caffarelli-Friedman monotonicity formula and its application to the regularity of the solutions of the two-phase Bernoulli problem, as well as its applications to the optimal partition problem and to the regularity of the solutions of elliptic PDEs.

*Part 4. Harmonic maps* (approx. 5 hours, Luca Spolaor). This part will consist of several (2-3) introductory lectures on the theory of harmonic maps.

#### An introduction to Khovanov homology and its applications (starting date: February 27)

**Lecturer:** Lukas Lewark (Universität Regensburg)

**Schedule:** February 27, 11 am -13 pm, February 28, 11 am -13 pm, March 1, 11 am -13 pm, March 3, 9-11 am – all lectures in Sala Riunioni

**Duration: **8 hours

**Abstract:** the Jones polynomial introduced in 1984 was the first so-called quantum invariant in knot theory. It associates to every knot a polynomial, which is not defined in a geometric but rather in a diagrammatic way.

Fifteen years later, Khovanov homology was introduced as a so-called categorification of the Jones polynomial. In spite of its combinatoric origins, Khovanov homology has an ever-growing number of geometric applications in knot theory. In this mini-course, we will first work through the detailed construction of Khovanov homology, following a modern version of Bar-Natan’s tangle setup. Then, we will see some of the applications, such as Rasmussen’s proof of the Milnor conjecture (regarding the unknotting number of torus knots). Previous knowledge of knot theory and homological algebra is helpful, but will not be required.

Additionally, courses taught at the Scuola Normale Superiore of Pisa can be found here.