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

#### Fall/Winter Term

#### Chromatic symmetric functions: recent advances

**Lecturers:** Michele D’Adderio (Università di Pisa), Alessandro Iraci (Università di Pisa), and Anna Vanden Wyngaerd (Université Libre de Bruxelles)

**Schedule:** The Course will start on October, 17

**Duration: **30 hours

**Syllabus:** Chromatic symmetric functions were introduced in the nineties by Stanley as an extension of chromatic polynomials of graphs, and they immediately attracted a lot of attention as they were shown to be related to Hecke algebras and Kazhdan-Lusztig polynomials.

In the last few years, in an attempt to make progress on the so-called Stanley-Stembridge conjecture (the most important open problem in this area, but probably also in the whole of algebraic combinatorics), a burst of activity led to the discovery of new interesting connections among chromatic symmetric functions, Hessenberg varieties, and LLT polynomials.

In this course, we present some of the most interesting developments that occurred in the last decade.

The prerequisites are little to none, so the course will be accessible to any student with mathematical maturity and curiosity.

#### Introduction to Mean Curvature Flow

**Lecturers:** Alessandra Pluda (Università di Pisa) and Yoshihiro Tonegawa (Tokyo Institute of Technology)

**Schedule:** November 14, 5 – 7 pm (Aula Magna); November 15, 2:30 – 4:30 pm (Aula Magna); November 16, 9 – 11 am (Aula Magna); November 22, 2:30 – 4:30 pm (Aula Magna); November 24, 3 -5 pm (Aula Riunioni); November 25, 2 – 4 pm (Aula Magna).

**Duration: **30 hours

**Syllabus:** a family of surfaces is called the Mean Curvature Flow (MCF) if the velocity of a surface is equal to the mean curvature at each point and time. It is one of the most important geometric evolution problems with many facets of studies such as analysis of singularities, notions of weak solution, solvability of initial value problems, and so forth, and the course touches upon some of the recent developments.

The course consists of two parts. The first part is on the MCF in the framework of Geometric Measure Theory called the Brakke flow. Starting from the definitions and preliminaries, many of the basic properties of Brakke flow as well as some advanced topics such as the general existence theory will be covered.

The second part focuses on classical results on MCF obtained with a PDE approach. The main topics are short-time existence, the maximum principle, evolution equations of geometric quantities, and the analysis of type I and II singularities in the special case of positive mean curvature. In particular, the case of planar curves will be analyzed in full detail.

Some familiarity with measure theory and parabolic equations is desirable but not necessary.

References:

*Tonegewa, Yoshihiro, Brakke’s Mean Curvature Flow: An Introduction, Springer Briefs in Mathematics, Springer, 2019

*Mantegazza, Carlo, Lecture notes on mean curvature flow, Progress in Mathematics,

Birkhäuser/Springer Basel AG, Basel, 2011.

*Mantegazza, Carlo, Novaga, Matteo and Pluda, Alessandra, Lectures on curvature flow of networks,

In “Contemporary research in Elliptic PDEs and Related Topics”, Springer INDAM series, vol. 33, 2019.

#### Winter/Spring Term

#### An introduction to Khovanov homology and its applications

**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.

#### Breaking Nonconvexity: Consensus-Based Optimization

**Lecturer:** Massimo Fornasier

**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 least 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 are 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 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

https://arxiv.org/abs/2103.15130

https://arxiv.org/abs/2211.12184

https://arxiv.org/abs/2001.11994

https://arxiv.org/abs/2001.11988

#### Combinatorics of the flag variety

**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.

#### Interdisciplinary Celestial Mechanics

**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.

#### L-functions

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

**Schedule:** The course will be held from March 2023

**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).

#### Mapping Class Groups and relatives: hierarchically hyperbolic spaces

**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

**Lecturers:** Bozhidar Velichkov (Università di Pisa), Roberto Ognibene (Università di Pisa), 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.