This page contains the list of Ph.D. courses that are taught during the academic year 2022 – 2023. In addition, you may visit the page of courses taught at the Scuola Normale Superiore, Pisa.

#### Fall/Winter Term

#### Introduction to Mean Curvature Flow

**Lecturers:** Alessandra Pluda, Yoshihiro Tonegawa

**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.

#### Chromatic symmetric functions: recent advances

**Lecturers:** Michele D’Adderio, Alessandro Iraci, Anna Vanden Wyngaerd

**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.

#### Winter/Spring Term

#### Monotonicity formulas in free boundary and geometric variational problems

**Lecturers:** Bozhidar Velichkov, Roberto Ognibene, Giorgio Tortone

**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 three 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.

#### L-functions

**Lecturer:** Davide Lombardo

**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).