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.

### List of Ph.D. courses:

#### An introduction to Cluster Algebras and related topics

**Lecturer:** Job D. Rock

**Duration: **10 hours

**Syllabus:** This 10-hour course will provide an introduction to cluster algebras and their applications. Cluster algebras were introduced by Fomin and Zelevinksy in the 2000s and have become a large subject of study, partially due to their connection to particle physics. We will begin with the basic definitions for cluster algebras. We will then cover some applications to physics and some alternative ways to study cluster algebras, such as cluster categories.

#### L functions

**Lecturer:** Davide Lombardo

Schedule: the course will be held from March, 2023 at Dipartimento di Matematica, Università di Pisa, and it will last 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 programme:**

- 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).