**Training Activity**

In agreement with the tutor,** the PhD student is required to present a study plan consisting of advanced courses/activities for 24 credits (ECTS) and 3 complementary training activities.** The study plan must be approved by the board.

The study plan must be presented, at least in part, within the end of January of the 1st year.** It must be completed (and possibly modified) within the end of January of the 2nd year**.

**Advanced courses**

Students can choose among the following activities:

a) **courses offered within the PhD program in Mathematics** of the University of Turin or other universities, including foreign universities;

b) **advanced courses offered within the Master degree in Mathematics** of the University of Turin;

c) courses in the form of** reading courses**, where the PhD student studies independently a subject under the guidance of a teacher, followed by a final exam;

d) study activities in the form of a **working group** of advanced level, i.e. a group of professors, researchers, and PhD students who study together a specific research topic;

e)** research schools** and similar activities.

Each course/study activity corresponds to a number of credits, defined, **in the case of courses of type a), b), c), or d),** **by the course holder** in agreement with the Coordinator; in **case e)** **by the Coordinator**, in agreement with the tutor.**Type e)** activities **can contribute** to the acquisition of** no more than 6 credits**.**For each course/activity it is required to pass a final exam**, defined by the person in charge of the activity (with the possible exception of type e) activities that do not have a final exam). These exams** must be completed within the first two years of the PhD program**.

Besides their study plan, graduate students **can follow courses or participate in research activities** in all the years of their studies.

PhD students are also expected **to actively participate** in the **seminars** of our department; some of them give lectures within these seminars.

**Complementary training**

PhD students are required to participate in at least three courses/seminars offered for the Complementary Training by the Doctoral School of the University of Turin, within the first two years of the PhD program.

One of these activities can be replaced by an outreach activity (terza missione).

Complementary training activities

**An introduction to inner model theory****Alessandro Andretta**

Hours: 30

Period: second term

I plan to sketch the construction of inner models up to a Woodin cardinal. We will cover various issues, like the comparison lemma, complexity of the well-ordering of the reals, and (weak) covering. Inner model theory is known for being somewhat difficult to learn. I will try, as much as possible, to give motivations and applications in other areas of set theory. In order to make the course as self-contained as possible, I will cover some standard prerequisites, such as basic facts about fine structure, large cardinals, and determinacy.

**An introduction to Iwasawa theory****Ignazio Longhi**

Hours: 30

Period: spring 2025**Contents**: profinite groups and rings; (commutative) Iwasawa algebras and their interpretation in terms of measures and distributions; Mahler's theorem and the Mahler/Amice transform; structure theorem for finitely generated modules over Iwasawa algebras and applications to growth control; the Iwasawa main conjecture for cyclotomic fields.

The idea is to start without assuming too much (some basic course in commutative algebra and algebraic number theory, but, for example, no real familiarity with inverse limit constructions) and spend some time in building a framework which can encompass much current research, a bit more general than the standard textbook material (so the Iwasawa algebra to be discussed would be Z_p Gamma, with Gamma isomorphic to Z_p^d and also some applications beyond the standard textbooks. Ideally I would also give a complete proof of the simplest case of the main conjecture, but if time should not suffice I might decide to be sketchy on some parts, since details can be found in textbooks.

**Einstein manifolds****Reto Buzano**

Hours: 20

Period: April - June 2025**Contents**: In this course we will study Ricci curvature (and the Einstein condition) from a variety of viewpoints, using analysis, geometry, topology, and the calculus of variations. After a very short review of the basic definitions and facts from Riemannian geometry (different notions of curvature, Einstein manifolds and space forms, Schur's theorem, etc.), we will first study Riemannian curvature functionals and their critical points. In the second part of the course, we study the Einstein condition as a nonlinear PDE, discussing analytical and topological aspects (local and global obstructions for solvability, regularity, uniqueness, etc.). If time permits, we might also touch the topic of compactness results for sequences of Einstein manifolds.

**Geometric aspects of the AdS/CFT correspondence****Dario Martelli**

Hours: 10

Period: June 2025**Contents**:The aim of this course is to introduce the students to a selection of contemporary topics of the AdS/CFT correspondence, where supersymmetry and geometry play a crucial role. Possible topics include: advanced geometric toolkits for the study of supergravity solutions; the interplay of extremization principles between (Riemannian) geometry and superconformal field theories; the holographic approach to the microscopic structure of black holes. A detailed program will be coordinated with the interested students.

**Index theory in geometry and mathematical physics****Alessandro Portaluri**

Hours: 30

Period: Jan-Feb 2025**Contents**: Symplectic vector spaces and Lagrangian subspaces. Some examples from classical mechanics

Homotopy and differential structure of the Lagrangian manifold

Maslov index for loops, Cayley transform and winding number after Arnol'd

Maslov index for general Lagrangian paths, crossing forms and intersection index

Maslov index for solutions of Lagrangian bvp with selfadjoint boundary conditions on bounded intervals. Stable and unstable spaces

Path of symmetric matrices, evolution of the inertia indices and spectral flow

Unbounded selfadjoint Fredholm operators and spectral flow

Sturm-Liouville type differential operators on bounded and unbounded interval for systems. Spectral properties

Spectral flow for solutions of a second order Lagrangian boundary value problem under general Lagrangian boundary conditions

Morse index theorems on bounded intervals with Lagrangian bc and on unbounded intervals: halfclinic, homoclinic and heteroclinic cases.

**Introduction to K3 surfaces and the Torelli theorem****Tommaso Pacini**

Hours: 15

Period: March-April 2025**Contents**: We will cover the following topics.

- Review of basic facts of complex geometry.

- Review of complex curves, introduction to complex surfaces.

- Elliptic surfaces, projective surfaces and the Kodaira embedding theorem.

- K3 surfaces and the Torelli theorem.

**Introduction to Loop Quantum Gravity****Lorenzo Fatibene**

Hours: 30

Period: Jan-Feb 2025**Contents**: We consider mathematical structures used in loop quantum gravity (LQG) starting from classical GR to the introduction of spin network Hilbert space and operators on it to represent geometric quantities.

From a physical viewpoint, LQG is a proposal for a generally covariant QFT.

From the mathematical viewpoint one sees in action classical field theories, bundle theory, holonomies of connections on manifolds, analysis on (bare) manifolds, group representation theory for SU(2), Clifford algebras, topological obstructions, and, finally, operators on Hilbert spaces. All in action to solve the specific problem of describing quantum gravity.

**Introduction to modular forms****Lea Terracini**

Hours: 20

Period: January-April 2025**Contents**: Introduction to classical modular forms, Hecke algebras and Atkin-Lehner theory. Modular curves as Riemann surfaces and their moduli-theoretic interpretation.

Group cohomology and Eichler-Shimura isomorphism. Automorphic forms and representations. Modular Galois representations.

**Kernel-based approximation and quasi-Monte Carlo integration****Roberto Cavoretto, Giacomo Elefante**

Hours: 20

Period: Nov 2024 - Mar 2025**Contents**:The teaching aims to provide knowledge on some topics of kernel-based approximation and quasi-Monte Carlo integration methods.

Kernel methods play an important role in many different areas of mathematics, science, and engineering. They are of particular interest in the field of multivariate scattered data interpolation and solution of partial differential equations using radial basis functions (RBFs). The goal is, on one hand, to introduce theory and, on the other, to show application of related numerical methods.

Quasi-Monte Carlo (qMC) and Monte Carlo (MC) methods for integration are a decisive step in overcoming the so-called “curse of dimensionality” when you deal with the approximation of integral in high-dimensions. Nevertheless, the advantages of the first, compared to the second, are many. MC method is a numerical method based on random sampling, whereas in qMC method these samples are replaced by well-chosen deterministic points. The aim is to select these points in order that the deterministic error is smaller than the probabilistic error bound of MC method.

**Logic, Models and Games****Mirna Dzamonja (University of East Anglia, UK)**

Hours: 20

Period: end of second term**Contents**: We discuss various logics that have been developed in mathematics and computer sciences, for various purposes. This includes of course the first order logic FO, the basis of classical model theory, but also abstract logics and logics that are weaker than FO and can be used to study finite models. Our purpose is to find the common ground and to develop a machinery which allows to compare logics. Moreover, keeping the paradigm case of FO as the grail of the universe of logics, we study what properties of this logic can and how be transferred to other cases. And if not, why this is the case.

We try to answer the question of what is a logic, after all.

The course will have elements of set theory, model theory, finite model theory and category theory. It is based on the M2 course taught at the Université Paris-Cité, a book in preparation written by the lecturer jointly with her students Ivory Fronteau, Baptiste Schilling and Denis Yilmaz, as well as on a number of research articles.

**Non-Markov random walks****Federico Polito**

Hours: 15

Period: second term**Contents**: The course will be focused on recent results on non-Markov random walks in connection with the theory of discrete-time semi-Markov chains.

First, some notions on finite-dimensional random walks and discrete-time Markov chains on discrete state-spaces will be recalled. We will then examine the theory of discrete-time semi-Markov chains. In this framework we will discuss semi-Markov random walks and, in particular, discrete-time renewal processes. We will also highlight the role of some generalized difference-operators of convolution type and some specific random time-changes in the construction of non-Markov random walks.

Furthermore, we note the PARID model of random graph (i.e. the Preferential Attachment with Random Initial Degrees model) exhibits a structure which very much resembles that of discrete-time-renewal processes. We will study it in relation to the problem of the asymptotic concentration of the degree sequence.

**Topics on Fano varieties and birational geometry****Cinzia Casagrande**

Hours: 10

Period: March-April 2025**Contents**:The minicourse, in the area of algebraic geometry, will be about (smooth, complex) Fano varieties and birational geometry in the sense of the Minimal Model program. The precise topics will be chosen also depending on the interests of the audience.

**Working Groups**Every academic year the research groups of the Math Department organize a few working groups on specific research topics, see for instance here for 2022-23.

Participation to such a working group can be part of the study plan of a graduate student.**Courses offered by the Master Degree in Mathematics****Courses offered by the Master in Mathematical and Physical methods for Space Sciences****Activities in nearby Universities:**- Politecnico di Torino, PhD program in Mathematical Sciences
- Università di Milano, PhD program in Mathematical Sciences, courses
- Joint PhD program in Mathematics of Milano Bicocca - Pavia - IndAM, courses
- Università di Genova, PhD program in Mathematics and Applications, courses

- Algebraic number theory
- An Introduction to Kernel Methods: Theory and Applications
- Fano varieties
- Geometrical Aspects of PDEs
- Geometrical Aspects of PDEs II
- Introduction to loop quantum gravity (LQG)
- Model theory of continuous structures
- Topics in Algebraic Geometry
- Topics in Mathematical Logic
- Topics on Lie groups, nilmanifolds and solvmanifolds