Below is a list of Ph.D. courses that will be taught during the academic year 2023-2024.

#### Fall/Winter Term

#### Computational Fluid Dynamics

**Lecturer:**

mattia.demichielivitturi@ingv.it

**Period:**

November 2023 – February 2024 (30 hours)

**Description:**

The course aims to introduce students to elements of computational fluid dynamics that have particular relevance in the simulation of environmental and geological processes, and in particular volcanic processes. Starting from conservation principles, the main partial differential equations (PDEs) describing the dynamics of the processes of interest will be derived in different forms (Lagrangian and Eulerian). The properties of these equations will be studied with a particular focus on the properties of the solutions of hyperbolic PDEs and to the techniques for their resolution by numerical methods. Finally, the finite volume method for the solution of PDEs will be presented, also through the use of open-source codes that implement this method.

#### (Deep) Learning Theory

**Lecturer:**

**Period:**

December 2023 – February 2024 (30 hours)

**Description:**

Deep learning has emerged as a powerful approach to solving complex problems in artificial intelligence, and understanding the underlying theory is crucial for practitioners and researchers alike. This postgraduate course offers an introduction to the theory behind deep learning, focusing specifically on mathematically rigorous results on the subject.

The course will start with an overview of the rudiments of statistical learning theory, such as loss functions, empirical risk minimization, kernel methods, generalization, and regularization. We will then thoroughly discuss the fundamentals of neural networks theory, covering topics such as architecture, activation functions, expressivity, approximation theorems, and training through (stochastic) gradient descent. The third part of the course will be devoted to some aspects of the optimization theory of neural networks. In particular, we will discuss the training dynamics of neural networks in the infinitely wide limit in two contrastive regimes: the neural tangent kernel regime and the mean-field regime. Depending on time and interest, other topics that might be covered are: (deep) reinforcement learning, generalization bounds for stochastic gradient descent and time-series learning with recurrent neural networks.

The class is open to advanced master students. While no previous knowledge of machine learning theory is expected, a solid background in analysis and probability will be necessary to reach an in-depth understanding of the topics treated in the course.

#### Nonlinear Analysis

**Lecturer:**

**Period:**

January – February 2024

**Description:**

The aim of this course is to give the flavor of the so-called “Topological methods in nonlinear analysis”. These methods had a large development in the eighties of the last century, and are still used to obtain existence and multiplicity results for solutions in nonlinear elliptic PDE.

We will start with some preliminary notions to give the variational framework (Euler Lagrange equation, the energy functional, etc). Then we will prove the main tool of the course: the “Deformation Lemma”, which establishes a link between critical points of the energy functional and the change of topology of the sublevels of the functional. This lemma, paired with a compactness condition, is the base of all topological methods.

From the Deformation Lemma, it follows immediately the celebrated Ambrosetti-Rabinowitz mountain pass theorem, as well as the Saddle and the Linking theorems, and we will apply these methods for the

existence of nonlinear elliptic PDE on bounded domains.

The final third of the course will focus on constrained problems (more precisely mass constraints and the natural constraint: the Nehari manifold). Finally, we will prove the Concentration-Compactness Lemma which is one of the fundamental tools to extend topological methods to equations on the whole space $\mathbb{R}^n$.

During the course, several examples of PDE and applications to these methods will be discussed.

#### p-adic Galois Representations

**Lecturer:**

**Period:**

January – February 2024 (20 hours)

**Description:**

The absolute Galois group $G_\mathbb{Q}$ of the field $\mathbb{Q}$ of rational numbers is a central and quite mysterious object of study in number theory. It has many interesting representations `coming from geometry’. For instance, representations coming from the action of $G_\mathbb{Q}$ on points of finite order on elliptic curves or, more generally, abelian varieties defined over $\mathbb{Q}$ are of fundamental significance. Already here an important distinction arises. Namely, for each prime $p$ the group $G_\mathbb{Q}$ contains subgroups that can be identified with the absolute Galois group $G_p$ of the field $\mathbb{Q}_p$ of $p$-adic numbers and we may restrict representations of $G_\mathbb{Q}$ to these subgroups. The observation is that representations coming from the action of $G_p$ on points of $p$-power order (the “$p$-adic representations”) behave completely differently from those coming from points of q-power order for a prime q different from p.

The study of the remarkable properties of $p$-adic Galois representations was initiated by such giants of 20th-century mathematics as Grothendieck, Tate, and Serre. It was put into a systematic framework by J-M. Fontaine. Understanding their deeper structure played a pivotal role in many central results of modern number theory such as Wiles-Taylor’s proof of Fermat’s Last Theorem or the development of the $p$-adic Langlands program, but it had profound applications to algebraic geometry as well. Recently, the work of P. Scholze and his collaborators has opened up whole new vistas and also enabled a better understanding of some classical topics.

In this class, we shall give a brief introduction to this fascinating subject. In the first part, which will be purely algebraic, we shall present some classical structural results from a recent point of view. The latter part will focus on representations coming from algebraic geometry.

As prerequisites, familiarity with the contents of the courses “Istituzioni di algebra” and “Teoria algebrica dei numeri” is recommended (but not compulsory). Toward the end, we shall use some algebraic geometry too.

#### Theory of currents

**Lecturer:**

**Period:**

November 2023 – February 2024 (30 hours)

**Description:**

The theory of rectifiable currents is a fundamental part of Geometric Measure Theory, and has been applied in various areas of mathematical analysis (starting from the homological Plateau problem, which was the original motivation for the development of the theory). In this course, I will describe the foundations and some more advanced aspects of the theory of currents (rectifiable and not). The course is structured as follows:

- Prerequisites from the geometric theory of measure theory (Hausdorff measures, rectifiable sets) and of multi-linear algebra (k-vectors and k-forms).
- Definitions and fundamental results of the theory of currents: polyhedral deformation and approximation theorems, isoperimetric inequality, slicing, Federer and Fleming compactness theorem.
- Applications and advanced results.
- Metric currents, varifolds.

(The treatment of the last points depends on the time available.)

**Prerequisites: **a good knowledge of the theory of integration with respect to an abstract measure and of some related results in Functional Analysis (weak topology and Riesz theorem) is required. Some knowledge of the basics of geometric measure theory (e.g., the area formula) is desirable but not strictly necessary.

#### Topological persistence: theory and applications

**Lecturer:**

davide.moroni@isti.cnr.it

maria.antonietta.pascali@isti.cnr.it

**Period:**

November 2023 – January 2024

**Description:**

The course deals with one of the most known and used tools from Computational Topology, for example in shape and data analysis: persistent homology. Firstly, we will recall basic notions of Algebraic Topology; hence, we will introduce the theory of persistent homology, including classical results (e.g. the stability theorem) and the definition of derived descriptors (e.g. vectorizations and/or kernels). The second part of the course will be devoted to recent advances in the field, such as topological machine learning, and to the description of a selection of applications, ranging from signal and image processing to graph and network classification, from 3D shape analysis to the monitoring of machine and deep learning processes.

Main references:

G. Carlsson, M. Vejdemo-Johansson; Topological data analysis with applications; Cambridge, 2022

H. Edelsbrunner; A Short Course in Computational Geometry and Topology, Springer, 2014

A. Hatcher; Algebraic Topology, Cambridge University Press, 2002 (available here)

#### Winter/Spring Term

#### 3-manifolds, decorated triangulations, and quantum invariants

**Lecturer:**

**Period:**

Second semester (30 hours)

**Description:**

“Naked” triangulations up to a determined finite set of elementary moves provide a basic combinatorial realization of 3-manifolds up to diffeomorphism. Classes of enhanced triangulations up to enhanced moves provide realizations of 3-manifolds equipped with additional structures (combing, framing, spin structures, hyperbolic structure, …), and also arise in the construction of quantum invariants defined as state-sums over suitable triangulations. This theme will be developed with particular attention toward so-called quantum-hyperbolic invariants.

#### Contact Geometry

**Lecturer:**

**Period:**

Second semester (30 hours)

**Description:**

The course will be an introduction to three-dimensional contact geometry and topology. I plan to cover the following topics:

- Examples of contact structures, Moser’s trick, Gray stability, Darboux’s theorem.
- Open book decompositions, Alexander’s theorem, Thurston-Winkelnkemper’s

proof of Martinet’s theorem. - Open books and contact structures, Giroux’s correspondence.
- Legendrian and transverse knots and links: front and Lagrangian projections,

Thurston-Bennequin and rotation numbers, examples. - Bennequin’s inequality, tight vs. overtwisted dichotomy.
- Characteristic foliations, convex surfaces, Giroux’s flexibility.
- Tight contact structures on solid tori, contact surgery.

Time permitting, I will also cover material from one or more of the following topics:

- Construction and classification results for symplectic fillings.
- Invariants of Legendrian and transverse knots and non-simplicity phenomena.
- Classification results for Legendrian and transverse knots.

#### Dynamics of Complex Systems

**Lecturer:**

**Period:**

April – May 2024 (30 hours)

**Description:**

A complex system is typically a system in which many subsystems interact, and whose evolution is substantially determined by this interaction.

The typical characteristic of these systems is the emergence of global behaviors of the system that are due to the interaction among the components and that are not explainable/predictable from the characteristics of the individual subsystems (emergent properties). Coupled maps exhibit a variety of behaviors reminiscent of those of complex systems: from regular ones such as synchronization, to “random” and uncorrelated evolution of certain components.

In this course, we will present some topics on the study of coupled maps with mean-field interactions and study, within the framework of ergodic theory, the mechanisms that explain certain behaviors of complex systems.

In the first part, we will introduce some general concepts of ergodic theory, with particular attention to the study of transfer operators of hyperbolic systems.

In the second part, these results will be extended to the study of the statistical properties of coupled maps. Rigorous results on these systems have been obtained in the thermodynamic limit when the number of coupled maps tends to infinity. In this limit, the dynamical system is of infinite dimension, and its evolution is prescribed by a self-consistent transfer operator. We will study the main techniques available to study these objects and discuss how to prove the existence of fixed points (corresponding to equilibria in the thermodynamic limit), their stability (i.e., whether nearby states tend to converge to the fixed point or not), and their stability under perturbation (i.e., how the fixed point changes when the equations defining the system change).

The course will be taught in collaboration with Matteo Tanzi (Laboratoire de Probabilité Statistique et Modelisation, CNRS-Université de Paris-Sorbonne).

**Contents:**

- Recall of ergodic theory.
- Transfer operators, invariant measures, convergence to equilibrium, functional-analytic approach.
- Statistical stability, Linear Response to perturbations.
- Interacting systems in mean field, self-consistent operators.
- Coupled maps systems, examples.
- Emergent properties of coupled systems, synchronization phenomena.
- Statistical properties of coupled systems: invariant measures, convergence to equilibrium.
- Statistical stability and linear response to perturbations.

#### Heegaard Splittings and Handle Numbers

**Lecturer:**

**Period:**

May 2024 (30 hours)

**Description: **

The 30 hour course Heegaard Splittings and Handle Numbers will build from the foundations of the classical theory of Heegaard splittings of closed oriented 3-manifolds to the contemporary study of circle valued Morse functions and Morse-Novikov numbers of 3-manifolds. The course will touch upon several threads of low-dimensional topology and geometric topology. In particular, topics will involve relationships with Dehn surgery, analogies with bridge numbers and thin positions of knots, connections to modern Floer theories, interactions with sutured manifold theory, and the structures of curve complexes and Kakimizu complexes.

A full description of the course can be found at this link.

#### Quotients and Moduli Spaces

**Lecturers:**

**Period:**

April – May 2024 (30 hours)

**Description: **

“Moduli spaces” are spaces whose points classify some algebro-geometric (or even topological, etc.) objects of interest. The most famous example is probably the moduli space of complex algebraic curves (i.e. compact Riemann surfaces), already envisioned by Riemann in the mid-800s. The study of the geometry of moduli spaces has been one of the central topics in algebraic geometry, geometric representation theory, etc, for a few decades now, and their construction often requires taking a quotient (i.e. orbit space) of the action of an algebraic group on some algebraic variety. This is often done via the machinery of GIT (geometric invariant theory), although there are more recent alternatives, that make use of algebraic stacks.

The first part of the course will cover some basics about moduli spaces and their construction, illustrated alongside relevant examples, and focusing on GIT and/or on the use of algebraic stacks. The second part will be devoted to examples.

The decision about the focus of the course and what examples to cover will be taken after a preliminary meeting with the prospective attendees of the course, which will take place presumably during November/December 2023.

#### Storia, tecnologie e teorie: strumenti in e per la ricerca in didattica della matematica

**Lecturers:**

**Period:**

Second semester (30 hours)

**Description:**

Il corso di dottorato tratterà, da diversi punti di vista, il tema degli “strumenti” in e per la ricerca in didattica della matematica. In particolare, saranno trattati i temi: della progettazione e implementazione di attività didattiche che fanno uso di strumenti digitali; del ruolo di teorie e metodologie; dell’utilizzo della storia della matematica in contesto didattico e per la ricerca didattica.

#### Topics in Modeling and Analysis in Materials Science

**Lecturer:**

**Period:**

Second semester (30 hours)

**Description:**

Present-day materials science is a vast interdisciplinary field that aims to develop substances and structures with novel physical properties. Designing new materials is a labor-intensive process that has been seen to greatly benefit from mathematical modeling, analysis, and computation. After a brief overview, this course will focus on the types of problems that arise in the context of hard and soft condensed matter systems undergoing a phase change. These problems are characterized by the presence of a non-convex (free) energy functional that drives the behavior of the material and determines its interesting properties. Mathematically, these problems give rise to challenging questions that can be tackled by the tools of the calculus of variations, analysis of nonlinear PDEs, and dynamical systems theory.

An outline of the topics covered:

- Overview of the challenges in mathematical materials science
- Phase transitions: their origin from statistical mechanics and phenomenological theories
- Kinetics of phase transformation: nucleation, growth, and coarsening
- Emerging complexity: micromagnetics as a prime example
- Soft matter: the emergence of microstructure in polymer systems