Below is a list of Ph.D. courses that will be taught during the academic year 2024-2025.

#### Fall/Winter Term

#### An introduction to Khovanov-Rozansky homologies and their applications

**Lecturer:**

**Period:**

January – March 2025 (20/24 hours)

**Description:**

A link is a (finite) collection of circles, possibly knotted and linked, in the 3-dimensional space. The study of links is a central topic in low-dimensional topology. An important tool to study knots and links is given by link homology theories. Khovanov introduced one of these theories in the early 2000s. A few years later, together with Rozansky, Khovanov defined a family of homology theories called Khovanov-Rozansky homologies.

The aim of these lectures is to provide an introduction to Khovanov-Rozansky homologies covering the basic definitions, important properties, several variants, and some applications. We chose to focus mostly on topological applications and, in particular, applications to the study of concordance.

The plan is to quickly cover some basic material on knot theory, and then start with Khovanov homology. We will introduce the $s$-invariant and see the main applications of this theory. Then, we will turn to the definition of the Khovanov-Rozansky homologies. We focus on two different definitions: the original via matrix factorizations and the definition via foams. Afterwards, an idea of how to prove functoriality for Khovanov-Rozansky homologies is given. We conclude with an overview of the applications of these theories and open problems.

Time permitting, in the final part of the course, we will cover some additional topics which may depend on the students’ interests.

**Prerequisites:**

Linear algebra, basic group theory (basic definitions: groups, subgroups, quotients, etc.), commutative algebra (ring, modules, tensor product, etc.), homological algebra (chain complexes, homotopy equivalence, short and long exact sequences, etc.). No prior knowledge about knot theory is required as we shall cover the basic facts needed.

**Website of the course:**

Please, visit here.

#### Dal problema di ricerca all’analisi dei dati: esperienze di ricerca qualitativa in didattica della matematica

**Lecturers:**

**Period:**

November 2024 – February 2025 (30 hours)

**Description:**

Durante il corso verranno presentate alcune esperienze di ricerca qualitativa in didattica della matematica delle docenti, che metteranno in luce i processi seguiti, a partire da un problema, per arrivare all’impostazione metodologica della ricerca, all’analisi dei dati e ai risultati. In particolare verrà affrontato il tema del posizionamento rispetto ad un quadro teorico. Nelle esperienze di ricerca discusse il contesto educativo studiato coinvolgerà sempre artefatti digitali per l’insegnamento-apprendimento della matematica.

#### Data sparsity, low-rank approximation, and applications

**Lecturers:**

**Period:**

November-January (40 hours)

**Description:**

Matrices and tensors are at the heart of many scientific and industrial computational problems. In this course we will survey numerical methods that leverage low-rank structures in matrices and tensors in order to reduce the consumption of computational resources while maintaining a satisfactory accuracy of the approximation. The course will cover the following topics:

- Deterministic and randomized algorithms for low-rank approximation of matrices and tensors
- Krylov methods for functions of matrices
- Trace and diagonal estimation of black box matrices
- Hierarchical matrices
- Low-rank tensor formats: Tucker and tensor train
- Dynamical low-rank approximation
- Riemannian optimization on low-rank matrix/tensor manifolds
- Applications to data science, evolution problems, complex networks, …

**Prerequisites**:

Basic tools of numerical analysis and numerical linear algebra (e.g. matrix factorization, numerical methods for linear systems)

#### List of advanced MSc courses offered during the first term

Analisi armonica (Harmonic analysis) taught by J. Bellazzini

Analisi Gaussiana (Gaussian analysis) taught by D. Trevisan

Calcolo delle variazioni B (Calculus of variations B: Regularity theory for minima of variational integrals) taught by A. Pratelli

Determinazione orbitale (Orbital determination) taught by G. Tommei

Curve Ellittiche (Elliptic curves) taught by T. Szamuely

Geometria – Analisi complessa B (Complex analysis) taught by F. Bianchi

Geometria Riemanniana (Riemannian Geometry) taught by D. Conti

Gruppi algebrici lineari (Linear algebraic groups) taught by A. Maffei

Metodi numerici per catene di Markov (Numerical methods for Markov chains) taught by B. Meini

Metodi numerici per la grafica (Numerical methods for graphics) taught by P. Boito

Modelli Matematici in Biomedicina e Fisica Matematica (Mathematical models in biomedicine and mathematical physics) taught by V. Georgiev

#### Winter/Spring Term

#### An introduction to convex bodies with applications

**Lecturer:**

**Period:**

The course will start in December or January (30 hours)

**Description:**

This course aims to provide participants with the main tools to understand and perform shape optimization in the framework of convex sets.

The course is divided into three parts. The first one is devoted to recalling some (easy) properties of convex sets and introducing some (not-so-easy) representations, such as the support function and the gauge function. In the second part, we will move on to the description of the space of convex bodies, namely the family of nonempty compact convex sets of $\mathbb R^n$, endowed with a suitable notion of distance. Among other topics, we will cover the Blaschke-selection theorem and the Brunn-Minkowski theorem.

We will conclude, in the third part, with some applications in shape optimization, with a numerical analysis flavor.

There are no prerequisites to attend the course.

Some references:

- book: “Bodies of Constant Width, An Introduction to Convex Geometry with Applications”, by Martini, Montejano, Oliveros
- book: “Convex bodies: the Brunn-Minkowski theory”, by Schneider
- book: “Shape Variation and Optimization”, by Henrot and Pierre
- article: “Numerical shape optimization among convex sets”, Bogosel, Appl Math Optim 87, 1 (2023).

#### An introduction to micro- and nanomagnetics

**Lecturer:**

**Period:**

Second term (30 hours)

**Description:**

Micromagnetics is a continuum modeling framework introduced in the 1930s by Landau and Lifshitz to describe the statics, dynamics and stochastics of the atomic spins in ferromagnetic materials. It has been demonstrated time and again to be successful in describing a great variety of magnetic phenomena in all sorts of ferromagnetic materials and devices. Nevertheless, with the advent of magnetic nanomaterials since the early 2000s new physical effects that give rise to additional intriguing physical phenomena need to be incorporated into consideration. This course is aimed at reviewing the state of the art of micromagnetic modeling and analysis, with the discussion of the emergent new challenges in the context of magnetic nanomaterials. The course will cover (to the extent that time permits):

- the origin of ferromagnetism, the classical micromagnetic energy functional and the basic magnetization structures in ferromagnets
- different representations of the stray field and magnetic domains
- one-dimensional domain walls and their stability
- flux closure domains, interior and boundary vortices
- dimension reduction: dots, wires and films
- antisymmetric exchange in non-centrosymmetric materials
- topological spin texture

#### An introduction to stochastic PDEs

**Lecturer:**

**Period:**

Second term (30 hours)

**Description:**

Stochastic PDEs are infinite dimensional problems that incorporate random influences and noise as much as stochastic differential equations incorporate random influences and noise in finite dimensional systems described by differential equations. Stochastic PDEs find applications in physics, finance, biology, and in general any model with spatially extended randomness.

The course aims to be a thorough mathematical introduction to the topic, starting from the basic tools (such as Gaussian measures in infinite dimension and semigroup theory) and the basic models (linear and semi-linear stochastic equations), and concluding with a selection of some of the most recent developments in the theory.

**Preliminaries:**

The requirements for the course are basic concepts in probability theory and stochastic analysis, basic notions of functional analysis, mainly properties of Hilbert and Banach spaces, and basic knowledge of PDEs. A prior knowledge of SDEs is helpful, but non-mandatory. Any other necessary piece of knowledge will be discussed during the course.

#### Derived categories and applications to the theory of Hall algebras

**Lecturer:**

**Period:**

April – June 2025

**Description:**

The derived category of a variety stands as one of the most refined invariants which encode its geometry. Furthermore, it serves as a prime example of triangulated categories, often regarded as “noncommutative” varieties. This course aims to provide an overview of the theory of triangulated categories and their properties, such as t-structures. The second part of the course delves into the application of this theory to explore an algebraic structure in geometric representation theory known as “Hall algebras.”

**Prerequisites:**

Prospective students would benefit from familiarity with the theory of abelian categories and some prior knowledge of basic algebraic geometry (e.g. varieties).

#### Elements of Geometric Analysis in finite and infinite dimensions

**Lecturer:**

**Period:**

Second term (30 hours)

**Description:**

In the first part of the course, we provide an overview of various results and problems arising from different synthetic formulations of Geometric Analysis. We start by introducing general tools from Analysis in Metric spaces, especially in relation to geodesics, Sobolev spaces, and differentiation of measures. Then we describe the main differences between the “commutative’’ Geometric Analysis of Euclidean spaces and the “noncommutative’’ Geometric Analysis of the so-called homogeneous Lie groups. A special attention will be devoted to the problem of computing the intrinsic area of submanifolds in a noncommutative homogeneous group, pointing out some recent open questions.

In the second part of the course, we mainly study some classes of infinite dimensional Lie groups. We introduce weak and strong Riemannian metrics, the Levi-Civita covariant derivative, the concept of curvature and geodesic distances in infinite dimensions. We also present some counterexamples and problems that only appear in infinite dimensions. An astonishing phenomenon is the vanishing of the geodesic distance, which P. Michor and D. Mumford conjectured to be related to the blow-up of the sectional curvature. Finally, some rigidity results concerning biLipschitz embeddings of homogeneous groups into Banach spaces will be discussed in relation to general versions of Rademacher’s differentiation theorem.

**Prerequisites:**

Basic facts about Banach spaces and basic notions of Differential Geometry, Measure Theory, and Multilinear Algebra.

#### Ergodic theory of geodesic flows

**Lecturer:**

**Period:**

March – June 2025 (30 hours)

**Description:**

The course will be an introduction to the dynamical properties of the geodesic flow on Riemannian negatively curved surfaces, a classical example of a hyperbolic flow which is a flourishing research area in dynamical systems.

The statistical distribution of the orbits of the geodesic flow was already studied in the first half of the last century, first for the case of surfaces with constant negative curvature and then in the general case of variable curvature, and it was established the existence of a very rich dynamics making the geodesic flow as a prototypical example of a chaotic system. However, it was not until 1998 that the first quantitative result appeared about the speed of the decay of correlations for the case of compact surfaces with variable curvature, a result lately sharpened by applying the modern techniques of the dynamical systems theory.

After recalling the classical results, the course will focus on the more recent results about the decay of correlations, discussing the case of surfaces with constant negative curvature in full detail and the extensions of the results to manifolds of higher dimensions.

Finally, we will introduce the thermodynamic formalism approach to dynamical systems and apply it to count the number of prime closed geodesics.

Time permitting, we will discuss some recent results on the ergodic properties of the horocycle flow, an example of parabolic flow, which depends on the geodesic flow.

**Prerequisites**:

It requires just a basic knowledge of differential geometry, dynamical systems, and functional analysis. The advanced notions of dynamical systems will be recalled in the course.

#### Harmonic maps and applications

**Lecturers:**

**Period:**

From March 2025 to May 2025 (30 hours)

**Description:**

This course will be an introduction to the study of Harmonic maps and to their applications to the theory of Minimal surfaces and Free-boundary problems. In the first part of the course, we will introduce the notion of Harmonic map and discuss existence theorems. Then, in the second part, we will focus on regularity properties, showing the Epsilon-regularity result by Schoen and Uhlenbeck. Finally, we discuss some applications to minimal surfaces and free-boundary problems.

**Prerequisites:**

Knowledge and use of Sobolev Spaces.

**References**:

- M. Giaquinta, L. Martinazzi;
*An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs,*Edizione della Normale 2005. - R. Schoen, K. Uhlenbech,
*A regularity theory for harmonic maps*, J. Diff. Geom. 17 (1982), 307-335. - J. Jost,
*Lectures on Harmonic maps*, Lecture notes in Mathematics, Fondazione CIME 1984.

#### High-performance solution of partial differential equations

**Lecturer:**

**Period:**

TBA

**Description:**

Partial differential equations (PDEs) are fundamental tools for modeling a wide range of scientific and engineering phenomena. This PhD course focuses on developing high-performance computational methods for solving real-world PDEs.

The course will explore the following key areas:

- Finite Element Method (FEM): We will quickly review the theoretical foundation of FEM, a powerful technique for discretizing PDEs into a system of algebraic equations
- deal.II Library: The course will leverage the capabilities of deal.II, a high-performance finite element library, for efficient implementation of FEM discretizations.
- Domain Decomposition Methods: We will explore strategies for parallelizing FEM computations by decomposing the computational domain into subdomains, enabling efficient utilization of high-performance computing resources.
- Parallel Linear Algebra Techniques: Techniques for solving large-scale linear systems arising from FEM discretizations on parallel architectures will be a key focus. This includes exploring libraries like PETSc or Trilinos, in conjunction with deal.II.
- Matrix-Free Geometric Multigrid Methods: We will investigate matrix-free geometric multigrid methods, which are powerful iterative solvers that exploit the geometric structure of the problem to achieve optimal scaling properties, with negligible storage requirements.

By combining these elements, the course equips students with the knowledge and skills necessary to develop and implement high-performance solutions for complex PDEs on modern computing platforms. Students will gain hands-on experience through programming exercises and case studies, enabling them to tackle real-world scientific and engineering challenges.

**Prerequisites:**

This course assumes a strong foundation in applied mathematics, including numerical analysis, linear algebra, and partial differential equations. Programming experience (C++ preferred) is also beneficial.

**Reference books:**

- Theory and Practice of Finite Elements, 2004, Alexander Ern, Jean-Luc Guermond
- Numerical Linear Algebra for High-Performance Computers, 1998, Jack J. Dongarra, Iain S. Duff, Danny C. Sorensen, Henk A. van der Vorst
- Domain Decomposition Methods – Algorithms and Theory, 2005, Andrea Toselli, Olof B. Widlund
- The Analysis of Multigrid Methods, 2000, James H. Bramble Xuejun Zhang

#### Nonlinear dispersive PDEs

**Lecturers:**

**Period:**

Second term (30 hours)

**Description:**

The aim of the course is to discuss the Nonlinear Schroedinger equation in the Euclidean space. The first part of the course (Bellazzini) concerns local and global Cauchy Theory in the energy space, the existence of standing waves, and their dynamical properties. The main tools are Strichartz estimates and variational methods for the study of critical points for the energy functional. The second part of the course (Forcella) discusses the asymptotics below the ground state energy, scattering using concentration compactness a là Kenig Merle and interaction Morawetz estimates.

#### List of advanced MSc courses offered during the second term

Algebra superiore A (Commutative algebra) taught by E. Sbarra

Geometria algebrica F (Toric varieties) taught by M. Talpo

Geometria differenziale complessa (Complex differential geometry) taught by G. Pearlstein

Topologia algebrica B (Algebraic topology B: Algebraic and combinatorial topology, with application to Hyperplane Arrangements and related spaces) taught by M. Salvetti

Teoria dei Nodi A (Knot theory A: Invariants of knots and links, theory of braids) taught by P. Lisca

Equazioni della Fluidodinamica (Equations of fluid mechanics) taught by L. C. Berselli

Equazioni ellittiche (Elliptic partial differential equations) by B. Velichkov

Analisi dei dati (Data analysis) taught by M. Romito and A. Papagiannouli

Meccanica Celeste (Celestial mechanics: Singularities and periodic orbits in the N-body problem) taught by L. Gronchi

Fisica Matematica (Mathematical Physics: Insights in the Theory of Dynamical Systems and in the Hamiltonian) taught by C. Bonanno

Meccanica Superiore (Advanced mechanics: Measure of chaos and ergodic methods in mechanics) taught by P. Giulietti

Meccanica Spaziale (Space mechanics: Motion of artificial satellites and space probes, interplanetary space missions) taught by G. Baù

Metodi numerici per equazioni alle derivate parziali (Numerical methods for partial differential equations) taught by L. Heltai