Current Ph.D. Courses

Lecturer:

Period:

From Monday, Dec 2, 2024 to Friday, Feb 28, 2025

Description:

The course will start in December of January. 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).

Lecturer:

Period:

From Wednesday, Jan 1, 2025 to Friday, Feb 28, 2025

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.

Lecturer:

Period:

From Monday, Nov 11, 2024 to Thursday, Dec 19, 2024

Description:

Introduction. A significant portion of equations encountered in continuum mechanics and physics are nonlinear. Many of these nonlinear systems can be expressed as compensated systems, comprising a linear PDE and a pointwise constraint (carrying the nonlinearity) of the form: $A(D)u= 0$, with $u(x) ∈ K$.

Focus and Goals. This course will delve into the geometric aspects of compensated systems in PDEs. Specifically, we will investigate the geometry and fine properties of measures µ that satisfy $A(D)µ= 0$. We will explore how the regularity properties of the operator $A(D)$ can be used to understand the geometry of these measures. In a manner analogous to how elliptic regularity is classically linked to compactness, we will see that geometric rigidity arises from compensated compactness.

Beyond the Main Objective. Time permitting, we will also discuss the variational aspects of mass concentration, which occurs when mass along a weakly convergent sequence accumulates on negligible sets. This can be viewed as a dynamical or variational process leading to the formation of measure-type singularities.

Core Concepts and Tools. The course will be largely self-contained, except for a few foundational concepts from classical geometric measure theory and Fourier analysis. The essential tools we will develop and utilize include:

• basic geometric measure theory (notions of convergence, Hausdorff dimension, Lp spaces, weak convergence, tangent measures, rectifiability, the Besicovitch–Federer rectifiability theorem)
• basic theory of partial differential equations (partial derivation, Laplace’s operator, Sobolev spaces, functions of bounded variation)
• Fourier transform and multiplier theory (properties of the Fourier transform, boundedness of singular integrals, the Mikhlin–Hörmander Theorem)

The course content will complement other topics in analysis, such as: $L^p$ elliptic regularity theory, the fine properties of functions of bounded variation, theory of currents, and various elements of the calculus of variations (variational rigidity of differential inclusions and the lower semicontinuity of variational integrals, etc.).

Further information Visit the website of the course.

References

[1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.

[2] Adolfo Arroyo-Rabasa. Characterization of generalized young measures generated by A-free measures. Arch. Ration. Mech. Anal., 242(1):235–325, 2021.

[3] Adolfo Arroyo-Rabasa, Guido De Philippis, Jonas Hirsch, and Filip Rindler. Dimensional estimates and rectifiability for measures satisfying linear PDE constraints. Geom. Funct. Anal., 29(3):639–658, 2019.

[4] Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011.

[5] Guido De Philippis. A-free measures and applications. In Geometric measure theory and free boundary problems, volume 2284 of Lecture Notes in Math., pages 5–35. Springer, Cham, [2021] 2021.

[6] Guido De Philippis and Filip Rindler. On the structure of A-free measures and applications. Ann. of Math. (2), 184(3):1017–1039, 2016.

[7] Loukas Grafakos. Classical Fourier analysis, volume 249 of Graduate Texts in Mathematics. Springer, New York, third edition, 2014.

Lecturers:

Period:

From Monday, Nov 18, 2024 to Friday, Feb 14, 2025

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
• Hierarchical matrices
• Low-rank tensor formats: Tucker and tensor train
• Model order reduction via reduced basis methods
• Dynamical low-rank approximation
• Riemannian optimization on low-rank matrix/tensor manifolds
• Applications to data science, evolution problems, PDEs, complex networks, ...

Prerequisites

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

Lecturers:

Period:

From Thursday, Jan 9, 2025 to Friday, Jan 31, 2025

Description:

The course will introduce some qualitative research experiences in mathematics education conducted by the instructors, highlighting the processes followed—from identifying a problem to establishing the research methodology, analyzing data, and reaching results. In particular the course will address the topic of positioning a study within a theoretical framework. The research experiences discussed will always involve an educational context that includes digital tools for teaching and learning mathematics.

Lecturer:

Period:

From Monday, Nov 4, 2024 to Friday, Feb 28, 2025

Description:

The goal of this PhD course is to build a general and flexible machinery of decreasing rearrangements satisfying fine Pólya-Szegö principles covering a variety of different settings. In particular, the set-up will be that of functions defined on metric measure spaces. This is motivated by the growing interest around geometric and functional inequalities in different frameworks such as, but not limited to, weighted Euclidean and Riemannian manifolds, metric spaces with synthetic curvature bounds, sub-Riemannian structures etc.

In the first part of the course, we shall focus on a key tool to build up this machinery, which is the theory of functions of bounded variations on metric measure spaces. We shall discuss the notion of total variation in this setting and settle important properties of this functional space.

In the second part, we shall consider notions of decreasing rearrangements on open intervals of the real line weighted by a positive weight and discuss their basic properties. Subsequently, we investigate the effect of decreasing rearrangements on Dirichlet and BV energies relying on the theory of functions of bounded variations. In particular, assuming an underlying isoperimetric principle, we will derive the so-called Pólya-Szegö inequality in this generality. Finally, time permitting and depending on the audience, we shall discuss relevant classes of examples where this theory applies and/or comment on the rigidity of the P´olya-Szeg½o inequality.

Prerequisites. Absolutely continuous functions and functions of bounded variations in 1D; Metric spaces and basic measure theory. No knowledge of functions of bounded variations in several variables or around non-smooth calculus in metric spaces will be assumed.

Schedule. The course will start in November 2024 and will be approximately 20 hours. Ideally, every week there will be a 2 hours lecture, but adjustments might be possible. The course should finish in February 2025.

Lecturer:

Period:

From Monday, Mar 17, 2025 to Friday, Apr 18, 2025

Description:

The topology of 4-dimensional manifolds presents unique phenomena that do not appear in any other dimension: for example, every compact manifold of dimension n different from 4 has always finitely many smooth structures, but this is not true in dimension 4. As the title says, the object of study of this class will be 4-manifolds, which we will approach both from the topological and from the smooth viewpoints. We will survey some popular techniques, invariants, constructions and obstructions. The last part of the course will be an overview of Heegaard Floer homology, a popular invariant useful for the study of exotic phenomena.

The first lesson will take place in Aula Seminari at the Department of Mathematics on 11th March 2025 at 9am.

Lessons will also be streamed online at this link.

Lecturer:

Period:

From Monday, Mar 3, 2025 to Thursday, Jul 31, 2025

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

Lecturer:

Period:

From Monday, Mar 3, 2025 to Thursday, Jul 31, 2025

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.

Prerequisites

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.

Lecturer:

Period:

From Monday, Mar 3, 2025 to Saturday, May 31, 2025

Description:

We will study the combinatorics and representation theory of 0-Hecke Algebras. Hecke algebras are deformations of the group algebra of the symmetric group, whose representation theory connects those of the symmetric group and the quantum groups. Hecke algebras associated with general Coxeter groups appear in diverse areas such as harmonic analysis, quantum groups, knot theory, algebraic combinatorics and statistical physics. The representation theory of the symmetric group is closely connected to the algebra of symmetric functions, Sym, through the so-called Frobenius characteristic map. Sym admits two notable generalizations: the algebra of quasisymmetric functions, QSym, and the algebra of noncommutative symmetric functions, NSym. We will explore the relationships between these algebras and representations of 0-Hecke algebras through a quasisymmetric characteristic map. These combinatorial Hopf algebras play a central role in contemporary algebraic combinatorics, aspects of which we will discuss throughout the course.

Prerequisites: Familiarity with linear and abstract algebra. Some knowledge of basic group representation theory would be helpful but is not strictly necessary. There are no combinatorial prerequisites for this course.

Duration: 30 hours, during the second semester.

Lecturer:

Period:

From Wednesday, Apr 9, 2025 to Thursday, May 29, 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, tilting, and more (approximately 20 hours).

During the second part of the course (approximately 10 hours), we will discuss the construction of perverse coherent sheaves à la van der Bergh and the derived McKay correspondence.

Prerequisites

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

Lecturer:

Period:

From Tuesday, Mar 11, 2025 to Friday, May 30, 2025

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.

Lecturer:

Period:

From Monday, Mar 3, 2025 to Wednesday, May 21, 2025

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.

Lecturers:

Period:

From Monday, Mar 3, 2025 to Friday, May 30, 2025

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.

Lecturer:

Period:

From Monday, Mar 3, 2025 to Thursday, Jul 31, 2025

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

Lecturer:

Period:

From Sunday, Feb 23, 2025 to Friday, May 30, 2025

Description:

Quantum mechanics is one of the central theories of physics, whose language describes a huge variety of systems, from elementary particles to superconducting materials and quantum computers. Fascinating mathematical problems arise in this context, posing hard challenges in functional analysis, partial differential equations and operator theory. This course is an introduction to the rigorous mathematical framework of quantum mechanics and functional analytic methods for the study of the Schrödinger equation. We will discuss the role of symmetries, properties of bound states, scattering theory and spectral questions. The course will conclude with an introduction to current problems in many-body systems.

Course Period: last week of February until the last week of May.

Lecturers:

Period:

From Monday, Mar 3, 2025 to Thursday, Jul 31, 2025

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.

Lecturer:

Period:

From Monday, Feb 17, 2025 to Wednesday, Apr 16, 2025

Description:

This course can serve as an introduction to fundamental discrete geometric structures; the beauty of the subject is the proximity of the fundamentals to open areas of research, and we will see related open problems and active areas of research throughout the course. We focus first on combinatorial aspects of polytopes: that is, we study the combinatorial structure of faces of polytopes, its face lattice; in the second part of the course we place the emphasis on metric and convex geometric properties, starting with (usual, Lesbegue measure) volume and refining and relating it to other central notions of convex geometry: mixed volumes, Ehrhart polynomials, covering radius. Applications in (linear, polynomial, combinatorial) optimization, algebra, tropical and toric geometry will be seen throughout the course.

Duration: 30 hours between February and April 2025