PhD in Mathematics in Rome

Course offered by: Sabino Di Trani (Sapienza)
Duration: 21h
Period: October - December 2025

Program

Available here.

Schedule

• October: Wednesday 8, 10:00-13:00; Tuesday 14, 11:00-13:00; Wednesday 22, 10:00-13:00

• November: Wednesday 5, 10:00-13:00; Tuesday 11, 11:00-13:00; Wednesday 19, 11:00-13:00; Tuesday 25, 11:00-13:00

• December: Tuesday 2, 11:00-13:00; Wednesday 10, 11:00-13:00.

All lectures will take place in Room L at the Department of Mathematics "Guido Castelnuovo", Sapienza.

Course offered by: Gianluca Pacienza (Institut Élie Cartan de Lorraine)
Duration: 8h
Period: February 2026

Program

TBD.

Schedule

• February: Monday 16, Wednesday 18, Monday 23, Wednesday 25.

All lectures will take place in Room L at the Department of Mathematics "Guido Castelnuovo", Sapienza, from 10:30 to 12:30.

Course offered by: Lorenzo Foscolo (Sapienza)
Duration: 20h
Period: February - March 2026

Program

The course will be an introduction to the theory of Riemannian holonomy groups and the geometry of manifolds with special holonomy. After an introduction to holonomy and Berger’s classification of holonomy groups of irreducible non-symmetric Riemannian manifolds, we will concentrate on the Ricci-flat holonomy groups SU(n), Sp(n), G2 and Spin(7). The second part of the course will focus on constructions of complete non-compact and compact Ricci-flat manifolds with special and exceptional holonomy.

Rough outline of the lecture contents:

• Berger’s classification of the Riemannian holonomy groups. (Principal bundles and connections, the Levi-Civita connection, parallel transport, Berger’s classification.)

• The Ricci-flat holonomy groups. (Calabi-Yau, hyperkähler, G2 and Spin(7) metrics, Ricci-curvature and topology, structure results for Ricci-flat manifolds.)

• Kähler Ricci-flat metrics. (Kähler and complex geometry, the Calabi Conjecture, compact Calabi-Yau and hyperkähler manifolds.)

• Bundle constructions of complete non-compact manifolds with special holonomy. (Calabi Ansatz, Bryant-Salamon’s asymptotically conical manifolds with exceptional holonomy.)

• Kummer-type constructions of compact manifolds with special holonomy. (The Kummer construction of hyperkähler metrics on the K3 surface, Joyce’s construction of compact manifolds with exceptional holonomy.)

Recommended reading

A good reference for the course is
D. Joyce, Compact manifolds with special holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2000.

Prerequisites

Basic knowledge in differential and Riemannian geometry.

Schedule

• February: Tuesday 3, Thursday 5, Tuesday 10, Tuesday 17, Tuesday 24, Thursday 26

• March: Tuesday 3, Thursday 5, Tuesday 10, Thursday 12

All lectures will take place in Room F at the Department of Mathematics "Guido Castelnuovo", Sapienza, from 14:30 to 16:30.

Course offered by: Giacomo Cherubini (INdAM)
Duration: 16h
Period: February - April 2026

Program

• Arithmetic functions; multiplicative functions, L-functions of degree 1. Examples.

• Congruence groups and Fuchsian groups. Definitions of modular forms and automorphic forms. Examples.

• Poisson summation in R^n. Spectral theorem for automorphic forms.

• The pretrace formula. Applications: Hyperbolic circle problem.

• Selberg's trace formula. Applications: Prime Geodesic Theorem.

• Kuznetsov formula and Kloosterman sums formula. The Linnik-Selberg conjecture on sums of Kloosterman sums.

Schedule

• February: Thursday 5, 19, 26; 

• March: Thursday 5, 12, 19, 26;

• April: Thursday 2.

All lectures will take place in Room F at the Department of Mathematics "Guido Castelnuovo", Sapienza, from 10:00 to 12:00.

Course offered by: Guido Pezzini (Sapienza)
Duration: 20+20h
Period: March-June 2026

Program

• Foundations of algebraic linear groups and their representations

• Linearly reductive groups

• Quotients and geometric invariant theory

Schedule

Every Tuesday, starting March 17, from 15:30 to 18:30 in Room F (Department of Mathematics "G. Castelnuovo", Sapienza)

Webpage

Further information on the course are available at the dedicated webpage.

Course offered by: Eugenio Landi (Sapienza)
Duration: 10h
Period: March 2026

Program

The course will be an introduction to the theory of model categories and abstract homotopy theory. After the basic definitions (model structures, homotopy categories, Quillen functors) we will study homotopy limits and colimits, some useful properties of model categories and their relation to (∞,1)-categories. Time permitting we will talk about operads and the homotopy transfer theorem. 

Schedule

March, Tuesday 3, 10, 17, 24, 31, from 9:00 to 11:00, in Room F of the Department of Mathematics "Guido Castelnuovo".

Lectures can be also attended remotely following this link.

Course offered by: Sachin Gautam (Ohio State University)
Duration: 8h
Period: March 2026

Program and References

Available here.

Schedule

March, Tuesday 3, 10, 17, 24, from 11:00 to 13:00, in Room L at the Department of Mathematics "Guido Castelnuovo".

Course offered by: Igor Sikora (Bilkent University)
Duration: 8h
Period: October 26-30, 2026

Program

TBD

Course offered by: Lucia Di Vizio (Université de Versailles-St Quentin)
Duration: TBD
Period: TBD

Program

Available here.

Course offered by: Efthymios Sofos (Tor Vergata)
Duration: 20h
Period: November 2025 - January 2026

Program

In the first half of the course we will cover the theory of the famous paper of Riemann on the zeta function and the prime numbers. This will include the Riemann hypothesis and the proof of the Explicit Formula and the prime number theorem. In the second half of the course we will see how this theory is used in research today in 2 topics in Arithmetic Geometry. The first topic is Serre's problem on the probability that a random Diophantine equation has a solution. The second topic is Goldfeld's conjecture on the probability that a randomly chosen elliptic curve with integer coefficients has infinitely many rational solutions.

Schedule

• November 4: Definition of the zeta function and basic properties – Poisson summation, functional equation for the zeta function

• November 11: Prerequisites on Euler's Gamma functions, integral functions of order 1 – infinite products (Weierstrass' theorem)

• November 18: Zero free region – number of zeros in boxes

• November 25: Explicit formula

• December 2: Proof of the Prime Number Theorem, statement and consequence of the Riemann Hypothesis – Selberg-Delange method, Large sieve for quadratic characters

• December 9: The field of p-adic numbers, Hilbert symbols, Hilbert reciprocity – Hasse's local-global principle for quadratic hypersurfaces

• December 16: Friedlander-Iwaniec's 2010 paper "Ternary quadratic forms with rational zeros" on Serre's problem

• January 5 and 13: Theory of elliptic curves

• January 20: Heath-Brown's 1993 paper "The size of Selmer groups for the congruent number problem" that is connected to averages of ranks of elliptic curves

All lectures will take place on Tuesday at 11:00-13:00 in Room D'Antoni at the Department of Mathematics, Tor Vergata.

References

• H. Davenport. Multiplicative Number Theory. Graduate Texts in Mathematics, vol. 74, Springer (1980)

• J.-P. Serre. Spécialisation des éléments de Br2(Q(T1,…,Tn)), C. R. Acad. Sci. Paris Ser. I Math., pages 397-402 (1990)

• D.R. Heath-Brown. The size of Selmer groups for the congruent number problem. Invent. Math. 111(1), pages 171-196 (1993)

Course offered by: Luca Francone and Martina Lanini (Tor Vergata)
Period: November 2025 - September 2026

Program

It is an informal gathering taking place every Friday consisting of different modules (sometimes offered as a reading course) on various topics which will depend on the audience interests (among the possible topics, quiver representations, quiver Grassmannians, King’s stability spaces, flag varieties, perverse sheaves, representations of affine Lie algebras, Kazhdan-Lusztig theory.

Schedule

Every Friday at 11:00-13:00, starting on November 7, 2025 in Room Dal Passo at the Department of Mathematics, Tor Vergata.

Course offered by: Sara Scaramuccia (Tor Vergata)
Duration: 20h
Period: January - February 2026

Program

The main topic of the course is to introduce computational aspects in applied topology, computational topology, in order to provide the main mathematical ingredients to the the rather new field of research called Topological Data Analysis - TDA. TDA aims at capturing the shape of data as a robust and multiscale descriptor to be integrated in data handling: classification, comparison, multidimensional reduction, etc. First, we deal with the constructions from data to simplicial complexes according to the kind of data: filtrations of data, point clouds, networks, and topological spaces. For each construction, we underline the possible dependence on a fixed scale parameter. Secondly, we introduce the necessary algebraic structures capturing topological informations out of a simplicial complex at a fixed scale, namely the simplicial homology groups. The so-obtained linear structures are then integrated into the multiscale framework of persistent homology where the entire persistence information is encoded in algebraic terms and the most advantageous persistence summaries available in the literature are discussed. Finally, we introduce the necessary metrics in order to state properties of stability of the introduced multiscale summaries under perturbations of input data. At the end, we give an overview of applications of persistent homology as well as a review of the existing tools in the broader area of TDA.

Website of the course: available here.

Schedule

From January 15 to February 17, 2026, every Tuesday & Thursday at 14:30-16:00 in Room 1200, Department of Mathematics, Tor Vergata.

Course offered by: Victor Turchin (Kansas State University)
Duration: 30h
Period: February - March 2026

Program

The goal of the course is to provide an introduction to the manifold functor calculus and its modern reformulation in the operadic language. The main application that we will consider is the delooping problem for many types of mapping spaces: disc embedding spaces relative to the boundary, more general spaces of disc maps avoiding a fixed type of a multi-singularity (such as non-k-equal immersions), disc concordance embeddings, etc. We will also discuss the smoothing theory approach to delooping of diffeomorphism and embedding spaces.

Schedule

From February 9 to March 30, 2026, every Monday & Thursday at 11:00-13:00 in Room D'Antoni, Department of Mathematics, Tor Vergata.

Course offered by: Angelica Cueto (Ohio State University)
Duration: 6h
Period: March 2026

Program

Tropical Geometry has been the subject of great amount of activity over the last two decades sparked by its application to enumerative geometry. Loosely speaking, it can be described as a piecewise-linear version of algebraic geometry. It is based on tropical algebra, where the sum of two numbers is their maximum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. The combinatorial structure of these tropical varieties retains a surprising amount of geometric information about their classical counterparts.

In this course, we will give an introduction to the subject, focusing on how it can be used to recover three classical results from  the 19th century: (1) Pluecker's Theorem on the 28 bitangent lines to smooth quartic complex plane curves, (2) Coble's Theorem on the 120 tritangent planes to smooth sextic curves in projective space, and (3) Cayley-Salmon's Theorem on the 27 lines on smooth cubic complex surfaces in projective space.

Schedule

From March 6 to March 20, 2026, every Friday at 11:30-13:00 in Room D'Antoni, Department of Mathematics, Tor Vergata.

Course offered by: Dahisy Lima (Federal University of ABC, Brazil)
Duration: 12-16h
Period: May - June 2026

Program

TBA

Course offered by: Sandro Verra (Roma Tre)
Duration: 20h
Period: 2 April - 28 May 2026

Program

• Geometry of linear systems of quadrics

• Determinantal equations of planar curves

• Vector bundles on curves, Picard group

• Surfaces in P^3: classical examples

• Cremona transformations in P^3: classical examples

• Geometry of the line and Segre classes

• The cubic hypersurface of P^4

• Rational cubic hypersurfaces in P^5

• Unirationality problems for hypersurfaces in P^n

Course offered by: Enrico Rogora (Sapienza - History), Benedetto Scoppola (Tor Vergata - Didactics), Lorenzo Tortora de Falco (Roma Tre - Logic)
Duration: 36h
Period: TBD

Theme 1. Didactics

• Euclid's algorithm in arithmetics (GCD) and number theory (continued fractions)

• Propotions with integer elements: the results from Book VII

• Powers and continued proportions

• Incommensurables

Theme 2. Logic

The “Logic” module (12 hours) of the institutional PhD course “Didactics, Logic and History of Mathematics” will be held at the Department of Mathematics and Physics of Roma Tre University (room to be determined) during the month of April 2026, following the provisional program attached below.

The course will consist of 4 meetings of 3 hours each, either on a weekly basis (from Monday 30 March to Friday 24 April) or on a biweekly basis (from Monday 13 April to Friday 24 April).

The instructors will be Lorenzo Tortora de Falco (Roma Tre University) and Lionel Vaux (Aix-Marseille Université).

Call for Interested Students: In order to better determine both the dates and the content of the course, interested students are asked to come forward, briefly presenting their academic background (with particular reference to any Logic courses they have already taken) by sending an email to: tortora@uniroma3.it BY MARCH 1, 2026.

Program of the Course

Part 1 (6h): General Proof Theory

1. • Satisfiability and provability. Fundamental theorems of first-order logic (Canonical Analysis)

   • Gentzen and the cut-elimination theorem

   • Proofs and programs: Curry–Howard correspondence

Part 2 (6h): Logical Proofs as Graphs (Linear Logic)

1. • Graphs and colorings: Yeo’s theorem and variants

   • Proof nets: a graphical language for Linear Logic proofs

   • Graph theory meets proof theory: correctness criteria

   • Cut elimination as graph rewriting

Theme 3. History

• Lagrange

• Ruffini-Abel's theorem

• Abel's work on equations and elliptic functions

• Galois' contribution

• Analytical solutions to equations of fifth degree

• Galois-Klein theory

Course offered (in Italian) by: Paolo Freguglia (L'Aquila)
Duration: 12h
Period: November - December 2025

Program and References

Available here.

Schedule

• November: Tuesday 11, 14:00-16:00; Wednesday 12, 9:00-11:00; Tuesday 18, 14:00-16:00; Wednesday 19, 9:00-11:00;

• December: Tuesday 2, 14:00-16:00; Wednesday 3, 9:00-11:00.

All lectures will take place in Room F at the Department of Mathematics "Guido Castelnuovo", Sapienza.

Course offered (in Italian) by: Enrico Rogora (Sapienza)
Duration: 12h
Period: November-December 2025

Program and References

Available here.

Schedule

• November: Tuesday 11, 11:00-13:00; Wednesday 12, 11:30-13:30; Tuesday 18, 11:00-13:00; Wednesday 19, 11:30-13:30;

• December: Tuesday 2, 11:00-13:00; Wednesday 3, 11:30-13:30.

All lectures will take place in Room F at the Department of Mathematics "Guido Castelnuovo", Sapienza.

Course offered by: Radu Ignat (University of Toulouse) @SBAI
Duration: 20h
Period: January - February 2026

Program

The course aims at presenting the modern mathematical tools used in the study of micromagnetics. This topic lies at the interface between Partial Differential Equations (PDE), Calculus of Variations and Mathematical Physics. More precisely, micromagnetics is the continuum theory of magnetic moments underlying the description of magnetic structures and it is based on a variational principle that is nonconvex, nonlocal and multiscale. The plan is to analyse the pattern formation in ferromagnetic materials, in particular, domain walls, vortices, skyrmions etc. This study raises fundamental mathematical questions (regularity, uniqueness, symmetry, stability, asymptotic analysis etc.) for which various techniques will be presented coming from elliptic PDEs, theory of harmonic maps, scalar conservation laws, Γ-convergence, theory of Ginzburg-Landau etc. More info may be found here. The course is supported by INdAM. Interested students are invited to contact the Lecturer (radu.ignat@math.univ-toulouse.fr).

Schedule

From 16/01/2026 to 14/02/2026, every Tuesday & Friday 14:00-16:00, room 1E, building RM004, SBAI Department, Sapienza.

Course offered by: Massimo Grossi and Angela Pistoia (Sapienza) @SBAI
Duration: 16h
Period: February 2026

Program

Topological degree in finite dimensions, index of a critical point, and the Poincaré-Hopf theorem. Applications to solutions of differential equations: the torsion problem and eigenfunctions of the Laplacian. Overview of solutions on Riemannian manifolds. The implicit function theorem in infinite dimensions. Bifurcation from eigenvalues. The case of a simple eigenvalue. Lyapunov-Schmidt finite-dimensional reduction and applications to nonlinear elliptic problems. Prerequisites: Knowledge of general topology, L^p and Sobolev spaces, and the variational formulation of boundary value problems. Particularly suitable for students with a Master's degree in Mathematics or Physics. Interested students are invited to contact the Lecturers (massimo.grossi@uniroma1.it, angela.pistoia@uniroma1.it).

Schedule

From 05/02/2026 to 26/02/2026, every Monday 16:00-18:00 & Thursday 09:00-11:00, room 1E, building RM004, SBAI Department, Sapienza.

Course offered by: Albert Fathi (ENS Lyon)
Duration: 8h
Period: February 2026

Program

We introduce a framework for the "viscosity theory of the Hamilton-Jacobi equation" on more general metric spaces avoiding any recourse to tangent spaces or PDE’s. This allows for more geometric analysis aspects. It also applies to the space of probability measures on a compact space, allowing for example a better analysis of the action of the "induced action of a geodesic flow" (or an Euler-Lagrange flow) on a candidate for tangent space to the space of probability measures of a Riemannian manifolds, bridging with optimal transport and the recent formidable advances on the geometric and analytic aspects of these probability spaces.

Schedule

• Monday, February 2: 11:00–13:00

• Wednesday, February 4: 10:30–12:30

• Monday, February 9: 11:00–13:00

• Wednesday, February 11: 10:30–12:30

All lectures will take place in Room F at the Department of Mathematics "Guido Castelnuovo", Sapienza.

Course offered by: Francescantonio Oliva (Sapienza) @SBAI
Duration: 20h
Period: February-March 2026

Program

Sobolev spaces: approximation by smooth functions, embedding theorems, and the trace theorem. Weak formulation of elliptic boundary value problems: existence of solutions to the Dirichlet problem through minimization or the Lax-Milgram theorem. Elliptic regularity theory. Basics on uniqueness and nonuniqueness phenomena. Heat equation. Conservation laws. This is a draft, it will be adapted based on the audience. Interested students are invited to contact the Lecturer (francescantonio.oliva@uniroma1.it).

Schedule

From 3/02/2026 to 3/03/2026, every Tuesday & Friday 10:30-12:30, room 1E, building RM004, SBAI Department, Sapienza.

Course offered by: Vito Crismale (Sapienza)
Duration: 16h
Period: March 2026

Program

• The main types of Fracture Energies and their mechanical meaning/derivation

• The framework of linearized theories for bulk part

• The appropriate functional setting

• Approximate discrete evolutions

• Quasistatic evolutions (Griffith model and approximate cohesive models)

Schedule

March, Wednesday 4, Friday 5, Monday 8, Wednesday 10, Friday 12, Monday 15, Wednesday 17, Friday 19, from 11:00 to 13:00 in Room F at the Department of Mathematics "Guido Castelnuovo".

Course offered by: Elvira Zappale (Sapienza) @SBAI
Duration: 20h
Period: March - May 2026

Program

Preliminaries on measure theory; Riesz's representation theorem, Radon-Nykodim derivative, covering thoerems. Functional spaces as BV, BD, BH. The global method for relaxation by Bouchittè-Fonseca-Mascarenhas and further developments of the theory. Intereseted students are invited to contact the Lecturer (elvira.zappale@uniroma1.it).

Course offered by: Gabriella Tarantello (Tor Vergata)
Duration: 20h
Period: March 2026

Program

Recent results will be presented on the description of the moduli space of Constant Mean Curvature (CMC) immersions of a closed surface into hyperbolic 3-manifolds. To this end, both analytical methods (blow-up techniques for solutions of Liouville-type equations) and geometric methods (Higgs bundles and Hitchin theory for self-dual equations) will be developed.

Schedule

• Wednesday, March 4 (Introductory colloquium to present the course), Room: D'Antoni

• Thursday, March 5, 12 at 14:00-16:30, Room: D'Antoni

• Friday, March 6, 13, 20, 27 at 14:00-16:30, Room: D'Antoni

• Tuesday, March 10 at 16:00-18:30, Room: De Blasi

• Monday, March 16, 23 at 14:00-16:30, Room: D'Antoni

• Monday, March 30 (Backup lecture, only if necessary) at 14:00-16:30, Room: D'Antoni

Course offered by: Luigi Chierchia, Michela Procesi (Roma Tre)
Duration: 20h
Period: March - May 2026

Program

• Stable and unstable manifolds. Linearization of holomorphic maps and small divisors (Siegel)

• Diffeomorphisms of holomorphic vector fields; Poincaré-Dulac Theorem in resonant and non-resonant cases

Tentative schedule

• Chierchia: 30/03 (14-16), 01/04 (9-11), 08/04 (9-11), 10/04 (9-11), 13/04 (14-16);

• Procesi: 27/04 (14-16), 29/04 (14-16), 04/05 (14-16), 06/05 (14-16), 11/05 (14-16).

Course offered by: Jean Barbier (ICTP)
Duration: 16h
Period: September 2025

Program

Assuming prior knowledge of basic concepts in statistics and probability (in particular, probability distributions, Kullback-Leibler divergence, statistical inference and the Bayes formula), as well as some fundamentals of information theory (in particular, Shannon entropy, the source coding theorem, mutual information, minimum mean-square error, MMSE), the proposed course consists of 12 hours of frontal lectures covering the following topics:

• Models in high dimensional inference;

• Information-theoretic versus algorithmic limits;

• Introduction to random matrix theory;

• Spectral phase transition in the spike Wigner model;

• Concentration-of-measure in Bayesian optimal inference;

• Introduction to the generalised approximate message-passing algorithm.

Additionally, the course includes 4 hours of tutorials covering:

• Simulation of the spectral PCA estimator and comparison against the Bayesian estimator obtained by Monte Carlo methods;

• Step-by-step derivation of the information-theoretic limit in the spike Wigner model using the replica method.

Course offered by: Fabrizio Zanello (University of Potsdam)
Duration: 8h
Period: December 2025

Website of the course: available here.

Program

In these lectures I will present the algebraic approach to Quantum Field Theory in its simplest framework, namely, on flat Lorentzian spacetimes. First, observables are introduced as elements of an abstract algebra, prior to any representation on some Hilbert space. Then I will outline the appropriate analytical framework, essentially based on microlocal analysis techniques, which allows to interpret a quantum field theory as a deformation quantization of its classical counterpart. Finally, with these notions at hand, I will approach the interaction picture from the point of view of the so-called causal perturbation theory and, if time permits, I will formulate in a mathematically rigorous way the renormalisation problem.

Schedule

• Thursday, December 4, 2025 (Room F)

• Tuesday, December 9, 2025 (Room L)

• Thursday, December 11, 2025 (Room L)

• Thursday, December 18, 2025 (Room L)

All lecture take place from 10:30 to 12:30 at the Department of Mathematics of Sapienza.

Course offered by: Pierpaolo Vivo (King's College London)
Duration: 20h
Period: January 12-27, 2026

Program

This is a blackboard course on foundation and modern applications of Random Matrix Theory (RMT). The course will cover the following topics:

• Simple classification of random matrix models. Gaussian and Wishart ensembles. Warmup calculations: semicircle and Marčenko–Pastur laws.

• Level spacing statistics: Poisson vs Wigner–Dyson.

• Coulomb gas method.

• Orthogonal polynomial technique and numerical checks.

• Largest eigenvalue of a random matrix. Comparison with Extreme Value Statistics for i.i.d. random variables. Tracy–Widom distribution and third-order phase transitions.

• The Replica method. Edwards–Jones formalism. Applications to full and sparse matrices (random graphs). Large deviation function of the largest eigenvalue using replicas. Replica theory of random linear systems with quadratic constraints and associated phase transitions.

• Free probability. Sum of free random matrices. Blue’s function. R and S transforms.

• RMT for high-dimensional data analysis (matrix denoising, spiked models with structured noise, dictionary learning).

In addition, Dr Vivo will deliver two research seminars on his recent activity that combines Statistical Mechanics methods with real-life applications to economics, biology, and legal systems. Dr Vivo will also make himself available to lead short master classes and tutorials on specific topics of interest for the postgraduate cohorts in the department, including hands-on and computational sessions, student-led journal clubs and seminars, collective critical analysis of a recent research paper from a peer-review perspective. Any interested attendee is invited to fill in this form.

Course offered by: Alessandro Ingrosso (Radboud Universiteit)
Duration: 16h
Period: January 12 - January 24, 2026

Program

The course will explore foundational and advanced topics in the Statistical Mechanics approach to Neural Networks and Machine Learning, covering the following topics:

• Equilibrium theory of learning in neural networks:

  1. the Gardner approach to learning in simple perceptrons

  2. learning with structured dataset

  3. beyond perceptrons: neural networks with one hidden layer and beyond

  4. kernel renormalization perspective on learning in neural networks

  5. a Random Matrix perspective on learning in large-width neural networks

• Dynamics and learning:

  1. online learning dynamics in narrow networks

  2. learning in the neural-tangent-kernel (NTK) regime in large-width networks

  3. recurrent networks: dynamical mean-field theory and dimensionality

Both theoretical lectures and interactive sessions are planned.

Course offered by: Yu Deng (University of Chicago)
Duration: 4+8h
Period: March 2026

Program

In this series of 4 lectures we discuss our recent proof (with Zaher Hani and Xiao Ma) on the long-time derivation of the Boltzmann equation, starting from hard sphere dynamics, under the Boltzmann-Grad scaling law. The proof relies on a layering argument, recursive formulas for cumulants, and construction and analysis of combinatorial structures called molecules.

To facilitate participation by students with different backgrounds, the course will be preceded by two preparatory sessions reviewing the classical theory.

The plan for each lecture is as follows.

Preparatory sessions

• Session 1: Sergio Simonella (Sapienza)

Introduction to kinetic limits. A historical and conceptual overview of the derivation of the Boltzmann equation, from early kinetic theory to Lanford's 1975 theorem.

• Session 2 - Informal discussion: Mario Pulvirenti (Sapienza)

Interactive Q&A meeting aimed at clarifying foundational aspects of the Boltzmann equation.

Lecture plan (Yu Deng)

• Lecture 1: Introducing the brief idea of the proof: dividing long time intervals into short "layers", and deriving recursive formulas for the cumulant at each layer, in terms of molecules.

• Lecture 2: Reducing the theorem to estimating integral expressions for each molecule, and introducing the basic cutting operation for analyzing the molecules.

• Lecture 3: The cutting algorithm I: identifying the recollisions and reducing the general multiple-layer case to the two-layer case.

• Lecture 4: The cutting algorithm II: completing the two-layer case by introducing several basic algorithms and find the correct combinations oof them by studying certain dichotomy.

Schedule

• Session 1: Thursday, March 12, 10:00-12:00

• Session 2: Tuesday, March 17, 11:00-13:00

• Lecture 1: Thursday, March 19, 10:00-12:00

• Lecture 2: Tuesday, March 24, 11:00-13:00

• Lecture 3: Thursday, March 26, 10:00-12:00

• Lecture 4: Tuesday, March 31, 11:00-13:00

All lectures will take place in Sala di Consiglio at the Department of Mathematics "Guido Castelnuovo", Sapienza.

Funded by the European Union (ERC CoG KiLiM, 101125162).

Course offered by: Stefan Teufel (Eberhard-Karls Universität Tübingen)
Duration: 8-10h
Period: March 2026

Program

The quantum Hall effect (QHE) refers to the fact that the Hall conductance of a two-dimensional electron gas takes on only quantised values, i.e. integer (or fractional) multiples of e²/h. First discovered in 1980, it has sparked significant experimental, theoretical and mathematical research, with Nobel Prizes being awarded to both experimentalists and theorists. In this series of blackboard lectures, I will discuss recent developments that demonstrate how certain experimentally relevant features of the QHE can be rigorously understood within microscopic models of interacting fermions. Most importantly, I will explain results that establish a rigorous Ohm’s law for macroscopic Hall currents in interacting, spectrally gapped systems at zero temperature. Familiarity with the fundamental mathematical principles of quantum mechanics is expected. However, I will introduce the relevant operator algebraic framework during the lectures.

Schedule

• Friday, March 20, 14:00-16:00

• Monday, March 23, 14:30-16:30

• Wednesday, March 25, 16:00-18:00

• Monday, March 30, 14:30-16:30

• (A fifth lecture will be scheduled, if required.)

All lectures will take place in Room F [TBC] at the Department of Mathematics "Guido Castelnuovo", Sapienza. Interested participants can contact monaco@mat.uniroma1.it for more information.

Funded by the Bando Professori Visitatori Sapienza 2025.

Course offered by: Adriano Barra (Sapienza) @SBAI
Duration: 20h
Period: April - May 2026

Program

The course follows a historical perspective and analyzes explicit mathematical models and methods related to information processing spontaneously achieved by networks of neurons (biological or artificial). After a brief reviewing of key concepts stemming in statistical mechanics, stochastic processes and statistical inference, we start from the first models for the emission of an electrical signal from a single neuron, to move toward the study of their interactions in simple neural architectures, analyzing the statistical learning capabilities that these networks enjoy. In particular, due to the Nobel Prize in Physics awarded in 2024 to John Hopfield and Geoffrey Hinton for their pioneering studies on neural networks, particular emphasis will be placed on their contributions and on the close connections that exist between them. The methodological leitmotif will be the statistical mechanics of complex systems (i.e. Parisi's theory, Nobel Prize in Physics in 2021) with its associated package of observables and typical tools (replicas, overlaps, cavity fields, etc.). Interested students are invited to contact the Lecturer (adriano.barra@uniroma1.it).

Schedule

From April 20 to May 8, 2026, every Monday, Wednesday and Friday from 10:00 to 13:00, room 1E, building RM004, SBAI Department, Sapienza. No lecture on May 1. Last two lectures from 11:00 to 13:00. 

Course offered by: Carlangelo Liverani (Tor Vergata)
Duration: 20h
Period: November - December 2025

Program

• The problem of micro versus macro

• Probability and dynamical systems

• Qualitative statistical properties of dynamical systems

• Strong dependence on initial conditions and hyperbolicity

• Quantitative statistical properties of dynamical systems

• Hyperbolic billiards

• N hard spheres in a box

Schedule

• Tuesday, November 11, 18, 25 at 14:00-16:00, Room: D'Antoni

• Thursday, November 13, 20, 27 at 14:00-16:00, Room: D'Antoni

• Monday, December 1, 8 at 11:00-13:00, Room: Dal Passo

• Wednesday, December 3, 10 at 14:00-16:00, Room: D'Antoni

All lectures will take place at the Department of Mathematics, Tor Vergata.

Course offered by: Lorenzo Dello Schiavo (Tor Vergata)
Duration: 20h
Period: January - February 2026

Program

The course (tentatively: 10 two-hour lectures) will be divided into two parts. The first part (7-8 lectures) will review the theory of Dirichlet forms and their relation to Markov processes. A Dirichlet form is an energy functional (e.g., the Sobolev energy ∫|∇u|^2 dx) with a corresponding generator (e.g., the Laplacian −Δ), and semigroup (e.g., the heat semigroup t ↦ e^(tΔ)). The semigroup is an integral operator that can be represented by a kernel pt(x,dy), the transition kernel of a Markov process on a general state space (e.g., Brownian motion). We will show that this correspondence is essentially bijective, and that almost-sure properties of the Markov process may be deduced from properties of the corresponding form. The second part (2-3 lectures) will explore in detail a single example.

References

• M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. De Gruyter Studies in Mathematics, vol. 19, De Gruyter (2011)

• Z.-M. Ma and M. Röckner. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Universitext, Springer (1992)

Schedule

From January 7 to February 5, 2026, every Wednesday & Thursday at 14:15-16:00 in Room Dal Passo, Department of Mathematics, Tor Vergata.

Course offered by: Luca Fresta (Roma Tre)
Duration: 20h
Period: February - March 2026

Program

• Complex and Grassmann Gaussian integration (Efetov’s supersymmetric method)

• Random Schrödinger operators: Wegner and Lifshitz tail estimates

• Random matrix theory: Wigner's semicircle law for GUE matrices

• Gaussian random band matrices (RBM)

• Supersymmetric H^(n|m) non-linear sigma models and probabilistic connections

Tentative schedule

10 lectures in room C309 (Palazzina C, 3rd floor) from 11:00 to 13:00 on the following dates: 10/02, 12/02, 24/02, 26/02, 11/03, 13/03 16/03, 18/03, 20/03, 23/03. 

Course offered by: Daniele Bertaccini (Tor Vergata)
Duration: 20h
Period: May 2026

Course offered by: Yoh Tanimoto (Tor Vergata)
Duration: 20h
Period: December 2025 - January 2026

Program

In principle, a mathematical proof should be a logical sequence of claims that uses only the axioms and follows the inference rules. In the last few years, this idea has been realized concretely, to the extent that one can write some recent research results in a programming language, in particular in Lean. In this course, one learns the basics of the formal language Lean, of its library mathlib and of how to write advanced mathematics in Lean.

Schedule

From 10 December 2025 to January 2026, every Wednesday & Thursday at 9:30-11:30 in Room Dal Passo, Department of Mathematics, Tor Vergata.

Course offered by: Vincenzo Bonifaci (Roma Tre)
Duration: 20h
Period: January 13 - February 12, 2026

Website of the course: available here.

Program

• Convex sets, convex hulls, Carathéodory’s theorem, polyhedra and polytopes, extreme points, Minkowski’s theorem

• Convexity of functions, subgradients, inequalities, convex conjugate

• Convex optimization problems, Lagrange duality, KKT optimality conditions

• Convex optimization algorithms: gradient, subgradient, mirror descent

• Convex optimization modeling software (e.g., CVXPY)

Course offered by: Edoardo Persichetti (Florida Atlantic University)
Duration: 20h during 2 weeks
Period: May - June 2026

Abstract

Code-based cryptography is the area of study which is interested in primitives based on hard problems from coding theory. In recent years, it has seen an exponential increase in importance and volume of research, since it is one of the main areas inside “post-quantum” cryptography, and thus a candidate to secure our communication against quantum threats. In this course, we investigate the main constructions pertaining to code-based cryptography, starting from its inception in the late 1970s, and arriving to modern day. The course will include a refresher of all relevant notions from coding theory and cryptography.

Program

1. Coding theory refresher

2. Cryptography refresher

3. McEliece and Niederreiter

4. Structured Codes

5. BIKE and HQC

6. Signatures