PhD in Mathematics in Rome

Course offered by: Samouil Molcho (Sapienza)
Duration: 20h
Period: February 2027

Abstract

The course aims to expose the students to the modern viewpoint on enumerative geometry, that is intersection theory on moduli spaces. After a review of the foundations of intersection theory, we will apply them to progressively more delicate moduli spaces, terminating with the moduli space of vector bundles on a curve, to extract interesting consequences. An effort will be made to coordinate with Carocci's course at Tor Vergata while keeping the courses independent.

Program

• Cycles, rational equivalence, Chow groups. Pushforward, Gysin pullback, projection formula. Relationship with cohomology.

• Vector bundles and Chern classes. Projective bundle formula.

• Enumerative geometry of Grassmannians.

• The Grothendieck–Riemann–Roch theorem.

• Thom–Porteous formulas.

• The Jacobian of a curve. Intersection Theory on the Jacobian.

• Intersection theory on the moduli space of vector bundles on a curve.

Prerequisites

The students should have familiarity with algebraic varieties, coherent sheaves and their cohomology — from the algebraic or complex geometry perspective. Familiarity with algebraic curves would be very helpful.

Bibliography

• W. Fulton, Intersection Theory.

• A. Beauville, Sur la cohomologie de certains espaces de modules de fibrés vectoriels.

• D. Gieseker, Degenerations of the moduli space of stable bundles.

• A. Beauville, Vector bundles on curves and generalized theta functions.

• D. Zagier, On the cohomology of moduli space of rank two vector bundles over curves.

Schedule

TBD

Exam

Oral exam.

Course offered by: Francesca Carocci (Tor Vergata)
Duration: 20h
Period: January – March 2027

Abstract

This course aims to introduce the concept of moduli spaces as well as some of the classical concepts/techniques/results often used for the construction and the local and global study of such moduli spaces, focusing in particular on the example of moduli of semi-stable vector bundles on algebraic curves. Wherever, for time reason, some details of the more technical proofs will be omitted, the class will be complemented by a series of exercises, with the aim of fixing the main ideas via computations of easy examples/counterexamples.

Program

• Part I (4–6 hours): Coherent/Locally free sheaves and their cohomology on smooth projective varieties; Serre duality and Riemann-Roch for locally free sheaves on curves; cohomology and base change.

• Part II (4–6 hours): Moduli spaces and Quot schemes: Introduction to moduli Functors; sheaves in flat families and Hilbert polynomial; Quot scheme, representability and projectivity.

• Part III (6 hours): M(r,d) construction: Semi-stability, Jordan-Hölder and Harder-Narasimhan filtration; Boundedness of families of semi-stable bundles; GIT and conclusion of the construction.

• Part IV (4 hours): Geometry of M(r,d): Tangent to M(r,d); Smoothness criterion at stable points; irreducibility.

Prerequisites

The students should have some familiarity with algebraic varieties and coherent sheaves on them, as well as have seen singular cohomology for a smooth manifold. Ideally, they would also have seen Chern classes of vector bundles and the computation of cohomology for line bundles on projective space and on curves. However, these will be quickly reviewed in the first part of the course.

Bibliography Course

• J. Le Potier, Lectures on Vector Bundles.

Bibliography Prerequisites

• R. Hartshorne, Algebraic Geometry, Chapter I/II (except 9), working over an algebraically closed field.

Schedule

TBD

Exam

Oral exam.

Course offered by: Sandro Verra (Roma Tre)
Duration: 20h
Period: April – May 2027

Abstract

Let C be a curve of genus g, embedded in the (g-1)-dimensional projective space by the canonical linear system. The course aims to study the geometry of this embedding and relate it to some remarkable properties in the theory of curves and their Jacobians.

Program

• Canonical models of curves in low genus.

• Canonical curves, and rational normal scrolls.

• Quadrics through a canonical curve.

• The theorems of Noether -- Enriques -- Kempf.

• Mukai's constructions of canonical curves.

• Paracanonical and Prym canonical curves.

Bibliography

 E. Arbarello, M. Cornalba, h. Griffiths, J. Harris, Geometry of Algebraic Curves, I & II. 

D. Mumford, Curves and their Jacobians

Course offered by: Efthymios Sofos (Tor Vergata)
Duration: 8h
Period: November 2026

Abstract

We will give an introduction to the circle method for quadratic forms. This is a general number theory method that uses ideas from analysis to count the number of integer solutions in Diophantine equations in many variables. It has been used to prove general results for Hasse principle and weak approximation as well as non geometric questions like the ternary Goldbach problem. No background required. 

Schedule

From November 4 to 25, 2026, every Wednesday at 10:00-12:00 in INdAM.

Exam

TBD

Information on the course is provided in Italian since the course will be delivered in Italian.

Corso erogato da: Enrico Rogora (Sapienza)
Durata: 20h
Periodo: Gennaio 2027

Abstract

Nel corso verranno presentate le tappe principali della costruzione della moderna teoria delle equazioni algebriche e gli intrecci con la teoria delle funzioni ellittiche, la teoria dei numeri e la teoria dei gruppi. L'esposizione farà ampio uso di brani originali tratti da articoli, monografie e scambi epistolari. Ad un taglio principalmente storico nell'esposizione si accompagnerà il riferimento alla trattazione moderna degli argomenti per apprezzare il valore del confronto tra visioni diverse del fare matematica.

Programma

• Il punto di vista a priori di Lagrange, l'idea di risolvente e lo schema di Lagrangiano per la ricerca delle soluzioni di un'equazione algebrica.

• L'analisi di Gauss del carattere delle soluzioni delle equazioni ciclotomiche e l'algoritmo più efficiente per determinarle. Le applicazioni alla determinazione degli strumenti esatti per la divisione della circonferenza in parti uguali.

• I diversi approcci alla dimostrazione di non esistenza delle soluzioni per radicali delle equazioni di grado superiore al quarto (Ruffini-Abel, Galois, Arnold).

• Il problema della determinazione dei punti di divisione di una curva ellittica e l'analisi di Abel della risolubilità delle equazioni di divisione.

• Il problema della trasformazione delle funzioni ellittiche e i lavori di Jacobi sull'equazione modulare e del moltiplicatore.

• La sintesi di Galois e l'analisi di Galois delle condizioni di riducibilità del grado dell'equazione modulare di ordine 5.

• I contributi di Betti, Hermite, Brioschi e Kronecker alla risoluzione per funzioni ellittiche dell'equazione generale di quinto grado.

• La geometria dell'icosaedro e l'approccio geometrico di Klein alla risoluzione dell'equazione di quinto grado.

Prerequisiti

Nessuno oltre i contenuti dei corsi della laurea triennale in Matematica.

Bibliografia

• J. Tignol, Galois Theory of Algebraic Equations.

• J. Gray, A History of Abstract Algebra.

• E. Rogora, Appunti del corso di Istituzioni di Matematiche Complementari.

• E. Rogora, La teoria delle equazioni algebriche in Italia nella seconda metà del XIX secolo, Atti del convegno su Ruffini, Accademia delle Scienze dei XL, 2023.

• A. Cogliati, E. Rogora, Il problema della risoluzione delle equazioni di quinto grado nella corrispondenza tra Brioschi e Klein, Atti del convegno su Brioschi, Accademia delle Scienze dei XL, 2024.

• C. Maffei, Il libro di Klein sull'Icosaedro, Atti del convegno su Brioschi, Accademia delle Scienze dei XL, 2024.

Calendario

Da determinarsi.

Esame

Esame orale.

Information on the course is provided in Italian since the course will be delivered in Italian.

Corso erogato da: Riccardo Bellé (Tor Vergata)
Durata: 20h
Periodo: Gennaio – Febbraio 2027

Abstract

Il corso intende fornire un’introduzione alla ricerca nella storia delle matematiche attraverso lo studio dell’algebra dall’Antichità al Rinascimento: da Diofanto alla matematica araba, dalla tradizione dell’abaco alla soluzione delle equazioni di terzo e quarto grado, fino all’analisi dell’Algebra di Rafael Bombelli, assunta come caso di studio privilegiato. Non si intende tuttavia proporre una storia lineare dell’algebra; una questione che accompagnerà trasversalmente le diverse lezioni è la seguente: in che misura è lecito usare categorie della matematica contemporanea nella ricostruzione della matematica del passato? In particolare, è storicamente legittimo utilizzare il termine “algebra” per descrivere pratiche, problemi, tecniche e forme di sapere sviluppatesi in contesti molto diversi e lungo un arco cronologico di oltre un millennio?

Accanto alla ricostruzione dei contenuti matematici, particolare attenzione sarà dedicata ai metodi della ricerca storico-matematica: lettura e interpretazione delle fonti primarie attraverso l’analisi di manoscritti e stampe antiche; trascrizione e confronto fra testimoni e problemi relativi all’edizione dei testi matematici. Le ultime otto ore avranno carattere prevalentemente laboratoriale e saranno dedicate all’esame diretto delle fonti manoscritte e a stampa dell’Algebra di Bombelli.

Programma

• Modulo 1: Che cos’è la storia della matematica? Questioni storiografiche e il caso di Diofanto.

• Modulo 2: Equazioni e al-jabr: dalla matematica araba alla tradizione d’abaco. 

• Modulo 3: La soluzione delle equazioni di terzo grado.

• Modulo 4: Bombelli, autore de L’Algebra.

• Modulo 5: Editare e interpretare Bombelli: dall’Algebra manoscritta all’edizione critica.

Prerequisiti

Non sono richieste conoscenze specialistiche pregresse di storia della matematica. La conoscenza del latino è utile ma non indispensabile, trattandosi per lo più di testi in volgare. Per le fonti greche e arabe si farà ricorso, quando necessario, a traduzioni ed edizioni moderne.

Bibliografia

• Diofanto, Arithmetica.

• al-Khwārizmī, Kitāb al-jabr wa-l-muqābala.

• Leonardo Pisano, Liber abbaci.

• Girolamo Cardano, Artis magnae sive de regulis algebraicis liber unus, Nürnberg, 1545.

• Rafael Bombelli, L’Algebra, Bologna, 1572.

• A. Fiocca, E. Leone (a cura di), L’inedito terzo libro de L’Algebra di Rafael Bombelli, Edizioni della Normale, Pisa, 2017.

Calendario

Da determinarsi.

Esame

La verifica finale consisterà in un breve lavoro di ricerca su una fonte matematica concordata con il docente. A partire dalla riproduzione di un manoscritto o di una stampa antica, il dottorando dovrà predisporre la trascrizione di un passo circoscritto, analizzarne il contenuto matematico, confrontarlo, ove possibile, con altri testimoni o redazioni e discuterne i principali problemi storici ed editoriali. Il lavoro sarà presentato e discusso oralmente, con particolare attenzione ai metodi utilizzati, alle fonti consultate e al rapporto fra analisi testuale e interpretazione matematica.

Information on the course is provided in Italian since the course will be delivered in Italian.

Corso erogato da: Lorenzo Tortora de Falco (Roma Tre)
Durata: 12h
Periodo: Aprile – Maggio 2027

Abstract

Lo scopo del corso è presentare la teoria della dimostrazione ed introdurre alcuni temi della ricerca contemporanea.

Programma

• Lezione 1: Soddisfacibilità e dimostrabilità. Nozione di struttura per un linguaggio del primo ordine, soddisfacibilità di una formula e di una teoria. Derivabilità nel calcolo dei sequenti di Gentzen. Teorema fondamentale dell'analisi canonica e conseguenze (completezza, compattezza, eliminabilità del taglio, Löwenheim-Skolem).

• Lezione 2: Gentzen e l'eliminazione del taglio. Trasformazione di una derivazione logica in una derivazione senza tagli nel calcolo dei sequenti. Motivazioni storiche e conseguenze notevoli.

• Lezione 3: Dimostrazioni e programmi: la corrispondenza di Curry-Howard. Relazione tra derivazioni della deduzione naturale del frammento minimale e termini del $\lambda$-calcolo semplicemente tipato. Interpretazione delle dimostrazioni come programmi ed eliminazione del taglio come esecuzione.

• Lezione 4: Introduzione alla Logica Lineare. L'approccio di Jean-Yves Girard (1987): decomposizione dei connettivi classici, dimostrazioni come grafi ed eliminazione del taglio come deformazione. Analisi di un frammento di Logica Lineare e dei suoi risultati principali.

Prerequisiti

Nessuno oltre una laurea magistrale in matematica (con qualunque indirizzo).

Bibliografia

• V. M. Abrusci, L. Tortora de Falco, Logica. Vol. 1: Dimostrazioni e modelli al primo ordine, Springer, 2014.

• V. M. Abrusci, L. Tortora de Falco, Logica. Vol. 2: Incompletezza, teoria assiomatica degli insiemi, Springer, 2018.

•  J.-Y. Girard, Proof Theory and Logical Complexity, Bibliopolis, 1987.

• J.-Y. Girard, Y. Lafont, P. Taylor, Proofs and Types, Cambridge University Press, 1989.

• J.-Y. Girard, Linear Logic, Theoretical Computer Science, Vol. 50(1), 1987.

Course offered by: Claudia Pinzari (Sapienza)
Duration:  20h
Period: April – May 2027

Abstract

The course explores how the structure of a group or a quantum group can be reconstructed from the sole knowledge of the category of its representations, extending Tannaka–Krein duality. Applications in Mathematical Physics and Quantum Field Theory are analyzed: from the Doplicher–Roberts theorem (where charge and the particle–antiparticle duality emerge from the local categorical structure) to models of two-dimensional conformal theory (Wess–Zumino–Witten) and weak Hopf algebras.

Program

• Lectures 1–4: Representations of compact groups, Tannaka–Krein duality, and examples (SU(2), finite groups). The language of tensor C*-categories, conjugate objects, intrinsic dimension, and fiber functors.

• Lectures 5–7: Superselection sectors and the Doplicher–Roberts Theorem for the gauge group. Compact quantum groups, explicit construction of SUq(2) , quantum dimension, and the Tannaka–Krein–Woronowicz Theorem.

• Lectures 8–10: Braided categories, braids, and fractional statistics in 2D. The conformal theory side: the Wess–Zumino–Witten model, affine algebras, fusion categories, and quantum gauge groups (weak Hopf algebras).

Prerequisites

Hilbert spaces and bounded operators, basic functional analysis, linear algebra, elementary group theory. (No prior knowledge of category theory or C*-algebras is assumed.)

Bibliography

• S. Neshveyev, L. Tuset, Compact Quantum Groups and Their Representation Categories, SMF, 2013.

• S. Doplicher, J. E. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989).

• C. Pinzari, The braided Doplicher-Roberts program and the Finkelberg-Kazhdan-Lusztig equivalence, arXiv:2602.08348.

Schedule

TBD

Exam

Oral exam (or seminar on agreed topics).

Course offered by: Piermarco Cannarsa (Tor Vergata)
Duration:  20h
Period: January – February 2027

Program

• Climatology and meteorology. The complexity of the climate system (1 hour). 

• Introduction to energy balance models for climate studies (2 hours). 

• The Ghil–Sellers model in one spatial dimension: well-posedness and asymptotic behavior (4 hours). 

• Spatially homogeneous energy balance models with vertical resolution: well-posedness, asymptotic behavior, equilibria, and sensitivity to physical parameters (6 hours). 

• Energy balance models with vertical resolution and horizontal transport: well-posedness, asymptotic behavior, existence of the warmest equilibrium (7 hours).

Prerequisites

Evolution equations in Hilbert spaces.

Bibliography Course

• H. Kaper, H. Engler, Mathematics & Climate, SIAM, Philadelphia, PA, 2013.

• M. Ghil, Climate stability for a Sellers-type model, J. Atmos. Sci., 33(1):3–20, 1976.

• J. Tort, J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 29(5):683–713, 2012.

• P. Cannarsa, V. Lucarini, P. Martinez, C. Urbani, J. Vancostenoble, Analysis of a two-layer energy balance model: long time behavior and greenhouse effect, Chaos, 33(11):113111, 2023; Comparison principles and long time behavior for a diffusive energy balance model with vertical resolution, Math. Models Methods Appl. Sci. (in press).

Bibliography Prerequisites

• L. C. Evans, Partial differential equations, 2nd ed., AMS, Providence, 2010.

• A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, New York, 1983.

Schedule

TBD

Exam

Oral exam.

Course offered by: Roberto Feola (Roma Tre)
Duration: 20h
Period: March 2027

Abstract

In this course we shall study the water waves system, which describes the evolution of an inviscid and irrotational fluid under the action of gravity and capillary forces. In particular, we will focus on the mathematical formulation of the problem following the Zakharov/Craig-Sulem approach. We shall first discuss some general properties of the Dirichlet-to-Neumann (DN) operator, including tame estimates in Sobolev spaces and analyticity under periodic boundary conditions in any spatial dimension. We will then introduce the basic tools of paradifferential calculus and use them to derive a paralinearization formula for the DN operator. These results will be applied to establish the well-posedness of the associated Cauchy problem in Sobolev spaces. Time permitting, we shall also discuss the long-time behavior of solutions by means of normal form techniques.

Program

• Introduction to the water waves model and its Hamiltonian structure.

• Elliptic estimates on the Dirichlet-Neumann operator.

•  Introduction to paradifferential calculus.

•  Paralinearization formula for the Dirichlet-Neumann operator and a priori energy estimates for the water waves equations.

Prerequisites

Conoscenze di base di analisi funzionale, Spazi di Sobolev, teoria classica e debole per PDE ellittiche e di evoluzione. Distribuzioni, analisi di Fourier. Conoscenze di base di sistemi dinamici e sistemi Hamiltoniani.

Bibliography

•  L. C. Evans, Partial Differential Equations (2nd ed.), AMS (2010).

•   J.-M. Bony, Cours d'analyse: Théorie des distributions et analyse de Fourier, Éditions de l'École Polytechnique (2001).

•   S. Alinhac, P. Gérard, Pseudo-differential Operators and the Nash-Moser Theorem, AMS (2007).

•   D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics, AMS (2013).

•   R. Feola, F. Iandoli, Local well-posedness for the quasi-linear Hamiltonian Schrödinger equation on tori, J. Math. Pures Appl. (2022).

Course offered by: Hirokazu Ninomiya (Meiji University) @ Sapienza
Duration:  6h
Period: November 2026

Abstract

This mini-course provides an introduction to pattern formation and propagation phenomena in reaction-diffusion systems. We begin with a brief review of reaction-diffusion equations and discuss basic mechanisms of pattern formation through the Chafee-Infante problem and Turing instability. We then turn to propagation phenomena, focusing on traveling wave solutions and entire solutions in one-dimensional problems. The method of sub- and supersolutions, which provides a useful tool for constructing and analyzing such solutions, will also be introduced. We next introduce singular limit problems arising in reaction-diffusion systems and explain how reduced equations and geometric evolution laws can emerge from asymptotic analysis. Finally, we discuss various types of traveling waves and entire solutions, as well as their connections with singularly perturbed problems and interface dynamics.

The course is part of the activities of the Thematic Period "PDE Models". Download the poster for the course here.

Program

• Introduction to reaction-diffusion systems

• Pattern formation: the Chafee-Infante problem and Turing instability

• Traveling waves and entire solutions in one dimension

• The method of sub- and supersolutions

• Singular limit analysis and interface dynamics

• Various traveling waves and entire solutions

Schedule

November 11, 12 and 19, 2026, from 14:00 to 16:00 in Room B of the Department of Mathematics Guido Castelnuovo, Sapienza Università di Roma.

The course will be followed by a seminar taking place on November 16, 2026, at 14:30 in Sala di Consiglio of the Department of Mathematics.

Course offered by: Vittoria Silvestri (Sapienza)
Duration: h
Period: March – April 2027

Abstract

This course offers a rigorous introduction to Brownian motion for PhD students, focusing on its construction, sample-path properties, Markov property, hitting times and local time. It explores connections with PDEs, potential theory, harmonic functions, and fractal geometry. The course aims at being self-contained, but students are expected to be familiar with continuous probability, conditional expectation and measure theory.

Program

• Definition and construction of Brownian Motion.

• Sample-Path Properties: Continuity, Scaling, and Variation.

• The Markov and Strong Markov Properties.

• Hitting Times, Reflection Principles, and First-Passage Problems.

• Brownian Motion as a Martingale: Stopping Times and Optional Stopping.

• Brownian Motion in Several Dimensions and Its Geometric Properties.

• Local Time and the Occupation Density Formula.

• Brownian Motion and the Heat Equation.

• Potential Theory, Harmonic Functions, and the Dirichlet Problem.

• Fractal Properties, Recurrence, Transience, and Further Topics.

Prerequisites

Advanced Probability, Measure Theory. Students who wish to attend the course but are not familiar with these topics will be given references to catch up (via email).

Bibliography

• P. Mörters, Y. Peres, Brownian Motion, Cambridge University Press, 2010.

• J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, GTM 274, Springer, 2016.

• I. Karatzas, S. E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., GTM 113, Springer, 1991.

• D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Grundlehren der mathematischen Wissenschaften 293, Springer, 1999.

Schedule

TBD

Exam

Oral exam.

Course offered by: Domenico Marinucci (Tor Vergata)
Duration: 20h
Period: January – February 2027

Abstract

The course will cover the spectral analysis of steady-state processes and isotropic random fields, before moving on to current research topics in excursion set geometry. Applications to cosmology, climate data, and neural networks will also be covered.

Program

• Lectures 1–2: Stationary processes, autocovariance, Herglotz–Bochner theorem, spectral density. 

• Lectures 3–4: Stochastic integral and spectral representation theorem. 

• Lectures 5–6: Isotropic random fields on the sphere, spherical Laplacian, Helmholtz equation, spherical harmonics and SO(3). 

• Lectures 7–10: Properties of spherical harmonics, Legendre polynomials, Schur's lemma and orthogonality. 

• Lectures 11–14: Geometric functionals, Kac–Rice formulas and Gaussian Kinematic Formula. Excursion volume and nodal length (connections with Yau's conjecture). 

• Lectures 15–16: Current developments: geometry of sphere-time fields and neural networks.

Prerequisites

At least one probability course at the master's level (laurea magistrale).

Bibliography

• R. J. Adler, J. E. T. Taylor, Random Fields and Geometry, Springer, 2009.

• D. Marinucci, G. Peccati, Random Fields on the Sphere, Cambridge University Press, 2011.

• I. Nourdin, G. Peccati, Normal Approximations with Malliavin Calculus, Cambridge University Press, 2012.

Schedule

TBD

Exam

Small thesis on a topic of choice (further study or completion of a proof).

Course offered by: Hubert Lacoin (IMPA & Roma Tre)
Duration: 20h
Period: April – May 2027

Abstract

Statistical mechanics is a probabilistic representation of a physical system at a microscopic scale. Amongst other things, it aims at providing qualitative explanations for phase transitions which are observable at a macroscopic level.

Statistical mechanics models which are better understood are so-called homogeneous models. They are defined on a regular lattice, starting with an energy functional that gives to each vertex roughly the same importance. On the other hand, real world materials always present inhomogeneities, due for instance to the presence of microscopic impurities. An important question is thus to know whether the introduction of a certain amount of disorder in a statistical model modifies its properties.

Directed polymer in a random environment is one of the simplest disordered models that one can define: it is an inhomogeneous version of the simple random walk on Z^d.

Yet it displays a very rich phenomenology, and in particular when d ≥ 3, a disorder induced phase transition which is now well understood. In this series of lectures we will explore this phase transition, from a localized low temperature phase to a diffusive high-temperature phase. While most of the course is dedicated to the study of a single model, the techniques introduced along the way (correlation inequalities, concentration inequalities, convex comparisons) are ubiquitous in mathematical physics and beyond.

Program

• Introduction of model and of the phase transition between weak strong disorder.

• Existence of weak and strong disorder under minimal conditions.

• Exploration of the weak disorder phase: proof of Invariance Principles and Local Limit Theorems for directed polymers.

• Proof of localization in the strong disorder phase.

• Proof that strong disorder holds at every temperature in dimension 1 and 2.

Prerequisites

Advanced probability, measure theory

Bibliography

For prerequisits:

• J.F. Le Gall, Measure Theory, Probability, and Stochastic Processes, 2022, Graduate texts in Mathematics (first twelve chapters)

For the course:

• F. Comets, Directed Polymers in Random Environments, École d'Été de Probabilités de Saint-Flour XLVI – 2016

• N. Zygouras Directed polymers in a random environment: a review of the phase transitions, Stochastic Processes and their Applications, Volume 177, 2024.

• Random polymers in random environment: an introduction to disordered systems, Quentin Berger, 2026 (Personal webpage, Lecture notes under construction)

Schedule

TBD

Exam

Oral exam on a topic agreed with the lecturer.

Course offered by: Sergio Simonella e Lingbing He (Sapienza)
Duration: 10+10h
Period: December 2026 – January 2027

Abstract

The course provides an introduction to kinetic theory from two complementary viewpoints. The first part addresses the emergence of the Boltzmann equation from microscopic particle dynamics. Starting from large-scale limits, we introduce the Boltzmann equation, molecular chaos, correlation functions, and the statistical mechanics framework underlying kinetic limits, discussing the BBGKY hierarchy, Lanford's theorem, long-time behaviour, and fluctuations.

The second part focuses on spatially homogeneous Boltzmann and Landau equations, reviewing basic properties, collision kernels, propagation and creation of regularity, the weak-coupling/grazing limit from Boltzmann to Landau, and spectral numerical methods.

Program

Part I: Microscopic dynamics and kinetic limits (S. Simonella, 10 hours)

• Large-scale limits; kinetic and hydrodynamic descriptions; introduction to the Boltzmann equation.

• Molecular chaos, correlation functions and statistical description of many-particle systems.

• Liouville equation and BBGKY hierarchy; the Boltzmann-Grad limit.

• Lanford's theorem and the rigorous derivation of the Boltzmann equation.

• Beyond the classical short-time theory: fluctuation theory and the Deng-Hani-Ma argument.Cycles, rational equivalence, Chow groups. Pushforward, Gysin pullback, projection formula. Relationship with cohomology.

Part II: Spatially homogeneous Boltzmann and Landau equations (L. He, 10 hours)

• Introduction to the Boltzmann and Landau equations: basic properties and derivation of collision kernels.

• Propagation of regularity for the Boltzmann equation I: Lp-moment approach.

• Propagation of regularity for the Boltzmann equation II: smoothing estimates for the Boltzmann collision operator.

• Weak-coupling/grazing-collision limit from Boltzmann to Landau equation.

• Spectral methods for Boltzmann and Landau equations: error estimates and numerical simulations.

Prerequisites

Basic graduate-level knowledge of mechanics, analysis, probability and ordinary differential equations. No previous course in kinetic theory and large scale limits is assumed.

Bibliography

Background

• P. Buttà, P. Negrini, Note del corso di Meccanica Razionale, 2012.

• G. Gallavotti, Statistical Mechanics: A Short Treatise, Springer, 1999.

Course References

• H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, 1991; 

• O. E. Lanford III, Time evolution of large classical systems, LNP 38, Springer, 1975; 

• T. Bodineau et al., Dynamics of dilute gases: a statistical approach, Proc. ICM 2022; 

• T. Bodineau et al., Derivation of the Boltzmann equation from hard-sphere dynamics, Séminaire Bourbaki, Exp. 1247 (2025–2026); 

• C. Cercignani et al., The Mathematical Theory of Dilute Gases, Springer, 1994; 

• C. Villani, A Review of Mathematical Topics in Collisional Kinetic Theory, 2002.

Schedule

Part I (Simonella): December 7-23, 2026; Part II (He): January 12-19, 2027

Exam

Short individual presentation on a topic or paper related to the course (agreed in advance), followed by a brief discussion.

Course offered by: Giorgio Cipolloni (Tor Vergata)
Duration: 20h
Period: March 2027

Abstract

We begin with classical random matrix ensembles and the study of their eigenvalues and eigenvectors, introducing fundamental questions about spectral distributions, eigenvalue spacings, and eigenvector (de)localization.

The central theme is the Wigner–Dyson–Mehta universality: for broad classes of random matrices, local eigenvalue statistics depend only on the symmetry class, i.e., they do not depend on the distributions of the matrix elements. We will present a proof of this phenomenon for Wigner matrices through a dynamical approach based on Dyson Brownian motion, explaining how relaxation to local equilibrium produces universal spectral statistics.

The course will conclude with selected connections to disordered and interacting quantum systems, including random Schrödinger operators and examples drawn from the Sachdev–Ye–Kitaev (SYK) model. Time permitting, we will also discuss connections between random matrix statistics and the zeros of the Riemann zeta function as well as machine learning.

Schedule

TBD

Exam

TBD

Course offered by: Alessandro Giuliani (Roma Tre)
Duration: 20h
Period: 2 or 3 lectures per week in January-February 2027

Abstract

The goal of the course is to provide an introduction to perturbative Renormalization Group for the infrared problem for ɸ^4 scalar field models. After an introduction to the notion of universality for second order phase transition in Ising and other related models in statistical mechanics, I will define the lattice ɸ^4 model in dimension d and focus on this prototypical example. I will describe naive and multiscale perturbation theory for the computation of its correlation functions at the critical point, and introduce a mathematically controlled version of the Wilsonian Renormalization Group equations. The nature of the solutions for dimensions larger or equal to four will be discussed.

Program

• Second order phase transitions in the Ising model and in the lattice ɸ^4 model in dimension d. The universality phenomenon.

• ɸ^4_d: naive perturbation theory (PT) for the model at the phase transition (critical theory). Divergences and their physical significance: how to cure them via formal resummations.

• Multiscale expansion and Wilsonian Renormalization Group (RG). Effective potentials and the RG map. Relevant, marginal and irrelevant operators.

• The Gallavotti-Nicolò tree expansion for the effective potential.

• Running coupling constants and their flow equation (the beta function).

• Solution of the beta function equation in dimensions larger and equal to four. Convergence to an infrared trivial RG fixed point [Time permitting: outline of the solution in dimension smaller than four: the Wilson-Fisher non-trivial fixed point.].

Prerequisites

Some basics of statistical mechanics, such as the definition of Gibbs measure, of the Ising model and of phase transition, are welcome. The course will provide a short introduction to these topics and concepts. No advanced method in statistical mechanics will be assumed and the mathematical part will be self-consistent. However, some prior background on the notion of phase transition in statistical mechanics is useful for a deeper understanding of the model and the techniques discussed. In the absence of such a background, a preliminary reading of the material suggested in the bibliography is recommended.

Bibliography

• S. Friedli, Y. Veleni: Statistical Mechanics of Lattice Systems, Chapters 2 (the Curie-Weiss Model), and 3 (the Ising Model)

• K. Huang, Statistical Mechanics, 2nd edition, Chapters 16 (Critical Phenomena), 17 (the Landau Approach), and 18 (Renormalization Group)

Course offered by: Davide Torlo (Sapienza)
Duration: 6-8h
Period: February 2027

Abstract

Ordinary differential equations (ODEs) describe many physical, biological, and chemical phenomena. To obtain accurate numerical approximations within reasonable computational times, high-order accurate time integration methods are frequently required. This course provides an introduction to cutting-edge algorithms used to perform numerical time integration and explores their core mathematical properties. Theoretical sessions are complemented by hands-on Python/Jupyter notebook practical exercises.

Material for the course is available at the dedicated webpage.

Program

• Environment Setup (Python, Jupyter, Colab).

• Foundations & Theory of ODEs.

• Low-Order Classical Solvers (Euler methods, stability).

• High-Order Classical Solvers (Runge-Kutta, IMEX, Multistep).

• Energy & Entropy Preserving Schemes.

• Iterative Explicit High-Order Solvers (DeC and ADER).

• Structure-Preserving & Positivity Schemes.

Prerequisites

Basics of numerical analysis (linear algebra, ODEs, polynomial interpolation, quadrature). Basic Python.

Bibliography

• E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer (1993).

• J. C. Butcher, Numerical Methods for Ordinary Differential Equations, Wiley (2008).

• P. Öffner, D. Torlo, Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes, Appl. Numer. Math. (2020).

Schedule

1–19 February 2027

Exam

Project.

Course offered by: Francesco Patrizi (Tor Vergata)
Duration: 20h
Period: January 2027

Abstract

B-splines and Non-Uniform Rational B-Splines (NURBS) are fundamental tools in geometric modeling and Isogeometric Analysis. However, their tensor-product structure makes refinement inherently global. Locally Refined (LR) B-splines provide a flexible framework for local refinement while retaining key properties. This course presents the theoretical foundations of LR B-splines, spline spaces over box partitions, dimension formulas, knot insertion, and automatic refinement strategies.

Program

• Spline spaces (univariate, bivariate and multivariate cases).

• Locally Refined B-splines over box partitions.

• Spanning properties and (local) linear independence.

• Automatic refinement strategies.

Bibliography

• T. Dokken, T. Lyche, K. F. Pettersen, Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Design (2013).

• F. Patrizi, T. Dokken, Linear dependence of bivariate Minimal Support and Locally Refined B-splines over LR-meshes, Comput. Aided Geom. Design (2020).

• T. Lyche, C. Manni, H. Speleers, Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement, In: Splines and PDEs: From Approximation Theory to Numerical Linear Algebra, Springer (2018).

Schedule

11–29 January 2027

Exam

TBD