ADVANCED TOPICS IN ANALYSIS (JOINT COURSE WITH SAPIENZA AND TOR VERGATA)

Module n. 1

title: "Stability and instability for nonlinear Schrödinger equations on tori"

Teachers:  Michela Procesi - Emanuele Haus

Period: Semester II

Nonlinear partial differential equations are effectively used to model waves since they manage
to describe their complexity and attempt to give a mathematically rigorous justification of
phenomena such as turbulence or the formation of solitary or recursive waves.
We will focus on the nonlinear Schrödinger equation (NLS) on the torus, which is a
paradigmatic example.
In the first part of the course, we will discuss the basic tools to study nonlinear Hamiltonian
systems in infinite dimension (with special attention to PDEs on compact manifolds).
Then, we will study the Cauchy problem for the NLS on the torus, and discuss almost global
existence, recursive solutions and weakly turbulent solutions (namely, solutions exhibiting a
large growth of Sobolev norms).

Module n. 2

Title: Introduction to Pde

Teacher: D. Bartolucci (Univ. Tor Vergata)

Schedule: 20h in March  2025. Dates: 3, 4, 6, 10, 11, 13, 17 ,18, 20, 24, 25, 27.

Program

ADVANCED TOPICS IN LOGIC, HISTORY AND PEDAGOGY( JOINT COURSE WITH SAPIENZA AND TOR VERGATA)

Module  1:  Laboratorio di Didattica:
Teacher: Prof. Benedetto Scoppola  (Univ. Tor Vergata)
Period: TBA
Program:
- algoritmo di euclide in aritmetica (massimo comun divisore) e nella teoria dei numeri
(frazioni continue)
- proporzioni ad elementi interi: i risultati del libro VII
- potenze e proporzioni continue
- incommensurabili.

Modulo 2. Logica:

Teacher: Prof. Lorenzo Tortora De Falco  (Univ. Roma Tre)
Period: TBA
Program:
- Soddisfacibilità e dimostrabilità. Il teorema fondamentale dell'analisi canonica.
- Gentzen e l'eliminazione del taglio
- Dimostrazioni e programmi: la corrispondenza di Curry-Howard
- Introduzione alla Logica Lineare.

Modulo 3. Storia:

Teacher: Prof. Enrico Rogora  (Univ. la Sapienza)
Period: TBA
Program:
- Lagrange
- Il teorema di Ruffini Abel
- Il lavoro di Abel sulle equazioni e sulle funzioni ellittiche
- Il contributo di Galois
- Le soluzioni analitiche delle equazioni di quinto grado
- La teoria di Galois-Klein

ADVANCED TOPICS IN ALGEBRA (JOINT COURSE WITH SAPIENZA AND TOR VERGATA)

Module n. 1
Title: Resolution of singularities
Teacher: Michael McQuillan  (Univ. Tor Vergata)
Period: November 2024
Program:
The goal of the course will be resolution of singularities of excellent algebraic spaces in characteristic 0. However, as far as I can, I'll try to keep things characteristic free. As such, the programme will be: 1) Resolution of complete local rings in characteistic zero. 2) Unipotent groups in arbitrary characteristic. 3) Resolution of complete local rings of small dimension in arbitrary charactertic. 4) How, independently of the characteristic, to go from complete local rings to excellent algebraic spaces. Items (1), (2) & (3) are linear algebra, albeit some knowledge of L.algebraic groups and their Lie algebras would be helpfull. I will, however, aim to cover some background abut excellent rings, with a view to explaining why the study of any other kind of rings is an invitation to pathology
 

GEOMETRY OF MODULI OF VECTOR BUNDLES ON CURVES AND SPECIAL VARIETIES

Description of the course

Let C be a smooth and irreducible complex curve of genus g. In the course we will present
some of the more important moduli spaces of vector bundles over C from the point of view of
their projective immersions. This will lead to highlight several varieties with special features,
which turn out to be examples of such moduli spaces or of Brill-Noether loci contained in
them. A remarkable example is the classical quartic hypersurface of Coble in P^6, which is the
projective model of the moduli space SU_C(2,O_C)  of semistable vector bundles of rank 2 and
trivial determinant over a non-hyperelliptic curve of genus 3.
After recalling the main notions concerning line bundles and Picard varieties of curves, in the
course we will focus on vector bundles over C of rank r >= 2.
We will study with particular attention the generalized theta divisor, which generates the
Picard group of SU_C(r,O_C)and governs the projective realizations of this moduli space, along
with the rich projective geometry related to it.
The course is intended to highlight, along with the analysis of the moduli spaces of bundles
themselves, classical and moderns aspects of the geometry which connects the theory of curves
with other remarkable subjects.

ADVANCED TOPICS IN ALGEBRA AND GEOMETRY (JOINT COURSE WITH SAPIENZA AND TOR VERGATA)

Module n. 1

Titlle: Factorization Homology

Title: Geometry of moduli of vector bundles on curves and special varieties

Teacher: Alessandro verra ( Univ. Roma Tre)

Schedule: Thursdays, 11:00 - 13:00, Room M6

First lecture: Thursday November 28

Description of the course

Let C be a smooth and irreducible complex curve of genus g. In the course we will present
some of the more important moduli spaces of vector bundles over C from the point of view of
their projective immersions. This will lead to highlight several varieties with special features,
which turn out to be examples of such moduli spaces or of Brill-Noether loci contained in
them. A remarkable example is the classical quartic hypersurface of Coble in P^6, which is the
projective model of the moduli space SU_C(2,O_C)  of semistable vector bundles of rank 2 and
trivial determinant over a non-hyperelliptic curve of genus 3.
After recalling the main notions concerning line bundles and Picard varieties of curves, in the
course we will focus on vector bundles over C of rank r >= 2.
We will study with particular attention the generalized theta divisor, which generates the
Picard group of SU_C(r,O_C)and governs the projective realizations of this moduli space, along
with the rich projective geometry related to it.
The course is intended to highlight, along with the analysis of the moduli spaces of bundles
themselves, classical and moderns aspects of the geometry which connects the theory of curves
with other remarkable subjects.

 

Abelian varieties: geometry and arithmetic

Program: The main goal of the course is the study of the geometric and arithmetic properties of abelian varieties. More precisely the focus will be on abelian varieties as complex tori, group schemes, as well as some arithmetic aspects such as canonical heights and the Mordell-Weil Theorem.
 

Tropical methods in moduli theory and algebraic geometry

The objective of the course is the application of recent techniques  from tropical geometry to the study of moduli spaces of various nature, and to selected problems in algebraic geometry. The course will also introduce  basic notions of classical moduli theory.

 

Geometry of Enriques Surfaces

General information for the course

First meeting. The first meeting will be in October.

Contact. Email: luca.schaffler at uniroma3 dot it
Level. PhD course.
Credits. 3.

Description. Historically, Enriques surfaces were the first example of non rational algebraic surfaces with geometric genus and irregularity equal to zero. Enriques surfaces form a 10 dimensional family, and they constitute a fundamental slice of the Enriques–Kodaira classification of algebraic surfaces. The objective of this course is to study the geometry of these objects, gradually leading to modern active research. In particular, we will focus on the study of elliptic fibrations and the non-degeneracy invariant, which is related to the realization of Enriques surfaces in projective space. Finally, we will consider Enriques surfaces from the point of view of moduli and compactifications, especially from the point of view of stable pairs and the minimal model program.
We will focus on the fundamental tools of this theory, concrete examples, and computations.
Meetings. One or two meetings per week on Monday and Thursday according to the schedule below. The class time is always from 16:00 to 18:00.
 We will meet in TBD.
Streaming. The course will be streamed. If you are interested in attending online, please email me at the above email address.
Type of examination. Attendance, participation, one written Homework, and a seminar (or a written report) on a related topic.
Prerequisites. An introduction to algebraic geometry, including schemes and sheaves.

Bibliographical references
François Cossec, Igor Dolgachev, Christian Liedtke. Enriques surfaces, I.
Igor Dolgachev, Shigeyuki Kondo. Enriques surfaces, II.
 

ADVANCED TOPICS IN MATHEMATICAL PHYSICS (JOINT COURSE WITH SAPIENZA AND TOR VERGATA)

Module n. 1
Title: The "ETH" approach to Quntum Mechanics
Teacher: Alessandro Pizzo (Univ. Tor Vergata)
Period: TBA
Program: TBA
 

Geometry and Mechanics

The goal of these lessons is to show the relashionships between Continuum Physics and Differential Geometry, starting from the fundamentals of Mechanics. In particular, we show how any typical theory of mathematical-physics is based on two layers: the physical layer – the phenomenon under investigation, and the mathematical model used to represent the physics. We shall discuss in detail some model problems, starting from the theoretical point of view, up to some noteworthy solutions; for each model, we shall show the dual role, mathematical and physical, of the notions that are used. Model problems will be selected together with the students, and chosen among:  
· Active Soft Matter;
· Liquid Crystals;
· Solid - Fluid interactions.
 
Synopsis

- A continuum body as a differentiable manifold.


- Geometric elements; change of densities.-
- Geometric meaning of Divergence and Gradient
- Tell the difference between tensors: strain tensor versus stress tensor.
- Pull back & push forward of scalar, vector and tensor fields.
- Principle of virtual power:  the notion of force as a power gauge.
- Power versus Energy.
- Dissipation principle.
- Frame invariance.
 

ADVANCED TOPICS IN NUMERICAL ANALYSIS AND INFORMATICS (JOINT COURSE WITH SAPIENZA AND TOR VERGATA)

Module n. 1
Title: "Advanced numerical methods for PDES:  the isogeometric paradigm”
Teachers: Carla Manni – Hendrik Spellers
Period: April-May 2025
Schedule: 20 h
Program: TBA