Advanced topics in Algebra and Geometry: "Lie Theory and Representation Theory (Algebre di Hecke)" (joint course with Sapienza and Tor Vergata)

Advanced topics in Algebra and Geometry:
"Lie Theory and Representation Theory (Algebre di Hecke)"

Docenti:
Martina Lanini (Roma Tor Vergata)      
Guido Pezzini (Roma Sapienza)

Periodo: Marzo-Aprile-Maggio 2024
Schedule: 40 hours

Abstract:
Hecke algebras are all over the place: they appear, for example, in algebraic combinatorics, representation theory, knot theory, harmonic analysis, equivariant K-theory, integrable models in statistical physics.
In this lecture series, articulated into two parts, each of them lasting about 20 hours (10 lectures), we will mainly focus on the algebraic (and, possibly, geometric) side. In the first part, we will discuss classical theory of Coxeter groups and define Hecke algebras via generators and relations, as well and their celebrated Kazhdan-Lusztig basis. In the second half of the course we will focus on a categorical approach to the study of Hecke algebras and deal with Soergel bimodules. Introduced by Soergel a couple of decades ago, these bimodules have been a central object of interest in geometric representation theory, but also investigated by a purely combinatorial viewpoint. Depending on the audience interests and background, we might discuss how Hecke algebras relate to representation theory (of Coxeter groups, of complex Lie/Kac-Moody algebras, of algebraic groups in positive characteristic, of quantum groups) and geometry (e.g. via intersection cohomology complexes on the flag varieties, or via homology of the Steinberg variety).

Advanced topics in Logic, History and Pedagogy( joint course with Sapienza and Tor Vergata)

Modulo 1. "Didattica"

Docente: Benedetto Scoppola (Roma Tor Vergata)

Period: 12h in april-may 2024, monday 14.00-17.00

Abstract:
The aim of the course is to present mathematical arguments using a laboratorial approach. The arguments will be:
Penrose tiling and quasicrystal. The history of quasicrystal is a very interesting example of the interaction between mathematics and natural sciences: in the ’70 Roger Penrose introduced the concept of quasicrystal, a non periodic structure with quinary symmetry. Right after this mathematical result, many groups of physicists found, with X-rays diffraction patterns, materials exhibiting quinary symmetry. Such patterns were prohibited by the fundamental theorem of crystallography, and represented a great puzzle for the theory of the structure of matter. The natural solution has been to identify these materials as quasicrystals. Some examples of quasiperiodic tiling of the plane will be provided with a laboratorial approach, and also some experiences of waves diffraction will be realized. Logic circuits and boolean mathematics. From an historical point of view this theme is extremely rich. One can describe to the students the circumstance in which in XX century the first computers, based on logic circuits, have been designed. One can also propose, in classical high school (liceo classico) the reading of Cicerone, who is one of the few witness of the dramatic development, in the context of the stoic philosophy of the III century b.C., of the propositional logic. An handcrafted material representing in an electromechanical way the logic circuits will be presented. We will construct a machine that is able to sum addend with an arbitrary number of bits.
Tides. This is one of the more striking example on how contradictory is the scientific progress: a chaotic mix of ancient (and nowadays lost) knowledge and superstition gave to the modern scientists the right ideas in order to understand this phenomenon. Some materials to help the students to understand the dynamical nature of the modern tidal theory will be presented, together with a mathematical quantitative description of the so called static tidal model, based on very simple mathematical instruments. Dynamical (dissipative) phenomena will be described in a more qualitative way.
Probabilistic models of vehicular traffic. It is possible to represent in a concrete way simple probabilistic models that are able to catch many interesting and unintuitive features of the vehicular traffic. Among them, the appearance of traffic in absence of bottleneck and the long range effect of the short range interaction among vehicles. The theoretical basis of the material that will be presented is one of the simplest among these models. The material gives the possibility to study the random fluctuation of a system in which the motion of each agent (vehicle) is governed by an extremely simple random model (coin tossing). The resulting theory is very simple but deep, and concrete random experiment may be realized in order to test the results obtained by probability theory

Modulo 2. "Logic"

Docente: Lorenzo Tortora de Falco (Roma Tre)

Period: 12h

Abstract:
Lezione 1: Soddisfacibilità e dimostrabilità.
Verrà introdotta la nozione di struttura per un linguaggio del primo ordine e la conseguente nozione di soddisfacibilità di una formula e di una teoria del primo ordine. Verrà introdotta la nozione di derivabilità in alcuni sistemi deduttivi concentrandosi principalmente sul calcolo dei sequenti di Gentzen. Le due nozioni (soddisfacibilità e derivabilità) verranno messe in relazione mediante il teorema fondamentale dell’analisi canonica, dal quale discendono i principali risultati sulla logica del primo ordine (teoremi di completezza, compattezza, eliminabilità del taglio, Löwenheim-Skolem).
Lezione 2: Gentzen e l’eliminazione del taglio.
Verrà presentata e discussa criticamente la tecnica introdotta da Gentzen negli anni ’30 del secolo scorso che permette di trasformare una qualsiasi derivazione logica del calcolo dei sequenti in una derivazione senza tagli. Verranno discusse le motivazioni che hanno portato a questo risultato ed alcune delle conseguenze notevoli che ha avuto in teoria della dimostrazione.
Lezione 3: Dimostrazioni e programmi: la corrispondenza di Curry-Howard.
Negli anni ’60 del secolo scorso fu messa in luce la stretta relazione che intercorre tra le derivazioni della deduzione naturale del frammento minimale della logica ed i termini del lambda-calcolo semplicemente tipato. Questa osservazione rinsaldò il legame già esistente tra Logica ed Informatica: attraverso questa corrispondenza una dimostrazione può vedersi come un programma la cui esecuzione corrisponde all’applicazione della procedura di eliminazione del taglio alla dimostrazione di partenza. Verrà presentata la corrispondenza di Curry-Howard e verrà discusso l’impatto che ha avuto nella teoria della moderna dimostrazione.
Lezione 4: Introduzione alla Logica Lineare.
Nella scia della corrispondenza di Curry-Howard, lo studio mediante strumenti matematici del processo computazionale di trasformazione delle dimostrazioni logiche (o di esecuzione dei programmi) ha portato Jean-Yves Girard ad introdurre, nel 1987, la Logica Lineare, un nuovo approccio alla teoria della dimostrazione in cui i connettivi della logica classica vengono decomposti e le dimostrazioni logiche diventano grafi la cui deformazione corrisponde al processo di eliminazione del taglio introdotto da Gentzen. Verrà presentato un frammento particolarmente semplice della Logica Lineare e verranno discussi alcuni risultati ottenuti grazie alla raffinatezza dell’analisi basata sulla Logica Lineare.

Modulo 3. "Episodi della teoria delle equazioni algebriche"

Docente: Enrico Rogora (Roma Sapienza)

Periodo: 12h (6 lezioni) che si terranno alla Sapienza, Dipartimento di matematica.
Schedule: 28, 29, 30 Novembre e 5, 6, 7 Dicembre 2023, ore 15-17

PROGRAMMA:
1. Lagrange, memorie sur la resolution des equations algebriche.
2. Gauss, disquisitiones arithmeticae.
3. Abel e Ruffini. Sulla non risolubilità per radicali dell'equazioni generale di quinto grado
4. Abel e Jacobi sulle equazioni di divisione e modulari.
5. Galois, sui collegamenti tra la teoria dei campi e la teoria dei gruppi.
6. Betti, Hermite, Brioschi e Kronecker Sulla soluzione dell' equazione di quinto grado con le funzioni ellittiche.

Time series analysis

1. We will recall the basic principles of applied and numerical Fourier analysis: Fourier series and transform, energy and power spectrum, mutual and autocorrelation and their numerical computation.
2. Impulse and harmonic response of a system.
3. Filtering of a time series.
4. Time series as sampling of a continuous signal.

DEFORMATION THEORY (graduate course)

PROGRAM:
Moduli problems and deformation theory. Infinitesimal  deformations. Examples. Formal smoothness. Deformations of nonsingular varieties. The local Hilbert functor. Deformations of locally free sheaves. Functors of Artin rings. Obstruction theory of  local rings and of functors of Artin rings. Smoothness criteria and applications. Versal, semiuniversal and universal deformations.   Schlessinger's conditions. Relation between automorphisms  and existence of (semi)universal deformations. Discussion of examples.

PREREQUISITES:   Familiarity with the main notions of commutative and homological algebra and with basic language of schemes.
USEFUL REFERENCES: R. Hartshorne: Deformation Theory, Springer GTM n. 257.
E. Sernesi: Deformations of Algebraic Schemes, Springer Grundlehren, b. 334.
 

Advanced topics in Analysis (joint course with Sapienza and Tor Vergata)

Istituzioni di analisi

Modulo 1. "Variational Calculus and Applications"

Docente: Emanuele N. Spadaro (Roma Sapienza)

Period_: The course runs bi-weekly (4 hours per week) from 1 April to 31 May 2024 (in case you need an introduction to the course from 28 February to April, please contact the lecturer for details).

Abstract:
The course introduces the variational methods underlying numerous problems in mathematical analysis and applied mathematics.
Beginning with classical methods of Variational Calculus developed since the 18th century, it will lead to a discussion of more recent results, such as the solution of Hilbert's 19th problem, notions of variational convergence and phase separation models in mathematical physics.
- Classical Problems of the Calculus of Variations and Examples of Applications
- Examples of existence and non-existence
- Euler Lagrange equations and differential equations in weak form
- Direct method of the Calculus of Variations
- Necessary and sufficient conditions for the semi-continuity of integral functionals
- Vector problems of the Calculus of Variations Convexity and quasi-convexity
- Regularity of minima and solution of Hilbert's 19th problem
- Variational convergence. Gamma convergence
- Application to asymptotic problems of the calculus of variations:
   -- Caccioppoli sets
   -- Phase transitions and the Modica Mortola functional

Modulo 2. "Introduction to PDE"

Docente: Daniele Bartolucci (Roma Tor Vergata)

Period: 20h (10 lessons), starting March 2024.

Abstract:
The aim of the course is to provide an introduction to the basic notions about Laplace-Poisson, Heat and Wave equations. There will be three lessons of two hours each a week. Lecture notes of the course will be available. The Lectures will be delivered in presence, possibly in mixed (online) form if needed.
Topics covered
- Laplace and Poisson equations. Harmonic functions. Fundamental solutions.
- Mean value formulas. Maximum principles, uniqueness. Mollifiers, convolutions and smoothing.
- Regularity and local estimates for harmonic functions. The Liouville Theorem, classification of solutions of the Poisson equation in RN" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">RN, Nge2" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">Nge2
- The Harnack inequality for harmonic functions. The Green function. The Green function on a ball. The Poisson Kernel.
- Variational (Energy) methods. The Dirichlet principle.
- The Heat equation. The fundamental solution. The Cauchy problem for the homogeneous and
non homogeneous equation. Mean value formula and the heat ball.
- Maximum principle for the heat equation. Uniqueness. Regularity of solutions of the heat equation.
- Transport equations. The Wave equation. D’Alambert formula (N=1), Euler-Poisson-Darboux equation, Kirchoff’s formula (N=3). Descent method, Poisson’s formula (N=2).
- Nonhomogeneous wave equations, retarded potentials. Energy methods, finite speed propagation.

Bibliography:
D. Bartolucci, Lecture notes of the course.
L.C. Evans, Partial Differential Equations. Second Edition. American Mathematical Society 2010.

Modulo 3. "Paradifferential operators and dynamics of non linear PDEs"

Docente: Roberto Feola (Roma Tre)

Period: 20h : April 3,5,8,10,12,15,17,19,22,24. Monday h 11:00-13:00; Wed h 14:00-16:00; Fri h 11:00-13:00.

Abstract: We shall discuss several modern tools of micro-local analysis with application to the study of nonlinear partial differential equations. The aim of the course is to provide a self-contained introduction to para-differential operators and show how they can be used to prove a priori energy estimates and build up local existence theory for some type of quasi-linear partial differential equations. Time permitting, we shall discuss some applications to normal form theory for PDEs on compact manifolds.
[Draft program: The course will be essentially divided into three parts. At first instance we will present some basic tools in harmonic analysis and we will provide an introduction to pseudo- differential symbols to discuss symbolic calculus: compositions, adjoints, quantizations. Then we will study the action of pseudo-differential operators in Sobolev spaces and generated flows.
In the second part of the course we shall introduce para-differential operators via quantizations of symbols with limited regularity. We will then prove the paralinearizations theorems to rewrite nonlinear expressions by para-differential expressions.
We will conclude with some applications to the Cauchy theory for some type of quasi-linear equations.]

Advanced Topics in Mathematical Physics and Probability (joint course with Sapienza and Tor Vergata)

Istituzioni di Fisica Matematica a Probabilita'

Modulo 1. "Renormalization Group and Critical Phenomena in Statistical Mechanics"

Docente: Alessandro Giuliani (Roma Tre)
Schedule: 20h in Jan-Feb 2024. Dates: Jan 09, 12, 19, 23, 26 and Feb 2, 6, 9, 13, 16, from 9:00 to 11:00

Abstract: The goal of the course is to provide an introduction to perturbative Renormalization Group in the context of phid4" id="MathJax-Element-1-Frame" role="presentation" style="position: relative;" tabindex="0">phi4d scalar field models in dimensions dge4" id="MathJax-Element-2-Frame" role="presentation" style="position: relative;" tabindex="0">dge4 . We will discuss the notions of: scaling dimensions; relevant, marginal and irrelevant operators; effective potentials; flow equation and beta function. We will show that the quartic interaction is irrelevant in dimensions dge5" id="MathJax-Element-3-Frame" role="presentation" style="position: relative;" tabindex="0">dge5 and marginal with negative beta function (sometimes called `marginally irrelevant'), in dimension d=4" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">d=4 . Time permitting, we will discuss the definition and construction of the non-trivial fixed point for the phid4" id="MathJax-Element-5-Frame" role="presentation" style="position: relative;" tabindex="0">phi4d model in dimension d=4−epsilon" id="MathJax-Element-6-Frame" role="presentation" style="position: relative;" tabindex="0"d=4−epsilon (Wilson-Fisher fixed point).

Modulo 2. "Introduction to Random Geometry"

Docente: Vittoria Silvestri (Roma Sapienza)

Schedule: 20 hours in March- April 2024. Dates: March 18, 20, 22, 25, 27 and April 2, 3, 8, 10, 12 from 4 pm to 6 pm. Room G-  Dip. Matematica, Univ. La Sapienza.
Abstract:
This course wants to give an overview of active research topics in the field of Random Geometry, with a focus on growth models. We will start by discussing discrete growth models such as the Eden model, Diffusion Limited Aggregation and Internal DLA. We will then move to the continuum for the remaining part of the course. After discussing conformal invariance of Brownian motion, we will focus on the class of randomly growing domains on the complex plane which can be described via Loewner dynamics. We will introduce several random aggregation models on the complex plane, which go under the name of Hastings-Levitov models and Aggregate Loewner Evolutions, of which we will study the large-scale features, presenting existing results and several open questions.
Keywords: Random aggregation, Diffusion Limited Aggregation, Schramm-Loewner Evolutions, Hastings-Levitov, Aggregate Loewner Evolutions.
Prerequisites: The course will be as self-contained as possible. However, basic notions of Probability and Analysis are necessary. Some knowledge of complex analysis, martingale theory and stochastic calculus is desirable but not required.

Modulo 3. "Smooth Ergodic Theory"

Docente: Oliver J. Butterley (Roma Tor Vergata)

Periodo: 20h (4 a settimana) nel periodo 29 April - 31 May 2024.
Abstract:
Smooth ergodic theory is the study of the statistical and geometric properties of measures invariant under a smooth transformation or flow. Some highlights of the history of the subject include: the work of Birkhoff and von Neumann on ergodicity; Hadamard and E. Hopf on geodesic flows for negatively curved surfaces; Kolmogorov, Arnold and Moser with a perturbative theory to construct obstructions to ergodicity in Hamiltonian systems; Anosov and Sinai on hyperbolic systems. The subject is broad and so in this short course we will focus on specific areas even though other areas are of equal relevance. Namely we will focus on smooth hyperbolic systems, identifying and studying invariant measures and obtaining statistical properties. We will also start along the road of using functional analytic techniques in order to work with these themes. This course is aimed at providing participants with a solid working knowledge in the basic concepts, important techniques and examples in smooth ergodic theory, particularly in the direction of hyperbolic systems. The course aims to be of interest to those with research interests in various flavours of ergodic theory and dynamical systems, and its applications to study problems in combinatorics, number theory, homegeneous dynamics, differential equations, probability theory.

 

Advanced topics in Numerical Analysis (joint course with Sapienza and Tor Vergata)

Module 1

Convex Optimization

Docente: Vincenzo Bonifaci (Roma Tre)

Period: The lectures will be in the period November 15 — December 20, 2023
Every Wednesday 14.00-16.00 and Friday 14.00-16.00 (except December 8)
Room TBA, Dipartimento di Matematica e Fisica, Università Roma Tre
via Lungotevere Dante, 376 — also accessible by walking from Largo San Leonardo Murialdo, 1
Lectures will also be streamed on the Microsoft Teams platform.

Abstract:
The aim of the course is to provide students with fundamental concepts in convexity and convex optimization, as well as their application to nonlinear optimization problems. The course will focus on how to recognize convexity, how to formulate convex relaxations of nonlinear optimization problems, and how to solve convex optimization problems. The course is addressed at an audience from all areas of mathematics. Planned topics include:

- Convex sets, convex hulls, polyhedra and polytopes, extreme points, Minkowski’s  theorem
- Convexity of functions, inequalities related to convexity, subgradients, conjugate functions
- Bregman divergence, generalized Pythagorean inequality, projections onto convex sets
- Convex optimization problems, Lagrange duality, Karush-Kuhn-Tucker optimality      conditions
- Convex optimization algorithms, gradient and subgradient methods, iteration complexity

Module 2.
Asymptotic Spectral Properties of Matrices arising from Differential Equations

Docenti:
Carlo Garoni (Roma Tor Vergata)
Mariarosa Mazza (Roma Tor Vergata)

Period: 27/11/2023 - 19/1/2024
Schedule: 27/11/2023, 13.30—17.00
04/12/2023, 13.30—17.00
11/12/2023, 13.30—17.00
18/12/2023, 13.30—17.00
16/01/2024, 10.30—13.30
19/01/2024, 10.30—13.30
all lessons in 1201 Aula "R. Dal Passo", Mathematics Department, Tor Vergata.

Abstract:
The theory of Generalized Locally Toeplitz (GLT) sequences was developed in order to solve a specific application problem, namely the problem of computing/analyzing the spectral distribution of matrices arising from the numerical discretization of differential equations (DEs). A final goal of this spectral analysis is the design of efficient numerical methods for computing the related numerical solutions. The purpose of this course is to introduce the reader to the theory of GLT sequences and to present some of its applications to the computation of the spectral distribution of DE discretization matrices. Particular attention will be paid on fractional DEs. The course will mainly focus on the applications, whereas the theory will be presented in a self-contained tool-kit fashion, without entering into technical details.
References:
[1] Garoni C., Serra-Capizzano S., Generalized Locally Toeplitz Sequences: Theory and Applications (Volume I), Springer, Cham, 2017.
[2] Garoni C., Serra-Capizzano S., Generalized Locally Toeplitz Sequences: Theory and Applications (Volume II), Springer, Cham, 2018.
[3] Garoni C., Serra-Capizzano S., Generalized locally Toeplitz sequences: a spectral analysis tool for discretized differential equations, Lecture Notes in Mathematics 2219 (2018) 161—236.
[4] Donatelli M., Mazza M., Serra-Capizzano S., Spectral analysis and structure preserving preconditioners for fractional diffusion equations, Journal of Computational Physics 307 (2016) 262—279.

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.