Courses of Study | Didactic Offer Doctorate XXXVII cycle
Courses of Study Didactic Offer Doctorate XXXVII cycle
Didactic Offer
The didactic offer was formulated as follows.
By clicking on the titles you can access the abstracts, which are however available further down.
A: attivo, in corso;
T: terminato.
Teoria dei Giochi
Abstract:
Game theory is the mathematical study of interaction among independent, self-interested agents. The main objective of the course is to understand the basic ideas behind the key concepts of Game Theory, such as equilibrium, rationality and cooperation.
The course introduces the analytical tools needed to understand how game theory is used, and illustrates these tools with some applications and examples (Prisoner's dilemma, Battle of sex, Bargaining, Ultimatum game, Centipede game, Repeated games).
Existence and uniqueness conditions are analysed.
Summary:
- Utility,decisionmakingandrationality
- Games,strategiesandtiming
- Dominance,iterateddominance,rationality
- Extendedformgameswithperfectinformation,backwardinduction 5. Nashequilibrium:pureandmixedstrategies.
- Subgameperfection,forwardinduction
- Repeatedgames,folktheorem
Bibliografia:
- Kokesen, L. and E. Ok. An Introduction to Game Theory. Online lecture notes, 2007.
- Myerson, Roger B. Game theory: analysis of conflict. Harvard university press, 1997.
- Patrone, Fioravante. Decisori (razionali) interagenti. Una introduzione alla teoria dei giochi. Edizioni plus, 2006.
- Tadelis, Steven. Game theory: an introduction. Princeton university press, 2013.
Metodi numerici avanzati per problemi differenziali
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Self-similar fractals: construction and dimensions
Introduction: Motivation. Background: measure theory fundamentals, topology, and metric spaces.
Iterated Function Systems: Contraction mapping theorem, Hausdorff metric, Iterated Function Systems and their attractors. Collage Theorem.
Dimensions of a self-similar fractal: Topological Dimension, Similarity Dimension and Hausdorff dimension.
Relevant examples: Cantor set, Sierpinski Gasket, Koch Snowake and Menger Sponge.
References
- Michael F. Barnsley, Fractals Everywhere, 2nd edition, Morgan Kaufmann, 2000.
- Gerald Edgar, Measure, Topology, and Fractal Geometry, 2nd edition, Springer, 2007.
- Kenneth Falconer, Fractal Geometry, Mathematical Foundations and Applications, 3nd edition, Wiley, 2014.
- Jonathan M. Fraser, Assouad Dimension and Fractal Geometry, Cambridge University Press, 2021.
Gruppi di permutazioni
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Teoria dei modelli di Sullivan e applicazioni geometriche
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Processi di diffusione e metodi probabilistici
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Elliptic and Parabolic Equations: from Fourier to Morrey
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Teoria della Formalità
Prerequisiti:
Geometria differenziale, algebre di Lie
Breve descrizione:
This course is focused on the theory of formal deformation quantization of Poisson manifolds, in the formalism developed by M. Kontsevich. The main topics covered are the basic theory of Poisson manifolds, star products and their classification, deformations of associative algebras and the formality theorem.
Operators and Semigroups associated to Forms
We consider operators with bounded measurable coefficients on arbitrary domains. The sesquilinear form technique provides the right tool to define such operators, and associates them with analytic semigroups on L2. We are interested in obtaining contractivity properties of these semigroups as well as Gaussian upper bounds on their associated heat kernels.
Introduzione al calcolo stocastico
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Il caos in dinamica topologica
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An introduction to Tensor Product of Groups
Abstract
The tensor product of groups is a powerful construction that arose in the realm of Algebraic Topology, but today extensively studied in Group Theory for its interesting properties. In literature, the tensor product of groups first appears in the context of abelian groups, as a special case of tensor product of modules. Later, a definition for (not necessarily abelian) groups has been introduced by R. Brown and Loday in their Van Kampen paper.
The aim of this course is to present general properties and open problems related to the study of the tensor product of groups.
Prerequisites
The participants should have some familiarity with rings and modules, and should also know some basic group theory. However, some preliminaries about modules and groups will be given.
An introduction to homogenization
CONTENTS:
Some variational elliptic problems.
Modeling of composite materials and structures.
Homogenization of elliptic equations: the convergence result via energy method.
Comparison results among different homogenization techniques.
Gruppi e Simmetria
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Teoria delle Categorie
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Equazioni ellittiche e paraboliche: da Fourier a Morrey
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Equazioni differenziali alle derivate parziali ellittiche del secondo ordine. Teoria L^p(R^N)
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Introduzione alla teoria dell'elasticità
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Complex thinking in mathematics education
The main objective of the intervention is to provide the necessary skills to understand the essentiality that the concept of complexity has in the paths inherent in the teaching of mathematics. Specifically, the focus is on complex thinking (Morin, 1993) and its articulation in the structure of mathematical thought, as the equivalent of the interaction between objects and processes (Sfard, 2009). In this context of reflection, mathematical culture is closely interconnected with humanistic culture when it is analyzed as a specific capacity of the mind to be critical / reflective towards the world. To support this thesis, examples of interdisciplinary teaching units will also be provided.
Homological Methods in Differential Geometry
ABSTRACT:
The course aims at discussing various standard homological techniques in differential geometry, with particular emphasis on fibered and foliated manifolds.
TENTATIVE PROGRAM:
Filtered complexes
Double complexes
Spectral sequences
Fibrations
Leray-Serre spectral sequence
Local coefficients for De Rham cohomology
Gauss-Manin connection
Gysin sequence
Thom isomorphism
Kunneth formula
Foliations
Foliated cohomology
Basic cohomology
Foliated spectral sequence
PREREQUISITES:
The participants have to know the basics of differential geometry (manifolds, submanifolds, vector fields, differential forms, (vector) bundles, etc.) and the basics of homological algebra (cochain complexes, cochain maps, cohomologies, homotopies, etc.).
Multi-objective Optimization
Abstract:
This course provides an introduction to the fundamental concepts of multi-objective optimization. Focusing in particular on the class of linear (integer) programming problems with multiple criteria, the main scalarization techniques for the generation of the optimal Pareto frontier are presented (e.g. weighted sum method, e-constraint method). For the class of bi-objective problems, the two- phase method is also introduced. The course includes examples, exercises and an introduction to the use of two open source solvers for multi-objective linear programming problems, Bensolve and Polyscip.
Table of contents:
- Motivation deriving from real applications - Introduction
- Fundamental concepts and definitions
- Multi-objective Linear Programming
- Multi-objective Combinatorial Optimization - Scalarization Techniques
- Two-phase methods
- Examples
- Software packages: Bensolve and Polyscip
Nonsmooth Optimization with applications in Integer Programming and Machine Learning
The course (five two-hour lectures) will be given on 3-4th May, 2022, at the Università di Salerno.
The subjects are:
- Essentials of numerical nonsmooth (nondifferentiable) optimization (Lectures 1 and 2): Numerical treatment of optimization problems where the objective function is nondifferentiable (it exhibiths kinks, that is discontinuities of the derivatives) is the main topic. Starting from basic concepts of convex analysis (subgradient, subdifferential), two main classes of algorithms will be considered: subgradient and cutting plane-bundle methods. Extensions to nonconvex nonsmooth functions will be examined as well.
- Lagrangian relaxation and applications (Lectures 3 and 4): The Lagrangian relaxation of hard combinatorial or integer programming problems will be described from a theoretical point of view. It allows to model such problems in terms of minimization (or maximization) of an appropriate nonsmooth function (the Lagrangian dual). The technique will be put in action in tackling several applications in areas such as Logistics and Network design.
- Nonsmooth optimization and Machine Learning (Lecture 5): Supervised and unsupervised classification problems will be introduced. The focus will be on modelling such problems via nonsmooth optimization.
Prerequisites:
Basic Courses of Calculus, Linear Algebra and Operations Research
An Introduction to Lie Algebras
CONTENTS
An Introduction to Remote Sensing and Geographic Information Systems (GIS)
The goal is to provide the rudiments of the tools necessary to exploit remote sensing data in the geophysical, environmental and territorial fields. We will start from the physical principles of remote sensing, the characterization of the response of the different land covers in different regions of the electromagnetic spectrum, the characteristics of a digital image, and the resolutions (spatial, spectral, radiometric, temporal) of remote sensing data. Those notions are essential to choose the most appropriate sensor / image. Then, we will move on to the joint analysis of remote sensing digital images and other cartographic data using a GIS software, which allows the integration of various geo-spatial data inside a project.
Gravitational lensing: from mathematical theory to astrophysical applications
- Introduction to General Relativity and Cosmology (2h)
- Basics of Gravitational Lensing: deflection angle and lens equation (2h)
- Mathematical Theory: amplification, images, singularities (4h)
- Lens models: axial, elliptic and multiple lenses (4h)
- Macrolensing: strong lensing, weak lensing (2h)
- Microlensing: basics, statistics, planetary microlensing (4h)
- Black holes: strong deflection limit, shadow and images (2h)
Noise spectroscopy: a window on the properties of matter
The electric noise spectroscopy is an experimental technique focused on the analysis of electric fluctuations produced by physical mechanisms intrinsic of conductor materials. This technique has shown its high potentials in identifying the kinetic processes and the dynamic behaviors of the charge carriers, arousing great interest in the field of Condensed Matter Physics. The study of the fluctuation properties allows the interpretation of very complex electrical transport phenomena, giving interesting information both from the point of view of the basic research and of the physics of applications. In this respect, the identification of possible strategies to reduce the intrinsic noise response is a mandatory request for the development of innovative and advanced technological devices. The success of such goal takes also advantage from the fact that the noise spectroscopy is a very sensitive and non-destructive experimental technique, having a large-scale applicability.
Disordered systems, replica method and complexity
After this year Nobel prize in Physics to Giorgio Parisi this series of lessons regards some of the arguments mostly touched by the prize motivations, namely disordered systems and complexity.
These lessons will have a theoretical nature, but I will also mention some experimental results the theory has to account for. Lessons are thought as being self-contained, therefore there is no need of any previous knowledge of the matter, besides a sufficient mathematical background; everything needed will be developed during the course. For this reason, the lessons are addressed to Mathematics and Physics students (but also students coming from different scientific disciplines) as well as to researchers of various areas which are interested in such themes.
In these lessons I will recall some basic concepts of Thermodynamics and Statistical Mechanics, such as the statistical ensembles, the phenomenon of phase transitions, spontaneous symmetry breaking, broken ergodicity, pure states, mean-field theories and universality. These concepts will be generalized to the realm of disordered systems, particularly the frustrated ones, namely the class of spin-glass systems, particularly the p-spin model. This model will be solved in mean-field using the replica method. The replica symmetric solution will be worked out and the solutions with different replica symmetry breakings, up to the Parisi solution. I will also discuss some other models close to spin-glasses, like the random energy model (REM) and some complex optimization problems.
Resistive Plate Gaseous Detectors in Particle Physics
The aim of this course is to provide the students with a detailed description of a specific particle detector widely used in Particle Physics, to detect cosmic radiation and particles produced in collisions at accelerators. The goal is to carefully describe the resistive plate gaseous detectors, their widespread use and performance. The lectures will deal with the signal detection based on creation and propagation of electric charge in a medium, with their characterization measurements, in terms of efficiency and time and space resolutions, and with their use as Particle IDentification via time-of-flight measurement technique. Finally, the description of interesting measurements obtained in experiments optimized to study proton and/or Heavy Ion collisions and in arrays to detect cosmic rays will be shown.
Josephson effect, superconductive devices, superconducting qubit for quantum technologies
Part 1: Josephson effect and non-linear dynamics.
The Josephson effect describes the passage of Cooper pairs between two superconductors separated by a non-superconducting layer. The course will focus on the properties of the single Josephson element, Josephson transmission lines and the non-linear phenomena that occur: solitons, chaos and coherent synchronization of oscillations. Finally, the operating principles and applications of a SQUID will be illustrated.
Part 2: Superconducting electronics
The focus will be on the "electronic" applications of superconductors, which are largely based on the exploitation of quantum properties on a macroscopic scale. Examples of applications of superconductors as magnetic sensors, radiation detectors, and as digital circuits will be analyzed in detail. The application fields of telecommunications (classical and quantum), astronomy, medicine, and materials science will be also illustrated.
Part 3: Superconducting qubits
Superconducting qubits are based on Josephson tunnel junctions, the only non-dissipative and highly non-linear circuit available at low temperatures. They can be easily coupled to other circuits, which makes them feasible for the implementation of reading operations and logic gates. In the lessons we will discuss the basic theory of quantum circuits, the main superconducting qubit models, and finally we will mention the decoherence problem with applications in quantum technologies.
Conceptual and Physical Foundations of Quantum Mechanics
Quantum mechanics is a well-established and successful scientific theory, even though its interpretation remains still controversial. This circumstance introduces most puzzling questions at the foundations of quantum mechanics, thus providing noticeable ways in which physicists are attempting to resolve them. Trying to clarify the state-of-the-art on the above-mentioned enigmatic topics, these lectures elucidate the basic concepts of quantum mechanics such as nonlocality, reality of the wavefunction and the measurement problem. Moreover, they will provide a discussion and a description of some of the most important mathematical results on recent work in quantum mechanics, including Bell's theorem, no-go theorems as well as the recent achievements christened as second quantum era.
Content of the lectures
- de Broglie hypothesis and its deeper meaning
- The postulates of quantum mechanics
- On the reality and completeness of quantum mechanics
- EPR paradox and its relevance
- Locality and realism of quantum mechanics
- No-go theorems
- Towards a modern quantum mechanics: quantum computing and quantum technologies
- Real quantum computers: the solid-state experience
Electrical, magnetic and thermal properties characterization techniques of superconducting materials relevant for applications
In these lectures, electrical, magnetic and thermal properties characterization techniques of superconducting materials will be presented. The experimental procedures and problems related to the studies about current conduction, magnetization, susceptibility, thermal conductivity and specific heat dependence on environmental parameters (temperature, applied magnetic field) will be the main topics of the lectures. A brief introduction on the fundamental physics of superconductors will be followed by a summary of the main measurement and analysis techniques. The focus will be on superconducting materials recognized as relevant for applications, thus also a brief excursus will be presented. Finally, the quench in technical superconductors will be analysed.
Formazione ed evoluzione delle galassie
- Large scale structures, clustering. Formation and structure of dark matter halos. Baryonic matter and dark matter. Gravitational lensing. Sub-halo mass function. 4h
- Classification of galaxies. Stellar populations and chemical evolution. Statistical properties of falaxies. Scaling laws. Interactions of galaxies and physical mechanisms in the transformation and evolution of galaxies (internal and environmental mechanisms). 6h
Quantum theory of solids
A course on theory of solids should always start with a definition of what a solid is. It is sensible to define a solid as a regular array of atoms in the sense of having, to a good approximation, translational invariance under one of the space groups. Solids have thus order. This order corresponds to a stable (local) minimum of the free energy. Every form of order (be it crystalline or magnetic) has its own particular set of elementary excitations with a characteritic symmetry and dispersion relation. In this course we will desribe the quantum mechanical theory of the order and elementary excitations of solids. The course will introduce the theory of spontaneous symmetry breaking, Landau description of phase transition, Bose condensation, magnons and Fermi-liquid.
Introduction to strongly correlated electron systems
The course is an introduction to the properties of materials whose behavior is dominated by electron-electron interactions. These systems often manifest an insulating behavior despite the band model prediction of a metallic ground state. Examples of classes of systems showing this characteristic behavior are given (nickel compounds, vanadium compounds, etc..), with particular attention to those types of metal-insulator transitions specifically driven by electronic correlations. The phenomenon is analyzed within the so-called Hubbard model, which is the simplest model explicitly taking into account the Coulomb interaction between electrons belonging to the same atom. The relevance of this model to the physics of the high critical temperature superconductors is also analyzed.
The study is then extended to the case of systems containing localized magnetic moments which form on rare earth or actinide atoms. Such systems present anomalies at low temperatures in the response functions due to the correlations between f-type shell electrons, as well as to the interaction between conduction electrons and localized moments giving rise to the Kondo effect and the RKKY interaction. The theoretical analysis will be presented in the context of the so-called Anderson model, which represents a generalization of the Hubbard model to the case of a two-band energy spectrum.
Non-equilibrium physics
One of the top research activities in modern condensed matter physics is to study the systems out of equilibrium. By the new experimental facilities for studying the real-time dynamics of non-equilibrium systems, the need for the corresponding theoretical tools gets more vital. One important approach to theoretically understand such systems is to exploit the theory of non-equilibrium Green’s functions (NEGF) which are fundamentally different from the equilibrium ones and still have so many issues to be explored and understood.
In this course we are going to give a detailed description of NEGFs. At first, we give a very brief review of the equilibrium many-body physics to recall some basic concepts and unify our notations. After that we start the NEGF theory and try to cover the following subjects: introduction to the non-equilibrium condition and describing the failure of the equilibrium many-body approach, Keldysh contour, EoM on the contour, NEGFs and correlators on the contour, Langrethe rules, Wick’s theorem, perturbative approximation and Feynman diagrams for NEGF, Dyson equation and self-energy for NEGF, mean-field approximations, conserving approximations for self-energy in non-equilibrium, Kadanoff–Baym equations, variational principle and Luttinger-Ward theorem, two-particle NEGFs, applications to transport problems with examples on QDs with e-e / e-ph interactions, application to transport in semiconductor physics out of equilibrium, etc.
References:
- G. STEFANUCCI, R. V. LEEUWEN , Non-equilibrium many-body theory of quantum systems, Cambridge University Press, 2013
- H. HAUG, A. JAUHO, Quantum Kinetics in Transport and Optics of Semiconductors, Springer, 2008
- Lecture notes and articles
Fundamentals of crystal growth and their characterization by scanning electron microscopy
The main aim of the course is to provide a theoretical and experimental framework for the epitaxial growth of single crystals and the main phenomena connected to it. Specifically, the course deals with the basic knowledge of the main single crystal growth techniques (Czochralski, Floating zone, micro pulling, etc.) and the diagnostic techniques typically used to characterize single crystals, with particular emphasis to scanning electron microscopy. Furthermore, the spectroscopic investigations for compositional and structural analyses employed within the scanning electron microscope will be illustrated.
Advanced methods for surface characterization
The processes occurring at the surface of a solid are of enormous importance in many fields of physics and chemistry. It is therefore very important to know, in addition to the volume properties, also the surface properties of the solid. The first step is certainly the determination of the elemental composition of the material. It may then be necessary to access more in-depth information such as the oxidation state of the elements, the crystalline structure, the presence of adsorbates, the electrical properties of the surface, etc. With the recent technological development, an enormous number of surface analysis methods are available. In the field of basic research, the study is essentially aimed at the knowledge of the properties of single crystals, thin films, nanostructures of various metallic materials, semiconductors, superconductors, insulators. The objective of the course will be dedicated to understanding the use and the mechanism underlying some of the most widely applied techniques in the field of matter physics (photoemission spectroscopy, Raman spectroscopy, low-energy electron diffraction, Auger spectroscopy).
X-ray diffraction in oriented materials
The course aims to provide basic knowledge on the crystallographic structure of crystalline substances and on the interaction of X radiation with matter. Particular attention will be paid to X-ray diffraction by crystalline materials, a phenomenon that is the basis of one of the most widely used microstructural characterization techniques.
The program of the course: X-rays: nature, production, properties. The interaction of X-rays with matter. Experimental methods for X-ray diffraction. The intensity of diffraction from solids. Structures of crystalline solids. Lattices of points and lattice planes. The reciprocal lattice and stereographic projections. Preferential orientations in polycrystalline solids.
