# Mathematics, physics and applications | Offerta didattica dottorato XXXV ciclo

## Mathematics, physics and applications Offerta didattica dottorato XXXV ciclo

### Offerta Didattica

L'offerta didattica è stata formulata come segue.

Cliccando sui titoli si accede agli abstract, che sono comunque disponibili più in basso.

A: attivo, in corso;

T: terminato.

### Numerical operations on oscillatory functions

#### Abstract

Oscillatory functions appear in various branches of current interest to mention only processes as vibration, rotation, oscillation, wave propagation in classical physics and engineering, or the

behavior of the wave functions in quantum physics. Our aim is describing novel mathematical and computational procedures for operations on these functions, e.g., numerical differentiation,

quadrature, interpolation, solving differential equations etc.

#### Content

- Introduction: where oscillatory functions do appear in practice and why their numerical treatment needs special methods.
- Exponential fitting, an efficient tool for approaching oscillatory functions; a few examples.
- Mathematical backgrounds of the exponential fitting:
- Reference differential equation
- Operator ${\cal L}$
- Functions $\eta_m(Z)\, m=-1,0,1,...$
- Various implementations:
- Numerical differentiation
- Quadrature
- Interpolation
- Solution of differential equations
- Applications

### Model Theory of Real-Valued Logic

#### Overview

The course will focus on [0, 1]-valued logics. These logics are objects of intense study in mathematical logic, and they are studied from two perspectives: (1) algebraic logic and (2) the continuous model-theory. Algebraic logic studies Łukasiewicz logic and its algebraic semantics via MV algebras and their associated structures while the continuous model theory studies continuous logics and its semantics via metric struc- tures. Both lines of research have evolved independently and have developed their own sets of problems. However, few researchers are aware that both perspectives trace their origin to the same mathematical work, namely, the groundbreaking research of C-C. Chang and A. Tarski in the 1950’s. Only recently, fundamental results have exposed deep connections.

This course will be a first attempt to close the gap separating both research lines. The logic group at the University of Salerno is one of the most active algebraic logic groups, and the course instructor is one of the leading researchers in continuous model theory; therefore the course presents an excellent opportunity to bring together students and researchers from both areas.

The prerequisite of the course is basic first-order logic and the proverbial *mathematical maturity*. We will start from the definition of metric structure and finish by presenting current research, as well as open problems.

#### Course Outline

Week 1: Overview. Chang’s [0, 1]-valued logic, Chang-Keisler logic with truth values in a uniform space, Chang Keisler continuous model theory, Krivine’s Real-Valied logic, Luxemburg’s nonstandard hulls, Henson’s model theory of metric spaces, Ben-Yaacov-Usvyatsov continuous logic, infinitary extensions.

Week 2: Fundamentals. Metric space structures, fundamentals of continuous logic (elementary exten- sions, elementary chains).

Week 3: Compactness and applications. Ultraproducts of metric space structures. The compactness the compactness theorem and applications, Beth definability theorem.

Week 4: Types. Spaces of types, the omitting types theorem and applications, countable categoricity, definable types.

Week 5: Stability. Stable formulas and stable theories, Krivine-Maurey stability, applications to functional analysis.

Week 6: Recent developments. Maximality theorems, Tao’s concept of metastable convergence, recent applications to functional analysis, recent applications to ergodic theory.

#### Prerequisites

The course assumes basic familiarity with first-order model theory.

#### References

The literature on continuous model theory is extensive. Since the area is evolving rapidly and we will be studying recent developments, we will not follow a particular book of paper. However, below is a list of some references that will be cited often during the course.

- I. Ben Yaacov and A. Usvyatsov. Continuous first order logic and local stability. Transactions of the American Mathematical Society 362 (2010).
- X.Caicedo.MaximalityofcontinuouslogicandŁukasiewiczlogicinBeyondFirstOrderModelTheory, CRC Press, 2017.
- X. Caicedo, E. Duen ̃ez and J. Iovino. Compactness, metastable convergence, and characterization of metric structures. Preprint.
- C.-C. Chang and H. J. Keisler. Continuous Model Theory. Annals of Mathematics Studies, 1966.
- C. W. Henson and J. Iovino. Ultraproucts in Analysis in Analysis and Logic, Cambridge University Press, 2002.
- J. Iovino. Applications of Model Theory to Functional Analysis. Dover Books in Mathematics, 2014.
- H. J. Keisler. Model theory for real-valued structures. Preprint.
- J.-L. Krivine. Sous-espaces de dimension finie des espaces de Banach re ́ticule ́s. Annals of Math., 116, p. 1-29 (1976).
- J.-L. Krivine. Sous-espaces et cones convexes dans les espaces Lp. The`se d’Etat (1967).
- J.-L. Krivine and B. Maurey. Espaces de Banach stables. Israel J. Math., 39, 4, p. 273-295 (1981).
- T. Tao. https://terrytao.wordpress.com.

### Algebra and Geometry in the Calculus of Variations

#### Abstract

The course aims at laying the algebraic and geometric foundations of the calculus of variations. From the algebraic side, it aims to show that differential (and integral) calculus is a part of commutative algebra and can be defined on an appropriate category of modules. Particular emphasis will be given to the algebraic version of the classical “integration by parts” which plays an important role in the calculus of variations. From the geometric side, the course aims at introducing the geometric portraits of PDEs (the so called diffieties) and the structure on them responsible for the main constructions in the calculus of variation (the so called variational bicomplex). In this language, variational principles, Euler-Lagrange equations, Helmoltz conditions, conservation laws, etc. appear as appropriate cohomologies of the variational bicomplex.

#### Tentative Program

##### Algebra

- differential operators over commutative algebras
- the symbol algebra
- the adjoint module of a module
- algebraic integration
- algebraic Green formula

##### Geometry

- infinite jet spaces
- PDEs and their infinite prolongation
- the Cartan distribution
- the variational bicomplex
- Euler-Lagrange calculus

#### Prerequisites

The participants have to know the basics of differential geometry (manifolds, submanifolds, vector fields, differential forms, (vector) bundles, etc.). The basics of homological algebra (cochain complexes, cochain maps, cohomologies, homotopies, etc.) are also useful but not mandatory.

#### Further remarks

The course might be useful for mathematicians interested in Algebra, Geometry, Analysis and/or Mathematical Physics, and for Physicists interested in Theoretical and High Energy Physics, particularly Classical and Quantum Field Theory, including General Relativity.

### Introduction to Character Theory

#### Abstract

Character theory provides a powerful tool for proving theorems about finite groups, but it is also interesting in its own right. After some preliminaries, we will study the properties of characters themselves and how these properties reflect and are reflected in the structure of the group.

#### Prerequisites

The participants have to have some familiarity with rings and modules and will need to know some basic finite group theory

### Variational methods for elliptic and parabolic problems

#### Abstract

The aim of this course is to introduce variational methods for studying elliptic and parabolic problems. After a short introduction to Hilbert spaces and operators on Hilbert spaces, we will give a crash course on semi- groups theory and its relationship to evolution equations. The sesquilinear form technique will be explained and applied to second order uniformly ellip- tic operators on L2-spaces. Using interpolation techniques one obtains results on Lp-spaces. We end this course by proving Gaussian upper bouds for heat kernels of uniformly elliptic operators and use them to obtain several spectral properties.

#### References

- A. Batkai, M. Kramar-Fijavz, A. Rhandi, Positive Operator Semigroups: From Infinite Dimensions. In Operator Theory: Advances and Applications, Birkh ̈auser 2017.
- E.B. Davies, Heat Kernels and Spectral theory, Cambridge University Press, 1989.
- K. J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.
- E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, 2005.

### Exact and heuristic approaches for optimization problems

#### Abstract

The course deals with exact and heuristic approaches for optimization problems. In particular, delayed column generation and genetic algorithms will be illustrated. Applications of these approaches will be also described considering wireless sensor network problems.

#### Prerequisites

The participants have to know the basics concepts of Operations Research (simplex method, duality).

### Set of Finite Perimeter and Geometric Variational Problems

#### Abstract

The main aim of the course is to provide a classical framework to study the existence, symmetry, regularity of minimizers of a broad class of geometric variational problems in which surface area is minimized under some volume constraint. The first part of the course will be devoted to the study of the basic properties of Radon measures.

We will focus mainly on Hausdorf measures and differentiation of measures. The second part of the course will concern the sets of finite perimeter and the coarea formula.

In the very end of the course we will employ the previous tools to the study of the Euclidean isoperimetric problem.

#### Prerequisites

The participants have to be familiar with Lebesgue measures and integration.

### Category theory: an Introduction

#### Abstract

The course aims to present some ideas and methods in the theory of categories.

### Convex Functions on Intervals and their applications

#### Abstract

By a mean (on an interval I) we understand any function M: I × I → I which verifies the following property of intermediacy, inf{s, t} ≤ M(s, t) ≤ sup{s, t}, for all pairs {s, t} of elements of I. An important mathematical problem is to investigate how functions behave under the action of means. The best-known case is that of midpointconvex (or Jensen convex) functions, which deal with the arithmetic mean. In the context of continuity (which appears to be the only one of real interest), midpoint convexity means convexity.

#### Tentative program

Convex Functions at First Glance. Young’s Inequality and Its Consequences. Smoothness. An Upper Estimate of Jensen’s Inequality. The Subdifferential. Integral Representation of Convex Functions. Conjugate (or Legendre transform of a) Convex Functions. The Integral Form of Jensen’s Inequality. The Hermite–Hadamard Inequality. Algebraic Versions of Convexity. The Gamma and Beta Functions.

#### Prerequisites

The necessary background is advanced calculus and linear algebra.

### PDEs and Boundary Value Problems with Applications

- Sturm-Liouvile eigenvalue problems. Example of S-L eigenvalue problems. Properties, completeness and positivity. General S-L problem.
- Boundary value problems in rectangular coordinates. The Heat equation. Five-stage method of solution. The vibrating string. D'Alembert's general solution.
- Application of Multiple Fourier series.
- BVPs in cylindrical coordinates. Bessel functions. Heat flow in the infinite cylinder. Heat flow in the finite cylinder.
- BVPs in spherical coordinates. Spherical symmetric solutions. Laplace's equation in spherical coordinates.
- Fourier transform and applications.

### Theoretical frameworks in Mathematics Education Research

#### Abstract

The course deals with the main theoretical frameworks developed within research in mathematics education, such as activity theory, commognition, semiotic mediation. Applications to e-learning environments will be particularly considered.

### 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.

### Introduction to Hodge Theory

#### Abstract

Hodge theory provides a way of studying the de Rham cohomology of a compact manifold by looking at harmonic forms, i.e. for differential forms that are in the kernel of the so called Hodge Laplacian, a self-adjoint linear operator associated to a Riemannian metric on the manifold. Indeed, the Hodge theorem guarantees that within the set of all closed forms representing each de Rham cohomology class there is exactly one harmonic representative. In order to deal with Hodge theory we will preliminarily need to introduce or recall some notions such as integration on manifolds, orientability, Riemannian metrics, Hodge star operator, de Rham cohomology.

#### Prerequisites

The participants will need to have a working knowledge of the Cartan calculus of differential forms on a manifold.

Familiarity with Riemannian metrics will be helpful but not strictly required.

### The Dirichlet problem for linear elliptic equations

#### Contents

Some functional space.

Embedding theorems.

Existence and regularity of the solution to Dirichlet problem.

#### Prerequisites

The participants have to know the basics concepts of Functional Analysis.

### Equazioni differenziali alle derivate parziali ellittiche del secondo ordine. Teoria L^p

Vedi allegato.

### Stationary Point Processes

#### Contents

The forward recurrence time.

Relationship between counts of events and times between events.

Second order properties of counts and times between events.

Examples of stationary point processes.

### Mathematical modelling for financial risks analysis

#### Abstract

The plan is to show the practical aspects of using regression models and different methods for economical, insurance and financial tasks.

### Physics of Strong Interactions

- Accelerators
- Strong Interactions
- Heavy Ion Physics

### Cosmology and particle physics

- Overview of the Standard Model
- Particle and Neutrino Physics
- Dark Matter and Dark Energy
- Extended theories of Gravity

### 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
- 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)

### Entanglement and quantum imformation principles

- Definition of entanglement
- Properties of entanglement measures
- Entropy as a measure of bipartite entanglement for pure states
- Concurrence as a measure of bipartite entanglement for mixed states
- Multipartite entanglement: open problems
- Quantum teleportation and its relation with the entanglement
- Shor's algorithm

### Complex Systems in Social and Health Sciences

- The Complex Systems and their Science
- Methods for studying and analyzing
- Social Systems and Networks
- Health Systems and Networks
- Examples of Applications

### Quantum Topological Systems

- Symmetries and effective Hamiltonians
- The ten-fold way to topological systems by symmetry perspective
- Topological insulators and Chern insulators
- Topological superconductors
- Introduction to quantum materials platforms for Majorana edge modes
- Basic introduction to Majorana devices: transport properties and Josephson junctions.

### Fundamentals of Nanotransistors

- The Transistor as a Black Box (2h)
- The MOSFET: A barrier-controlled device (2h)
- Poisson Equation and the Depletion Approximation (2h)
- Gate Voltage and Surface Potential (2h)
- The Mobile Charge: Bulk MOS and Extremely Thin SOI (2h)
- 2D MOS Electrostatics (2h)
- The Virtual Source Model (2h)
- The Landauer Approach to Transport (2h)
- The Ballistic MOSFET (2h)
- Carrier Scattering and Transmission Theory of the MOSFET (2h)

### Josephson effect, superconducting devices, superconducting qubits

- Josephson effect and non-linear dynamics
- Superconducting electronics
- Superconducting qubits

### Electric noise spectroscopy: a window inside condensed matter properties

- Electric noise and fluctuations: general concepts and measurement procedures (2h)
- Investigation of fluctuation mechanisms in advanced innovative systems (2h)
- Electric noise properties of magnetic compounds (2h)
- Noise in superconducting materials and devices (2h)
- The spectroscopy of charge carrier fluctuations in emergent photovoltaic devices (2h)

### Synthesis of single crystals and characterization by x ray diffraction and scanning electron microscopy

- Phase diagrams and Synthesis of single crystals (3h)
- Microscopy and related techniques:
- Optical and Scanning Electron Microscopy (2h)
- Energy dispersion spectroscopy, Wavelength Dispersion Spectroscopy (2h)
- High Resolution X ray Diffraction (2h)
- X-ray fluorescence, Transmission Electron Microscopy, Electronic Diffraction (2h)
- A Selection of surface sensitive techniques: Auger electron spectroscopy, X ray reflectivity, tunnel scanning and atomic force microscopy (2h)
- Other fundamental techniques for characterization of materials (ARPES, XPS, Neutrons) (2h)

### Electrical and thermal properties characterization techniques of superconducting materials relevant for applications

- Fundamental properties of superconductors and materials relevant for applications
- Electrical transport measurement techniques
- Specific heat measurement techniques
- Thermal transport and thermo-electric effects
- Quench in technical superconductors

### Thin film deposition and patterning

- Thin films growth process (Substrate choice, condensation, nucleation, Modes of thin-film growth)
- Physical Vapor Deposition techniques
- Heterostructures based on thin films.
- Thin film patterning techniques: optical, e-beam and focused ion beam lithography