## Courses 2022/2023

**CORSI/COURSES 2022/2023**

**Curriculum in**

**MATEMATICA PER L'INGEGNERIA /MATHEMATICS FOR ENGINEERING**

**PhD courses**

**Numerical Methods for Boundary Integral Equations**

Lecturer: Chiara Sorgentone (Sapienza)

Duration: 2CFU

Period: 29/11/2022, 01/12/2022, 15/12/2022, 20/12/2022, 14:00-17:00 Aula Seminari, RM004

Program: In this course we will introduce numerical methods for boundary integral equations, mainly for the Laplace equation and Stokes flow. The main topics to be discussed include:

- Theory, derivation and main mathematical properties of boundary integral equations. Starting with the Laplace equation, then moving on to Stokes equations. Single layer and double layer formulations;
- Numerical discretization of boundary integral equations;
- Quadrature rules, including singularity and quasi-singularity treatments;
- Error estimates for layer potentials in 2D and 3D.

Title:

**On the connection between non-local operators and probability**

Lecturer: Mirko D’Ovidio (Sapienza - SBAI)

Duration: 15 hours

Period: 2 hours a week starting from January 30th, 10:30-12:30, Room 7, RM018

Abstract: We discuss some basic and advanced facts about initial and boundary value problems involving non-local operators. In particular, we show some stimulating connections between non-local Cauchy problems, non-local boundary value problems and stochastic processes. Non-local boundary value problems also include non-local dynamic boundary conditions. We discuss the probabilistic representation of the solutions together with the associated functionals.

Title:

**Energetic relaxation of structured deformations, a multiscale geometrical basis for variational problems in continuum mechanics**

Lecturer: José Matias (University Of Lisbon)

Duration: 12 hours

Period: March 29, 10:00-12:00, room 4, RM0018; April 13, 20, 27, 16:00-18:00, Aula Seminari, RM004; May 4, 11, 16:00-18:00, Aula Sseminari, RM004

Abstract: This course will cover the material on the book [12]. Broadly speaking, first order structured deforma- tions introduced in [9] provide a mathematical framework to capture the effects at the macroscopic level of geometrical changes at submacroscopic levels. This theory was broadened by [17] in order to allow for geo- metrical changes at the level of second order derivatives (second order structured deformations). The theory of structured deformations was further enriched in [11] in order to consider different levels of microstructure, that is, hierarchical structural deformations.

Starting from this mechanical formulation of the theory, and upon describing the needed mathematical framework, namely recalling some basic properties of spaces of bounded variation and spaces of bounded hessian, the variational formulation for first order structured deformations in [7] is presented as well as two different variational formulations for second order structured deformations, in [3] and in [10]. A variational formulation for hierarchical structured deformations in [5] is also presented. Different applications in this context will be discussed, namely:

(1) Dimension reduction in the context of structured deformations [14] and [8];

(2) Derivation of explicit formulae for the relaxed energy densities [4] and [16];

(3) Optimal design in the context of structured deformations [15];

(4) Homogenization in the context of structured deformations [1];

(5) Upscaling and spatial localization of non-local energies [13].

Finally, some open problems and possible generalizations of the theory will be discussed:

(1) Mechanical framework: L∞ theory of first order structured deformations [9], second order struc- tured defomations [17], and hierarchical systems of structured defomtations [11]. Examples and approximation theorems.

(2) Mathematical framework: some results on measure theory, BV spaces and Γ-convergence that will be needed for the mathematical formulation. Later on, some other mathematical preliminaries will be needed, namely Reshetnyak -type continuity theorems, BH space and the Global method for relaxation [2].

(3) The variational formulation of Choksi-Fonseca for first order structured defomations [7]. Results and sketch of the proofs.

(4) Some applications: (a) Relaxation of purely interfacial energies; (b) Optimal design of fractured media; (c) Relaxation of non-local energies; (d) Hierarchical systems of first order structured deformations. (e) Homogenization in the context of structured deformations.

(5) Variational settings for second order structured deformations [3] and [10] .

(6) Outlook for future research.

References:

[1] M. Amar, J. Matias, M. Morandotti, and E. Zappale: Periodic homogenization in the context of structured deformations. ZAMP, 73, 173 (2022)

[2] G. Bouchitté, I. Fonseca, and L. Mascarenhas: A global method for relaxation. Arch. Rational Mech. Anal., 145 (1998), 51–98.

[3] A. C. Barroso, J. Matias, M. Morandotti and D. R. Owen: Second-order structured deformations: relaxation, integral representation and examples. Arch. Rational Mech. Anal., 225 (2017), 1025–1072.

[4] A. C. Barroso, J. Matias, M. Morandotti, and D. R. Owen: Explicit formulas for relaxed energy densities arising from structured deformations. Math. Mech. Complex Syst., 5(2) (2017),

[5] A. C. Barroso, J. Matias, M. Morandotti, D. R. Owen and E. Zappale The variational modeling of hierarchical structured deformations. Submitted to J. Elasticity (2022).

[6] R. Choksi, G. Del Piero, I. Fonseca, and D. R. Owen: Structured deformations as energy minimizers in models of fracture and hysteresis. Mathematics and Mechanics of Solids 4 (1999), 321–356.

[7] R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103.

[8] G. Carita, J. Matias, M. Morandotti, and D. R. Owen: Dimension reduction in the context of structured deformations. J. Elast. 133 Issue 1 (2018), 1–35.

[9] G. Del Piero and D. R. Owen: Structured deformations of continua. Arch. Rational Mech. Anal., 124 (1993), 99–155.

[10] I. Fonseca, A. Hagerty, and R. Paroni: Second-order structured deformations in the space of functions of bounded hessian. J. Nonlinear Sci., 29(6) (2019), 2699–2734.

[11] L. Deseri and D. R. Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations at multiple submacroscopic levels. J. Elasticity, 135 (2019), 149–182.

[12] J. Matias, M. Morandotti and D. R. Owen Energetic relaxation of structured deformations. A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics. Book to be published by SpringerBriefs on PDEs and Data Science.

[13] J. Matias, M. Morandotti, D. R. Owen, and E. Zappale: Upscaling and spatial localization of non-local energies with applications to crystal plasticity, Math. Mech. Solids, 26 (2021), 963–997.

[14] J. Matias and P. M. Santos: A dimension reduction result in the framework of structured deformations. Appl. Math. Optim. 69 (2014), 459–485.

[15] J. Matias, M. Morandotti, and E. Zappale: Optimal design of fractured media with prescribed macroscopic strain. Journal of Mathematical Analysis and Applications 449 (2017), 1094–1132.

[16] M. Šilhavý: The general form of the relaxation of a purely interfacial energy for structured deformations. Math. Mech. Complex Syst., 5(2) (2017), 191–215.

[17] D. R. Owen and R. Paroni: Second-order structured deformations. Arch. Rational Mech. Anal. 155 (2000), 215–235.

Title:

**Convexity notions arising in the supremal setting**

Lecturer: Elvira Zappale

Duration: 12 hours

Period: Room 5, RM018, 31/3, 14/4, 21/4, 28/4, 5/5, 12/5, 9:00-11:00

Abstract: The course will consist of 5 or 6 lectures where I will first introduce supremal functionals, basic concepts of direct methods of Calculus of variations, with a particular emphasis on relaxation and then I will present several notions necessary and/or sufficient for the lower semicontinuity of supremal functionals and related results ensuring a power law approximation using integral energies. Also I will compare the classical notios of convexity used in the integral setting with their counterparts in the supremal context and other possible notions. I will conclude with a quick overview of the nonlocal setting.

**Mini-courses**

Title

**: Integrable systems – Methods of Mathematical Physics in interaction**

Lecturer: Cornelia Schiebold (Mid Sweden University)

Duration: 6 hrs

Period: AULA 1B, Building RM002: March 7 12:00-14:00, March 14 12:00-14:00, March 21 12:00-14:00

Abstract: ntegrable systems in infinite dimension refer to an area in mathematical physics which is devoted to the study of a certain group of partial differential equations, many of them soliton equations like the classic Korteweg-de Vries equation and the nonlinear Schrodinger equation. One of the striking features is the existence of solutions with particle character, called solitons, remarkable in view of nonlinearity of the governing equations. Methodologically, integrable systems are a meeting point (melting pan) for methods from very diverse parts of mathematics. The main idea of this mini course is to highlight interactions of some of the main approaches to integrable systems, the inverse scattering method and an operator theoretic approach in the first place, and symmetry methods like Backlund transformations, recursion operators and hierarchies to a minor extent. Throughout we will emphasise the recent topic ofnon-commutative integrable systems, like vector- and matrix soliton equations, where many fundamental questions are still open. Notably, the construction of solutions is not interesting only under the mathematical viewpoint, but also under the physical one. Indeed, very important applications of soliton equations are in nonlinear optics, for instance.

The lectures are going to be reasonably self-contained. Some familiarity with PDE's and functional analysis is certainly helpful, but not required. An overview on the basic notions used during the course are provided when needed. Needed material as well as references are provided by the Lecturer.

Title:

**Stochastic Homogenization and large-scale regularity**

Lecturer: Felix Otto (MPI-MIS Leipzig)

Duration: 10 hrs

Period: Room 10, RM018: 10/02/2023 14:00-16:00, 15/02/2023 10:00-12:00, 17/02/2023 14:00-16:00, 22/02/2023 10:00-12:00, 24/02/2023 14:00-16:00

Abstract: In this mini-course, I will introduce the concept of large-scale regularity in case of a linear elliptic equation (or system) with heterogeneous coefficients. It is based on a smallness (on average) of the potentials of the harmonic coordinates, and proceeds via an intrinsic Campanato iteration. I will then apply this to the case of a random heterogeneous coefficient field, sampled from a stationary and ergodic ensemble. I will try to be self-contained and closely follow Theorem 1 and Lemma 1 in Gloria, Neukamm, and Otto ``A regularity theory for random

elliptic operators'', Milan J Math 2020.

**Useful courses from undergraduate programmes**

Lecturer: Francesca Pitolli (Sapienza -SBAI)

Duration : 20 hours (first part) + 40 (second part) hours

Period: Primo semestre, martedì h. 9:15-10:45 (Aula 25, via Eudossiana); giovedì h. 15:00-18:30 (Aula 15, via Eudossiana)

Abstract: Prima parte: Metodi numerici per la soluzione di problemi differenziali, metodi di Runge-Kutta, metodi alle differenze finite (2CFU)

Seconda parte: Approssimazione ai minimi quadrati per l'identificazione di un modello e la stima dei parametri. Soluzione di sistemi lineari sovradeterminati. Decomposizione ai valori singolari e sue applicazioni. Problemi inversi mal posti e tecniche di regolarizzazione. Soluzione di sistemi lineari sottodeterminati. Analisi delle componenti principali e sue applicazioni (4CFU)

Per ogni argomento verranno svolte delle esercitazioni in cui si utilizzeranno i metodi numerici illustrati a lezione per risolvere alcuni problemi applicativi.

**Curriculum in**

**ELETTROMAGNETISMO/ELECTROMAGNETISM**

**PhD courses**

**Analytical Techniques for Wave Phenomena**

Lecturer: Paolo Burghignoli

Duration: 6 CFU

Period:

*Facoltà di Ingegneria, Via Eudossiana 18, Roma*;

*7/2/2023, 14:00-18:00, Aula 33; 8/02/2023, 9:00-13:00 and 14:00-18:00, Aula 33;*

*14/02/2023, 14:00-18:00, Aula 33; 15/02/2023, 9:00-13:00 and 14:00-18:00, Aula 8 (morning) and 33 (afternoon)*

Abstract: The course aims at providing Ph.D. students with analytical tools useful in applied research on general wave phenomena. The unifying theme is that of complex analysis, of which a compact, self‐contained introduction is presented. Fundamental techniques asymptotic techniques are then illustrated, including ray optics and the Laplace and saddle‐point methods for the asymptotic evaluation of integrals. Applications are focused on the analysis of both time‐harmonic and transient waves excited in planar layered structures by canonical sources. As concerns the former, different wave species will be defined and physically discussed (space, surface, leaky, lateral waves). As concerns the latter, the Cagniard‐de Hoop method will be introduced.

Title:

**Nanophotonics and Plasmonics**

Lecturer: Concita Sibilia

Duration: 2 CFU

Period: Second semester, Mon and Wed 10:00-12:00, SBAI reading room, RM009

Abstract: The part of seminars related to Nanophotonics aims to introduce to students some exciting concepts that differ from conventional wave optics, with particular emphasis to the role of the evanescent fields in many practical applications, such as near field optical microscopy. The field of plasmonics (interaction of light with electrons in metals) has attracted a great deal of interest over the past two decades, but despite the many fundamental breakthroughs and exciting science it has produced, it is yet to deliver on the applications that were initially targeted as most promising. The seminars proposed examine the primary fundamental hurdles in the physics of plasmons that have been hampering practical applications and highlights some of the promising areas in which the field of plasmonics can realistically deliver.

Title:

**Basics of Nonlinear Optics**

Lecturer: Concita Sibilia

Duration: 2 CFU

Period: Second semester, Mon and Wed 10:00-12:00, SBAI reading room, RM009

Abstract: Nonlinear Optics (NLO) is the study of phenomena that occur as a consequence of the modification of the optical properties of a material system by the presence of light. Basics and more recent applications of NLO to new light sources and devices will be presented in a series of seminars.

Title:

**Radiazione Termica e Segnatura Infrarossa**

Lecturer: Roberto Li Voti

Duration: 4 CFU

Period: dal 9 gennaio al 13 marzo 2023, 17:30-20:00, online, https://meet.google.com/hdg-gijh-gaz

Abstract: Il corso tratterà i seguenti argomenti: emissione spontanea, radiazione termica, legge di Planck, legge si Stephan -Bolzmann. Emissività, teoria del corpo nero e radiatori selettivi. Spettro elettromagnetico, radiazione luminosa, radiazione nell'infrarosso. Parametri caratteristici di un elemento radiante: radianza, radianza spettrale, potenza, potenza spettrale. Propagazione del segnale luminoso in aria, bande di assorbimento atmosferico. Effetti atmosferici: assorbimento, autoemissione, diffusione, deflessione, turbolenza. Tecniche numeriche per la valutazione della emissività variabile dei materiali; e tecniche nondistruttive fotoacustiche e fototermiche per la caratterizzazione di materiali.

**Useful courses from undergraduate programmes**

Title:

**Laboratorio Di Applicazioni Industriali Delle Tecnologie Ottiche Fototermiche E Fotoacustiche**

Lecturer: Roberto Li Voti

Duration: 3 CFU

Period: Second semester

Abstract:The course will provide the theoretical bases of the photothermal, photoacoustic, radiometric, and infrared techniques for nondestructive evaluation and testing of materials (nanomaterials and metamaterials). Many applications will be introduced in different fields: industry, environment, energy, but also biology, medicine, agrifood. Final comparisons will be introduced among the diagnostic techniques. The course contains also some experimental activities in laboratory and the relative data analysis and data processing.

Title: Optics (LM)

Lecturer: Eugenio Fazio

Duration: 6 CFU

Period: 2nd semester

Title: Molecular dynamics and atomistic simulations (LM)

Lecturers: Giuseppe Zollo

Duration: 6 CFU

Period: First semester

Title: Laser Fundamentals (LM)

Lecturers: Concita Sibilia

Duration: 6 CFU

Period: Second semester

Title: Quantum Information, I e II (Master)

Lecturer: Fabio Bovino (docente master sapienza)

Duration: 5+5 CFU

Period: Second semester

Title: Esperienze di Laboratorio per il modulo di Ottica/Optics (?)

Lecturers: Alessandro Belardini

Duration: 2 CFU

Period: Second semester

Title: Advanced Electromagnetics and Scattering (LM)

Lecturers: Fabrizio Frezza

Duration: 6 CFU

Period: Second semester

Title: Microwaves

Lecturers:Marta Cavagnaro, Fabrizio Frezza, Alessandro Galli (LM)

Duration: 9 CFU

Period: First semester

Title: Artificial materials, metamaterials and plasmonics for electromagnetic applications (LM)

Lecturers: Fabrizio Frezza

Duration: 6 CFU

Period: First semester

**Curriculum in**

**SCIENZA DEI MATERIALI / MATERIALS SCIENCE**

**PhD courses**

Title:

**Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy**

Lecturers: Carlo Mariani, Settimio Mobilio, Francesco Offi, Alessandro Ruocco

Duration: 32 hours

Period:

- from 20/02/2023 to 10/10/2023, Mon and Wed 15:30-17:30, room 57, Dipartimento di Fisica, Roma Tre;
- from 13/03/2023 to 03/04/2023: Mon and Wed 16:00-18:00, room 2, Dipartimento di Fisica, Sapienza;
- from 12/04/2023 to 12/05/2023: Wed and Fri, 15:30-17:30, room 57, Dipartimento di Fisica, Roma Tre.

**Useful courses from undergraduate programmes**

Lecturer: Carlo Mariani

Planned period: October-December 2022

Title: Chimica Fisica dello Stato Solido e dei Materiali Nanostrutturati (6 CFU) (Corso di Laurea in Chimica Industriale)

Lecturer: Danilo Dini

Planned period: October-December 2022

Title: Sistemi di produzione ed accumulo dell'energia (9 CFU) (Corso di Laurea in Chimica Industriale)

Lecturer: Maria Assunta Navarra

Planned period: October-December 2022

Title: Fabbricazione e caratterizzazione di nanostrutture (6 CFU) (Corso di Laurea in Ingegneria delle Nanotecnologie)

Lecturer: Carlo Mariani - Ernesto Placidi

Planned period: March-June 2023

Title: Laboratorio Macromolecole (9 CFU) (Corso di Laurea in Chimica Industriale)

Lecturer: Andrea Martinelli

Planned period: March-June 2023

Title: Physics Laboratory II (9 CFU) (Corso di Laurea in Fisica)

Lecturer: Carlo Mariani

Planned period: March-June 2023

Title: Microscopie e tecniche di nanocaratterizzazione (9 CFU) (Corso di Laurea in Ingegneria delle Nanotecnologie)

Lecturer: Marco Rossi

Planned period: March-June 2023

Title: Chimica dei materiali polimerici (6 CFU) (Corso di Laurea in Chimica Analitica)

Lecturer: Ilaria Fratoddi

Planned period: March-June 2022

Title: Scanning Probe Microscopy (3 CFU) (Corso di Laurea in Nanotechnology Engineering)

Lecturer: Daniele Passeri

Planned period: March-June 2023