## Offerta formativa erogata 2024/2025

Elenco delle attività formative previste per i dottorandi del primo anno.

---------------------

Spectral Geometry

- data presunta: 03-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 24

- docente del corso: Luigi Provenzano (Sapienza) e Davide Buoso (U. Piemonte Orientale)

- abstract: In the first part of the course we will provide a brief introduction to the spectrum of the Laplacian on Euclidean domains and Riemannian manifolds, along with a few basic examples, and their relations with physical phenomena (waves and vibrations). Then, we will focus on some classical problems in spectral geometry, such as eigenvalue bounds and isoperimetric inequalities for the eigenvalues. The questions that we will address are the following: how does the geometry and the topology of the ambient space influence the spectrum (the eigenvalues)? On the other hand, what information can give the knowledge of the spectrum on the geometry and topology of the ambient space? We will present a few classical techniques which have been adopted throughout the years to address these questions. In the final part of the course (if time allows) we will consider some recent developments on old and new problems, and we will present some open questions.

- Exam: Short seminar/report on a research paper (possibly close to the student's interests)

--------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, I

- data presunta: 02-03/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- Abstract: Introduction to the photoelectron spectroscopy: theoretical background, the three-step model, atoms and molecules, low-dimensional solid systems, experiments with angular resolution, time-resolved experiments. Instrumentation: charged particles, Auger electron spectroscopy and resonant photoemission. Surfaces and low-dimensional systems, electronic properties. Core-level photoemission and surface core-level shifts. Angular resolved photoemission, electronic band structure. Band structure of exemplary 1D and 2D systems.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, II

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- abstract: Electromagnetic radiation sources, synchrotron radiation, theoretical background, storage rings, beamlines, photoemission. Introduction to the free-electron laser: a coherent source of radiation from UV to X rays. X ray absorption spectroscopy, theoretical background of absorption. Multiple scattering theory: a method for the observation of the electronic states and spectroscopy measurements. EXAFS and XANES/NEXAFS: fundamentals and applications. X ray elastic and anelastic scattering. High energy photoemission, application to buried interfaces/materials.

Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------

Advances in Raman spectroscopy: from traditional vibrational spectroscopy to surface enhanced approaches

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 10

- docente del corso: Angela CAPOCEFALO

- abstract: The aim of the course is to provide doctoral students with a thorough understanding of Raman spectroscopy, covering both the traditional technique and the more advanced surface enhanced Raman spectroscopy. After introducing the fundamentals of the Raman scattering, the experimental aspects of the technique will be examined, including the description of the measurement apparatus and the analysis and interpretation of data. Surface enhanced Raman spectroscopy will be then introduced, discussing the different mechanisms underlying signal amplification. The plasmonic properties of commonly used nanostructured metal substrates and the recent advances in the technique, such as tip-enhanced Raman spectroscopy will be presented. Finally, innovative applications of the technique in various research fields such as sensing, nanomedicine, materials science, and cultural heritage will be discussed.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------

Neural Networks & Machine Learning

- data presunta: 01-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Adriano Barra qualifica

- abstract: The course is meant to provide theoretical tools (both mathematical and computational) to allow the students to orient themselves in the proliferation of neural network techniques and machine learning algorithms that are nowadays broadly used in the processing of data and signals both in the world of research as well as in industry. Specifically, once shared the main mathematical methodological bases (a quick review of elements of probability and statistics), after a succinct historical introduction (e.g. the Turing machine, Rosenblatt's perceptron and AI’s winter time), modern neural networks will be addressed, both those biologically inspired (e.g. the Hopfield model and its variations on the theme) as well as those not-biologically inspired (Boltzmann machines and feed-forward networks), with the related algorithms for learning and automatic recognition (e.g., contrastive divergence and back -prograpation). The ultimate aim of the course is to share the salient concepts with the students and, at the same time, to provide them with the key tools, so that they can keep growing within the field of Artificial Intelligence and Machine Learning: this transfer of information will be supplied both from a formal/mathematical point of view (e.g. showing during the course clear methods for setting up a relevant problem and solving it appropriately) and from a logical/deductive point of view (e.g. understanding what it is reasonable to be addressed by modern techniques of Machine Learning). To this end the course program is divided into two main sections. The former is to ensure that we share basic scientific knowledge (obviously a necessary pre-requisite to guarantee that we understand information processing in neural networks from a mathematical perspective later on). The latter is completely dedicated to neural networks: after a succinct description (always in mathematical terms) of the key mechanisms inherent to the neuron and the propagation of information between neurons, "networks of neurons" will be built (in other words they will explain " what are” -mathematically speaking- these neural networks) and we will study their emergent properties (i.e. those not immediately deducible by looking at the behavior of the single neuron): specifically, we will try to see how these networks are able to learn and abstract by looking at supplied examples from the external world and how, subsequently, they use what they have learned to respond appropriately, if stimulated, to the external world. We will also understand how these neural networks can sometimes make mistakes, and why. Ideally at the end of the course the students should be able to independently continue in-depth study of this discipline and benefit from it accordingly during their careers

- Exam: Oral

--------------------

Introduction to Optical Spectroscopic Techniques and Applications to Low Dimensional Semiconductors

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Elena Stellino

- abstract: The course aims to introduce three of the most common techniques in optical spectroscopy (Infrared, Raman, and photoluminescence) by providing a comprehensive approach that begins with theoretical foundations and then leads the students to handle real experimental cases. The key objectives of the course include: • gain familiarity with the theoretical framework underpinning the presented spectroscopic techniques, covering phenomenology, classical models, and some aspects of the quantum approach; • identify what kind of physical information can be extracted from the theoretical models; • understand the working principles governing the setups used in experiments; • engage in actual experimental work within research laboratories specialized in optical spectroscopies. This involves sample preparation, data acquisition, and the use of softwares for the data analysis. The course comprises frontal lessons (50%) and supervised laboratory experiences with data analysis activities (50%). The laboratory sessions will focus on samples belonging to the class of 2D semiconductors, exposing students to one of the most studied research topics in the field of material science.

modalità di accertamento finale: Students, organized into groups, must prepare written reports for each teaching module detailing the laboratory experience, scientific case, experimental setup, data analysis, and result interpretation. At the course's conclusion, each student must present and discuss a scientific article from the literature that utilizes one of the discussed techniques.

- Exam: Evaluation will be based on three group reports and one individual presentation, resulting in a grade from 0 to 30. The final mark is an average of these four scores.

--------------------

The Dirichlet problem for elliptic equations with rough data

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Francescantonio Oliva

- abstract: We will first consider Dirichlet problems associated to elliptic equations whose principal operator is in divergence form with bounded coefficients. We briefly present the weak setting for these equations with a measurable function f belonging to a suitable Lebesgue space or even a Radon measure as a source datum. In accordance with the regularity of f we introduce the concept of weak, distributional and renormalized solution and we prove their well-posedness. In the second part, we deal with source terms of the form f(x)h(u) which can also depend on the solution u itself. We deal with the case of a function h(s) which has a finite limit at infinity, continuous and possibly blowing up at the s=0; as a prototypical example one should have in mind a negative power. For these equations we show existence, regularity, and uniqueness of finite and infinite energy solutions. If the time allows, we could also deal with the case of equations involving first order terms with natural growth with respect to the gradient. Depending on the attendees background knowledge, the course will mainly focus on the first and/or the second part.

- Exam: Seminar

--------------------

An introduction to cluster algebras

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 16

- docente del corso: Giovanni Cerulli Irelli

- abstract: We review the theory of cluster algebras intiated by Fomin and Zelevinsky in 2001 and its connection with the rapresentation theory of associative algebras, following Derksen, Weyman and Zelevinsky.

- Exam: Seminar

--------------------

Analytical Techniques for Wave Phenomena

- data presunta: 09-10/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 36

- docente del corso: Paolo Burghignoli

- 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 for the asymptotic evaluation of integrals are then illustrated, including the Laplace and saddle-point methods. Applications are focused on the analysis of time-harmonic waves excited in planar layered structures by canonical sources and on scattering from half planes and spheres. As concerns the former, different wave species will be defined and physically discussed (space waves, surface waves, leaky waves, lateral waves). As concerns the latter, the Wiener-Hopf method and the Watson transformation will be introduced.

- Exam: Oral discussion of course's topics.

--------------------

Nanophotonics and Plasmonics

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

--------------------

Basics of Nonlinear Optics

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

--------------------

Experiences in Optics

- data presunta: 11/2024-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Alessandro Belardini

- abstract: The course gives the theoretical basis of optics, geometrical optics and physical optics, polarisation, diffraction, interference, use of simple optical elements such as lenses, prisms, polarisers, waveplate. After the theoretical introduction, the course provides a series of optics laboratory experiences were the students can experimentally verify the laws of optics that they have studied. The experiences are divided into three groups. The first concerns geometric optics, in particular Snell's law. The second and third groups concern physical optics, in particular polarization, interference and diffraction.

- Exam: Oral discussion of course's topics.

--------------------------------

Numerical methods for simulations of electromagnetic wave – matter interactions

- data presunta: First semester

- numero ore: 20

- docente del corso: Emilija Petronijevic (Sapienza)

- abstract: The course gives practical basis for the numerical investigation of interaction between matter and electromagnetic waves in different spectral ranges. After the theoretical introduction focusing on finite difference time domain, the course makes use of a commercial solver to show how different materials, in micro-and nanoscaled geometries, tailor electromagnetic wave distribution. The course then provides two simulation experiences: the first treats a single nanostructure, and the second periodically organized nanostructures. Both experiences treat transmission and absorption of the waves, near- and far-field spatial and spectral properties, electromagnetic behavior at resonances, and the influence of the excitation wave polarization.

- Exam: discussion of a course topic

---------------------

Modelling and Simulations of Collective Dynamics

- data presunta: Second semester

- numero ore: 20

- docente del corso: Marta Menci (Università Campus Bio-Medico di Roma)

- abstract: The study of collective dynamics is attracting the interest of different research fields, both due to their wide range of applications and to their ability to model self-organization. The emergence of global patterns from local interactions can be easily observed in flock of birds, schools of fish, human crowds, but also cells exhibit collective behaviors in different biological processes characterizing the human body (e.g. in embryogenesis, wound healing, immune response, tumor growth). The main feature of collective cells migration is that the emergent behavior is also driven by chemical stimuli, and not only by mechanical interactions. This course aims to give participants a brief but complete introduction to the research field of modelling and simulation of collective dynamics. Starting with a survey of influential works of the literature, recent mathematical developments and new directions and applications will be presented. A specific focus will be on different numerical techniques proposed to simulate the different kind of equations involved in the presented models.

- Exam: final project on a specific topic

Elenco delle attività formative previste per i dottorandi del secondo anno:

------------------------

Spectral Geometry

- data presunta: 03-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 24

- docente del corso: Luigi Provenzano (Sapienza) e Davide Buoso (U. Piemonte Orientale)

- abstract: In the first part of the course we will provide a brief introduction to the spectrum of the Laplacian on Euclidean domains and Riemannian manifolds, along with a few basic examples, and their relations with physical phenomena (waves and vibrations). Then, we will focus on some classical problems in spectral geometry, such as eigenvalue bounds and isoperimetric inequalities for the eigenvalues. The questions that we will address are the following: how does the geometry and the topology of the ambient space influence the spectrum (the eigenvalues)? On the other hand, what information can give the knowledge of the spectrum on the geometry and topology of the ambient space? We will present a few classical techniques which have been adopted throughout the years to address these questions. In the final part of the course (if time allows) we will consider some recent developments on old and new problems, and we will present some open questions.

- Exam: Short seminar/report on a research paper (possibly close to the student's interests)

--------------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, I

- data presunta: 02-03/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- abstract: Introduction to the photoelectron spectroscopy: theoretical background, the three-step model, atoms and molecules, low-dimensional solid systems, experiments with angular resolution, time-resolved experiments. Instrumentation: charged particles, Auger electron spectroscopy and resonant photoemission. Surfaces and low-dimensional systems, electronic properties. Core-level photoemission and surface core-level shifts. Angular resolved photoemission, electronic band structure. Band structure of exemplary 1D and 2D systems.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, II

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- abstract: Electromagnetic radiation sources, synchrotron radiation, theoretical background, storage rings, beamlines, photoemission. Introduction to the free-electron laser: a coherent source of radiation from UV to X rays. X ray absorption spectroscopy, theoretical background of absorption. Multiple scattering theory: a method for the observation of the electronic states and spectroscopy measurements. EXAFS and XANES/NEXAFS: fundamentals and applications. X ray elastic and anelastic scattering. High energy photoemission, application to buried interfaces/materials.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

---------------------------

Advances in Raman spectroscopy: from traditional vibrational spectroscopy to surface enhanced approaches

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 10

- docente del corso: Angela CAPOCEFALO

- abstract: The aim of the course is to provide doctoral students with a thorough understanding of Raman spectroscopy, covering both the traditional technique and the more advanced surface enhanced Raman spectroscopy. After introducing the fundamentals of the Raman scattering, the experimental aspects of the technique will be examined, including the description of the measurement apparatus and the analysis and interpretation of data. Surface enhanced Raman spectroscopy will be then introduced, discussing the different mechanisms underlying signal amplification. The plasmonic properties of commonly used nanostructured metal substrates and the recent advances in the technique, such as tip-enhanced Raman spectroscopy will be presented. Finally, innovative applications of the technique in various research fields such as sensing, nanomedicine, materials science, and cultural heritage will be discussed.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

---------------------

Neural Networks & Machine Learning

- data presunta: 01-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Adriano Barra

- abstract: The course is meant to provide theoretical tools (both mathematical and computational) to allow the students to orient themselves in the proliferation of neural network techniques and machine learning algorithms that are nowadays broadly used in the processing of data and signals both in the world of research as well as in industry. Specifically, once shared the main mathematical methodological bases (a quick review of elements of probability and statistics), after a succinct historical introduction (e.g. the Turing machine, Rosenblatt's perceptron and AI’s winter time), modern neural networks will be addressed, both those biologically inspired (e.g. the Hopfield model and its variations on the theme) as well as those not-biologically inspired (Boltzmann machines and feed-forward networks), with the related algorithms for learning and automatic recognition (e.g., contrastive divergence and back -prograpation). The ultimate aim of the course is to share the salient concepts with the students and, at the same time, to provide them with the key tools, so that they can keep growing within the field of Artificial Intelligence and Machine Learning: this transfer of information will be supplied both from a formal/mathematical point of view (e.g. showing during the course clear methods for setting up a relevant problem and solving it appropriately) and from a logical/deductive point of view (e.g. understanding what it is reasonable to be addressed by modern techniques of Machine Learning). To this end the course program is divided into two main sections. The former is to ensure that we share basic scientific knowledge (obviously a necessary pre-requisite to guarantee that we understand information processing in neural networks from a mathematical perspective later on). The latter is completely dedicated to neural networks: after a succinct description (always in mathematical terms) of the key mechanisms inherent to the neuron and the propagation of information between neurons, "networks of neurons" will be built (in other words they will explain " what are” -mathematically speaking- these neural networks) and we will study their emergent properties (i.e. those not immediately deducible by looking at the behavior of the single neuron): specifically, we will try to see how these networks are able to learn and abstract by looking at supplied examples from the external world and how, subsequently, they use what they have learned to respond appropriately, if stimulated, to the external world. We will also understand how these neural networks can sometimes make mistakes, and why. Ideally at the end of the course the students should be able to independently continue in-depth study of this discipline and benefit from it accordingly during their careers

- Exam: Oral

----------------------------

Advances in Raman spectroscopy: from traditional vibrational spectroscopy to surface enhanced approaches

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Elena Stellino

- abstract: The course is meant to provide theoretical tools (both mathematical and computational) to allow the students to orient themselves in the proliferation of neural network techniques and machine learning algorithms that are nowadays broadly used in the processing of data and signals both in the world of research as well as in industry. Specifically, once shared the main mathematical methodological bases (a quick review of elements of probability and statistics), after a succinct historical introduction (e.g. the Turing machine, Rosenblatt's perceptron and AI’s winter time), modern neural networks will be addressed, both those biologically inspired (e.g. the Hopfield model and its variations on the theme) as well as those not-biologically inspired (Boltzmann machines and feed-forward networks), with the related algorithms for learning and automatic recognition (e.g., contrastive divergence and back -prograpation). The ultimate aim of the course is to share the salient concepts with the students and, at the same time, to provide them with the key tools, so that they can keep growing within the field of Artificial Intelligence and Machine Learning: this transfer of information will be supplied both from a formal/mathematical point of view (e.g. showing during the course clear methods for setting up a relevant problem and solving it appropriately) and from a logical/deductive point of view (e.g. understanding what it is reasonable to be addressed by modern techniques of Machine Learning). To this end the course program is divided into two main sections. The former is to ensure that we share basic scientific knowledge (obviously a necessary pre-requisite to guarantee that we understand information processing in neural networks from a mathematical perspective later on). The latter is completely dedicated to neural networks: after a succinct description (always in mathematical terms) of the key mechanisms inherent to the neuron and the propagation of information between neurons, "networks of neurons" will be built (in other words they will explain " what are” -mathematically speaking- these neural networks) and we will study their emergent properties (i.e. those not immediately deducible by looking at the behavior of the single neuron): specifically, we will try to see how these networks are able to learn and abstract by looking at supplied examples from the external world and how, subsequently, they use what they have learned to respond appropriately, if stimulated, to the external world. We will also understand how these neural networks can sometimes make mistakes, and why. Ideally at the end of the course the students should be able to independently continue in-depth study of this discipline and benefit from it accordingly during their careers.

modalità di accertamento finale: Students, organized into groups, must prepare written reports for each teaching module detailing the laboratory experience, scientific case, experimental setup, data analysis, and result interpretation. At the course's conclusion, each student must present and discuss a scientific article from the literature that utilizes one of the discussed techniques.

- Exam: Evaluation will be based on three group reports and one individual presentation, resulting in a grade from 0 to 30. The final mark is an average of these four scores.

--------------------------

On the Dirichlet problem for elliptic equations with rough data

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Francescantonio Oliva

- abstract: We will first consider Dirichlet problems associated to elliptic equations whose principal operator is in divergence form with bounded coefficients. We briefly present the weak setting for these equations with a measurable function f belonging to a suitable Lebesgue space or even a Radon measure as a source datum. In accordance with the regularity of f we introduce the concept of weak, distributional and renormalized solution and we prove their well-posedness. In the second part, we deal with source terms of the form f(x)h(u) which can also depend on the solution u itself. We deal with the case of a function h(s) which has a finite limit at infinity, continuous and possibly blowing up at the s=0; as a prototypical example one should have in mind a negative power. For these equations we show existence, regularity, and uniqueness of finite and infinite energy solutions. If the time allows, we could also deal with the case of equations involving first order terms with natural growth with respect to the gradient. Depending on the attendees background knowledge, the course will mainly focus on the first and/or the second part.

- Exam: Seminar

-------------------------

An introduction to cluster algebras

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 16

- docente del corso: Giovanni Cerulli Irelli

- abstract: We review the theory of cluster algebras intiated by Fomin and Zelevinsky in 2001 and its connection with the rapresentation theory of associative algebras, following Derksen, Weyman and Zelevinsky.

- Exam: Seminar

----------------------

Analytical Techniques for Wave Phenomena

- data presunta: 09-10/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 36

- docente del corso: Paolo Burghignoli

- 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 for the asymptotic evaluation of integrals are then illustrated, including the Laplace and saddle-point methods. Applications are focused on the analysis of time-harmonic waves excited in planar layered structures by canonical sources and on scattering from half planes and spheres. As concerns the former, different wave species will be defined and physically discussed (space waves, surface waves, leaky waves, lateral waves). As concerns the latter, the Wiener-Hopf method and the Watson transformation will be introduced.

- Exam: Oral discussion of course's topics.

----------------------

Nanophotonics and Plasmonics

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

------------------------

Basics of Nonlinear Optics

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

-----------------

Experiences in Optics

- data presunta: 11/2024-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Alessandro Belardini

- abstract: The course gives the theoretical basis of optics, geometrical optics and physical optics, polarisation, diffraction, interference, use of simple optical elements such as lenses, prisms, polarisers, waveplate. After the theoretical introduction, the course provides a series of optics laboratory experiences were the students can experimentally verify the laws of optics that they have studied. The experiences are divided into three groups. The first concerns geometric optics, in particular Snell's law. The second and third groups concern physical optics, in particular polarization, interference and diffraction.

- Exam: Oral discussion of course's topics.

-----------------------

Numerical methods for simulations of electromagnetic wave – matter interactions

- data presunta: First semester

- numero ore: 20

- docente del corso: Emilija Petronijevic (Sapienza)

- abstract: The course gives practical basis for the numerical investigation of interaction between matter and electromagnetic waves in different spectral ranges. After the theoretical introduction focusing on finite difference time domain, the course makes use of a commercial solver to show how different materials, in micro-and nanoscaled geometries, tailor electromagnetic wave distribution. The course then provides two simulation experiences: the first treats a single nanostructure, and the second periodically organized nanostructures. Both experiences treat transmission and absorption of the waves, near- and far-field spatial and spectral properties, electromagnetic behavior at resonances, and the influence of the excitation wave polarization.

- Exam: discussion of a course topic

-----------------

Modelling and Simulations of Collective Dynamics

- data presunta: Second semester

- numero ore: 20

- docente del corso: Marta Menci (Università Campus Bio-Medico di Roma)

- abstract: The study of collective dynamics is attracting the interest of different research fields, both due to their wide range of applications and to their ability to model self-organization. The emergence of global patterns from local interactions can be easily observed in flock of birds, schools of fish, human crowds, but also cells exhibit collective behaviors in different biological processes characterizing the human body (e.g. in embryogenesis, wound healing, immune response, tumor growth). The main feature of collective cells migration is that the emergent behavior is also driven by chemical stimuli, and not only by mechanical interactions. This course aims to give participants a brief but complete introduction to the research field of modelling and simulation of collective dynamics. Starting with a survey of influential works of the literature, recent mathematical developments and new directions and applications will be presented. A specific focus will be on different numerical techniques proposed to simulate the different kind of equations involved in the presented models.

- Exam: final project on a specific topic

---------------------

Spectral Geometry

- data presunta: 03-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 24

- docente del corso: Luigi Provenzano (Sapienza) e Davide Buoso (U. Piemonte Orientale)

- abstract: In the first part of the course we will provide a brief introduction to the spectrum of the Laplacian on Euclidean domains and Riemannian manifolds, along with a few basic examples, and their relations with physical phenomena (waves and vibrations). Then, we will focus on some classical problems in spectral geometry, such as eigenvalue bounds and isoperimetric inequalities for the eigenvalues. The questions that we will address are the following: how does the geometry and the topology of the ambient space influence the spectrum (the eigenvalues)? On the other hand, what information can give the knowledge of the spectrum on the geometry and topology of the ambient space? We will present a few classical techniques which have been adopted throughout the years to address these questions. In the final part of the course (if time allows) we will consider some recent developments on old and new problems, and we will present some open questions.

- Exam: Short seminar/report on a research paper (possibly close to the student's interests)

--------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, I

- data presunta: 02-03/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- Abstract: Introduction to the photoelectron spectroscopy: theoretical background, the three-step model, atoms and molecules, low-dimensional solid systems, experiments with angular resolution, time-resolved experiments. Instrumentation: charged particles, Auger electron spectroscopy and resonant photoemission. Surfaces and low-dimensional systems, electronic properties. Core-level photoemission and surface core-level shifts. Angular resolved photoemission, electronic band structure. Band structure of exemplary 1D and 2D systems.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, II

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- abstract: Electromagnetic radiation sources, synchrotron radiation, theoretical background, storage rings, beamlines, photoemission. Introduction to the free-electron laser: a coherent source of radiation from UV to X rays. X ray absorption spectroscopy, theoretical background of absorption. Multiple scattering theory: a method for the observation of the electronic states and spectroscopy measurements. EXAFS and XANES/NEXAFS: fundamentals and applications. X ray elastic and anelastic scattering. High energy photoemission, application to buried interfaces/materials.

Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------

Advances in Raman spectroscopy: from traditional vibrational spectroscopy to surface enhanced approaches

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 10

- docente del corso: Angela CAPOCEFALO

- abstract: The aim of the course is to provide doctoral students with a thorough understanding of Raman spectroscopy, covering both the traditional technique and the more advanced surface enhanced Raman spectroscopy. After introducing the fundamentals of the Raman scattering, the experimental aspects of the technique will be examined, including the description of the measurement apparatus and the analysis and interpretation of data. Surface enhanced Raman spectroscopy will be then introduced, discussing the different mechanisms underlying signal amplification. The plasmonic properties of commonly used nanostructured metal substrates and the recent advances in the technique, such as tip-enhanced Raman spectroscopy will be presented. Finally, innovative applications of the technique in various research fields such as sensing, nanomedicine, materials science, and cultural heritage will be discussed.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------

Neural Networks & Machine Learning

- data presunta: 01-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Adriano Barra qualifica

- abstract: The course is meant to provide theoretical tools (both mathematical and computational) to allow the students to orient themselves in the proliferation of neural network techniques and machine learning algorithms that are nowadays broadly used in the processing of data and signals both in the world of research as well as in industry. Specifically, once shared the main mathematical methodological bases (a quick review of elements of probability and statistics), after a succinct historical introduction (e.g. the Turing machine, Rosenblatt's perceptron and AI’s winter time), modern neural networks will be addressed, both those biologically inspired (e.g. the Hopfield model and its variations on the theme) as well as those not-biologically inspired (Boltzmann machines and feed-forward networks), with the related algorithms for learning and automatic recognition (e.g., contrastive divergence and back -prograpation). The ultimate aim of the course is to share the salient concepts with the students and, at the same time, to provide them with the key tools, so that they can keep growing within the field of Artificial Intelligence and Machine Learning: this transfer of information will be supplied both from a formal/mathematical point of view (e.g. showing during the course clear methods for setting up a relevant problem and solving it appropriately) and from a logical/deductive point of view (e.g. understanding what it is reasonable to be addressed by modern techniques of Machine Learning). To this end the course program is divided into two main sections. The former is to ensure that we share basic scientific knowledge (obviously a necessary pre-requisite to guarantee that we understand information processing in neural networks from a mathematical perspective later on). The latter is completely dedicated to neural networks: after a succinct description (always in mathematical terms) of the key mechanisms inherent to the neuron and the propagation of information between neurons, "networks of neurons" will be built (in other words they will explain " what are” -mathematically speaking- these neural networks) and we will study their emergent properties (i.e. those not immediately deducible by looking at the behavior of the single neuron): specifically, we will try to see how these networks are able to learn and abstract by looking at supplied examples from the external world and how, subsequently, they use what they have learned to respond appropriately, if stimulated, to the external world. We will also understand how these neural networks can sometimes make mistakes, and why. Ideally at the end of the course the students should be able to independently continue in-depth study of this discipline and benefit from it accordingly during their careers

- Exam: Oral

--------------------

Introduction to Optical Spectroscopic Techniques and Applications to Low Dimensional Semiconductors

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Elena Stellino

- abstract: The course aims to introduce three of the most common techniques in optical spectroscopy (Infrared, Raman, and photoluminescence) by providing a comprehensive approach that begins with theoretical foundations and then leads the students to handle real experimental cases. The key objectives of the course include: • gain familiarity with the theoretical framework underpinning the presented spectroscopic techniques, covering phenomenology, classical models, and some aspects of the quantum approach; • identify what kind of physical information can be extracted from the theoretical models; • understand the working principles governing the setups used in experiments; • engage in actual experimental work within research laboratories specialized in optical spectroscopies. This involves sample preparation, data acquisition, and the use of softwares for the data analysis. The course comprises frontal lessons (50%) and supervised laboratory experiences with data analysis activities (50%). The laboratory sessions will focus on samples belonging to the class of 2D semiconductors, exposing students to one of the most studied research topics in the field of material science.

modalità di accertamento finale: Students, organized into groups, must prepare written reports for each teaching module detailing the laboratory experience, scientific case, experimental setup, data analysis, and result interpretation. At the course's conclusion, each student must present and discuss a scientific article from the literature that utilizes one of the discussed techniques.

- Exam: Evaluation will be based on three group reports and one individual presentation, resulting in a grade from 0 to 30. The final mark is an average of these four scores.

--------------------

The Dirichlet problem for elliptic equations with rough data

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Francescantonio Oliva

- abstract: We will first consider Dirichlet problems associated to elliptic equations whose principal operator is in divergence form with bounded coefficients. We briefly present the weak setting for these equations with a measurable function f belonging to a suitable Lebesgue space or even a Radon measure as a source datum. In accordance with the regularity of f we introduce the concept of weak, distributional and renormalized solution and we prove their well-posedness. In the second part, we deal with source terms of the form f(x)h(u) which can also depend on the solution u itself. We deal with the case of a function h(s) which has a finite limit at infinity, continuous and possibly blowing up at the s=0; as a prototypical example one should have in mind a negative power. For these equations we show existence, regularity, and uniqueness of finite and infinite energy solutions. If the time allows, we could also deal with the case of equations involving first order terms with natural growth with respect to the gradient. Depending on the attendees background knowledge, the course will mainly focus on the first and/or the second part.

- Exam: Seminar

--------------------

An introduction to cluster algebras

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 16

- docente del corso: Giovanni Cerulli Irelli

- abstract: We review the theory of cluster algebras intiated by Fomin and Zelevinsky in 2001 and its connection with the rapresentation theory of associative algebras, following Derksen, Weyman and Zelevinsky.

- Exam: Seminar

--------------------

Analytical Techniques for Wave Phenomena

- data presunta: 09-10/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 36

- docente del corso: Paolo Burghignoli

- 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 for the asymptotic evaluation of integrals are then illustrated, including the Laplace and saddle-point methods. Applications are focused on the analysis of time-harmonic waves excited in planar layered structures by canonical sources and on scattering from half planes and spheres. As concerns the former, different wave species will be defined and physically discussed (space waves, surface waves, leaky waves, lateral waves). As concerns the latter, the Wiener-Hopf method and the Watson transformation will be introduced.

- Exam: Oral discussion of course's topics.

--------------------

Nanophotonics and Plasmonics

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

--------------------

Basics of Nonlinear Optics

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

--------------------

Experiences in Optics

- data presunta: 11/2024-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Alessandro Belardini

- abstract: The course gives the theoretical basis of optics, geometrical optics and physical optics, polarisation, diffraction, interference, use of simple optical elements such as lenses, prisms, polarisers, waveplate. After the theoretical introduction, the course provides a series of optics laboratory experiences were the students can experimentally verify the laws of optics that they have studied. The experiences are divided into three groups. The first concerns geometric optics, in particular Snell's law. The second and third groups concern physical optics, in particular polarization, interference and diffraction.

- Exam: Oral discussion of course's topics.

--------------------------------

Numerical methods for simulations of electromagnetic wave – matter interactions

- data presunta: First semester

- numero ore: 20

- docente del corso: Emilija Petronijevic (Sapienza)

- abstract: The course gives practical basis for the numerical investigation of interaction between matter and electromagnetic waves in different spectral ranges. After the theoretical introduction focusing on finite difference time domain, the course makes use of a commercial solver to show how different materials, in micro-and nanoscaled geometries, tailor electromagnetic wave distribution. The course then provides two simulation experiences: the first treats a single nanostructure, and the second periodically organized nanostructures. Both experiences treat transmission and absorption of the waves, near- and far-field spatial and spectral properties, electromagnetic behavior at resonances, and the influence of the excitation wave polarization.

- Exam: discussion of a course topic

---------------------

Modelling and Simulations of Collective Dynamics

- data presunta: Second semester

- numero ore: 20

- docente del corso: Marta Menci (Università Campus Bio-Medico di Roma)

- abstract: The study of collective dynamics is attracting the interest of different research fields, both due to their wide range of applications and to their ability to model self-organization. The emergence of global patterns from local interactions can be easily observed in flock of birds, schools of fish, human crowds, but also cells exhibit collective behaviors in different biological processes characterizing the human body (e.g. in embryogenesis, wound healing, immune response, tumor growth). The main feature of collective cells migration is that the emergent behavior is also driven by chemical stimuli, and not only by mechanical interactions. This course aims to give participants a brief but complete introduction to the research field of modelling and simulation of collective dynamics. Starting with a survey of influential works of the literature, recent mathematical developments and new directions and applications will be presented. A specific focus will be on different numerical techniques proposed to simulate the different kind of equations involved in the presented models.

- Exam: final project on a specific topic

Elenco delle attività formative previste per i dottorandi del secondo anno:

------------------------

Spectral Geometry

- data presunta: 03-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 24

- docente del corso: Luigi Provenzano (Sapienza) e Davide Buoso (U. Piemonte Orientale)

- abstract: In the first part of the course we will provide a brief introduction to the spectrum of the Laplacian on Euclidean domains and Riemannian manifolds, along with a few basic examples, and their relations with physical phenomena (waves and vibrations). Then, we will focus on some classical problems in spectral geometry, such as eigenvalue bounds and isoperimetric inequalities for the eigenvalues. The questions that we will address are the following: how does the geometry and the topology of the ambient space influence the spectrum (the eigenvalues)? On the other hand, what information can give the knowledge of the spectrum on the geometry and topology of the ambient space? We will present a few classical techniques which have been adopted throughout the years to address these questions. In the final part of the course (if time allows) we will consider some recent developments on old and new problems, and we will present some open questions.

- Exam: Short seminar/report on a research paper (possibly close to the student's interests)

--------------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, I

- data presunta: 02-03/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- abstract: Introduction to the photoelectron spectroscopy: theoretical background, the three-step model, atoms and molecules, low-dimensional solid systems, experiments with angular resolution, time-resolved experiments. Instrumentation: charged particles, Auger electron spectroscopy and resonant photoemission. Surfaces and low-dimensional systems, electronic properties. Core-level photoemission and surface core-level shifts. Angular resolved photoemission, electronic band structure. Band structure of exemplary 1D and 2D systems.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

--------------------------

Radiation-Matter Interaction, Photoemission and Photoabsorption Spectroscopy, II

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Carlo MARIANI (PO Sapienza), Settimio MOBILIO (emerito, Roma Tre), Francesco OFFI (PA, Roma Tre), Alessandro RUOCCO (PA, Roma Tre)

- abstract: Electromagnetic radiation sources, synchrotron radiation, theoretical background, storage rings, beamlines, photoemission. Introduction to the free-electron laser: a coherent source of radiation from UV to X rays. X ray absorption spectroscopy, theoretical background of absorption. Multiple scattering theory: a method for the observation of the electronic states and spectroscopy measurements. EXAFS and XANES/NEXAFS: fundamentals and applications. X ray elastic and anelastic scattering. High energy photoemission, application to buried interfaces/materials.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

---------------------------

Advances in Raman spectroscopy: from traditional vibrational spectroscopy to surface enhanced approaches

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 10

- docente del corso: Angela CAPOCEFALO

- abstract: The aim of the course is to provide doctoral students with a thorough understanding of Raman spectroscopy, covering both the traditional technique and the more advanced surface enhanced Raman spectroscopy. After introducing the fundamentals of the Raman scattering, the experimental aspects of the technique will be examined, including the description of the measurement apparatus and the analysis and interpretation of data. Surface enhanced Raman spectroscopy will be then introduced, discussing the different mechanisms underlying signal amplification. The plasmonic properties of commonly used nanostructured metal substrates and the recent advances in the technique, such as tip-enhanced Raman spectroscopy will be presented. Finally, innovative applications of the technique in various research fields such as sensing, nanomedicine, materials science, and cultural heritage will be discussed.

- Exam: oral presentation of a current research topic which uses the methods presented in the course.

---------------------

Neural Networks & Machine Learning

- data presunta: 01-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Adriano Barra

- abstract: The course is meant to provide theoretical tools (both mathematical and computational) to allow the students to orient themselves in the proliferation of neural network techniques and machine learning algorithms that are nowadays broadly used in the processing of data and signals both in the world of research as well as in industry. Specifically, once shared the main mathematical methodological bases (a quick review of elements of probability and statistics), after a succinct historical introduction (e.g. the Turing machine, Rosenblatt's perceptron and AI’s winter time), modern neural networks will be addressed, both those biologically inspired (e.g. the Hopfield model and its variations on the theme) as well as those not-biologically inspired (Boltzmann machines and feed-forward networks), with the related algorithms for learning and automatic recognition (e.g., contrastive divergence and back -prograpation). The ultimate aim of the course is to share the salient concepts with the students and, at the same time, to provide them with the key tools, so that they can keep growing within the field of Artificial Intelligence and Machine Learning: this transfer of information will be supplied both from a formal/mathematical point of view (e.g. showing during the course clear methods for setting up a relevant problem and solving it appropriately) and from a logical/deductive point of view (e.g. understanding what it is reasonable to be addressed by modern techniques of Machine Learning). To this end the course program is divided into two main sections. The former is to ensure that we share basic scientific knowledge (obviously a necessary pre-requisite to guarantee that we understand information processing in neural networks from a mathematical perspective later on). The latter is completely dedicated to neural networks: after a succinct description (always in mathematical terms) of the key mechanisms inherent to the neuron and the propagation of information between neurons, "networks of neurons" will be built (in other words they will explain " what are” -mathematically speaking- these neural networks) and we will study their emergent properties (i.e. those not immediately deducible by looking at the behavior of the single neuron): specifically, we will try to see how these networks are able to learn and abstract by looking at supplied examples from the external world and how, subsequently, they use what they have learned to respond appropriately, if stimulated, to the external world. We will also understand how these neural networks can sometimes make mistakes, and why. Ideally at the end of the course the students should be able to independently continue in-depth study of this discipline and benefit from it accordingly during their careers

- Exam: Oral

----------------------------

Advances in Raman spectroscopy: from traditional vibrational spectroscopy to surface enhanced approaches

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 30

- docente del corso: Elena Stellino

- abstract: The course is meant to provide theoretical tools (both mathematical and computational) to allow the students to orient themselves in the proliferation of neural network techniques and machine learning algorithms that are nowadays broadly used in the processing of data and signals both in the world of research as well as in industry. Specifically, once shared the main mathematical methodological bases (a quick review of elements of probability and statistics), after a succinct historical introduction (e.g. the Turing machine, Rosenblatt's perceptron and AI’s winter time), modern neural networks will be addressed, both those biologically inspired (e.g. the Hopfield model and its variations on the theme) as well as those not-biologically inspired (Boltzmann machines and feed-forward networks), with the related algorithms for learning and automatic recognition (e.g., contrastive divergence and back -prograpation). The ultimate aim of the course is to share the salient concepts with the students and, at the same time, to provide them with the key tools, so that they can keep growing within the field of Artificial Intelligence and Machine Learning: this transfer of information will be supplied both from a formal/mathematical point of view (e.g. showing during the course clear methods for setting up a relevant problem and solving it appropriately) and from a logical/deductive point of view (e.g. understanding what it is reasonable to be addressed by modern techniques of Machine Learning). To this end the course program is divided into two main sections. The former is to ensure that we share basic scientific knowledge (obviously a necessary pre-requisite to guarantee that we understand information processing in neural networks from a mathematical perspective later on). The latter is completely dedicated to neural networks: after a succinct description (always in mathematical terms) of the key mechanisms inherent to the neuron and the propagation of information between neurons, "networks of neurons" will be built (in other words they will explain " what are” -mathematically speaking- these neural networks) and we will study their emergent properties (i.e. those not immediately deducible by looking at the behavior of the single neuron): specifically, we will try to see how these networks are able to learn and abstract by looking at supplied examples from the external world and how, subsequently, they use what they have learned to respond appropriately, if stimulated, to the external world. We will also understand how these neural networks can sometimes make mistakes, and why. Ideally at the end of the course the students should be able to independently continue in-depth study of this discipline and benefit from it accordingly during their careers.

modalità di accertamento finale: Students, organized into groups, must prepare written reports for each teaching module detailing the laboratory experience, scientific case, experimental setup, data analysis, and result interpretation. At the course's conclusion, each student must present and discuss a scientific article from the literature that utilizes one of the discussed techniques.

- Exam: Evaluation will be based on three group reports and one individual presentation, resulting in a grade from 0 to 30. The final mark is an average of these four scores.

--------------------------

On the Dirichlet problem for elliptic equations with rough data

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Francescantonio Oliva

- abstract: We will first consider Dirichlet problems associated to elliptic equations whose principal operator is in divergence form with bounded coefficients. We briefly present the weak setting for these equations with a measurable function f belonging to a suitable Lebesgue space or even a Radon measure as a source datum. In accordance with the regularity of f we introduce the concept of weak, distributional and renormalized solution and we prove their well-posedness. In the second part, we deal with source terms of the form f(x)h(u) which can also depend on the solution u itself. We deal with the case of a function h(s) which has a finite limit at infinity, continuous and possibly blowing up at the s=0; as a prototypical example one should have in mind a negative power. For these equations we show existence, regularity, and uniqueness of finite and infinite energy solutions. If the time allows, we could also deal with the case of equations involving first order terms with natural growth with respect to the gradient. Depending on the attendees background knowledge, the course will mainly focus on the first and/or the second part.

- Exam: Seminar

-------------------------

An introduction to cluster algebras

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 16

- docente del corso: Giovanni Cerulli Irelli

- abstract: We review the theory of cluster algebras intiated by Fomin and Zelevinsky in 2001 and its connection with the rapresentation theory of associative algebras, following Derksen, Weyman and Zelevinsky.

- Exam: Seminar

----------------------

Analytical Techniques for Wave Phenomena

- data presunta: 09-10/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 36

- docente del corso: Paolo Burghignoli

- 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 for the asymptotic evaluation of integrals are then illustrated, including the Laplace and saddle-point methods. Applications are focused on the analysis of time-harmonic waves excited in planar layered structures by canonical sources and on scattering from half planes and spheres. As concerns the former, different wave species will be defined and physically discussed (space waves, surface waves, leaky waves, lateral waves). As concerns the latter, the Wiener-Hopf method and the Watson transformation will be introduced.

- Exam: Oral discussion of course's topics.

----------------------

Nanophotonics and Plasmonics

- data presunta: 03-04/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

------------------------

Basics of Nonlinear Optics

- data presunta: 04-05/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Concita Sibilia

- 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.

- Exam: Oral discussion of course's topics.

-----------------

Experiences in Optics

- data presunta: 11/2024-02/2025

- modalità di erogazione: Ex-cathedra

- numero ore: 20

- docente del corso: Alessandro Belardini

- abstract: The course gives the theoretical basis of optics, geometrical optics and physical optics, polarisation, diffraction, interference, use of simple optical elements such as lenses, prisms, polarisers, waveplate. After the theoretical introduction, the course provides a series of optics laboratory experiences were the students can experimentally verify the laws of optics that they have studied. The experiences are divided into three groups. The first concerns geometric optics, in particular Snell's law. The second and third groups concern physical optics, in particular polarization, interference and diffraction.

- Exam: Oral discussion of course's topics.

-----------------------

Numerical methods for simulations of electromagnetic wave – matter interactions

- data presunta: First semester

- numero ore: 20

- docente del corso: Emilija Petronijevic (Sapienza)

- abstract: The course gives practical basis for the numerical investigation of interaction between matter and electromagnetic waves in different spectral ranges. After the theoretical introduction focusing on finite difference time domain, the course makes use of a commercial solver to show how different materials, in micro-and nanoscaled geometries, tailor electromagnetic wave distribution. The course then provides two simulation experiences: the first treats a single nanostructure, and the second periodically organized nanostructures. Both experiences treat transmission and absorption of the waves, near- and far-field spatial and spectral properties, electromagnetic behavior at resonances, and the influence of the excitation wave polarization.

- Exam: discussion of a course topic

-----------------

Modelling and Simulations of Collective Dynamics

- data presunta: Second semester

- numero ore: 20

- docente del corso: Marta Menci (Università Campus Bio-Medico di Roma)

- abstract: The study of collective dynamics is attracting the interest of different research fields, both due to their wide range of applications and to their ability to model self-organization. The emergence of global patterns from local interactions can be easily observed in flock of birds, schools of fish, human crowds, but also cells exhibit collective behaviors in different biological processes characterizing the human body (e.g. in embryogenesis, wound healing, immune response, tumor growth). The main feature of collective cells migration is that the emergent behavior is also driven by chemical stimuli, and not only by mechanical interactions. This course aims to give participants a brief but complete introduction to the research field of modelling and simulation of collective dynamics. Starting with a survey of influential works of the literature, recent mathematical developments and new directions and applications will be presented. A specific focus will be on different numerical techniques proposed to simulate the different kind of equations involved in the presented models.

- Exam: final project on a specific topic