PhD Mathematics and Models

General info

Announcements

Call for applications

Regulations

News

Call for 1 Ph.D. fellowship funded by MOQS 26-04-2021
2020 Calls for application: PhD Mathematics and Models 28-08-2020
Next Ph.D. colloquium: Prof. Protter, Columbia University 08-06-2020

More News

The graduate school in Mathematics and Models of the University of L’Aquila offers a doctorate programme in the most relevant areas of mathematics.

The program is open to national and international students.

The doctorate council is made up of 25 Professors of the University of L’Aquila, and the coordinator is Prof. Davide Gabrielli.

Objectives

The program aims at training students to get a deep knowledge of the theoretical features of a mathematical subject
and to develop a qualified preparation to apply mathematical models in a diversified range of scientific areas.

The doctorate is one of the programmes of the INdAM-DP-COFUND-2015

Algebra: (theory of groups and of the representations, commutative algebra and applications)
Geometry (riemannian, complex and algebraic)
Partial differential equations with applications
Dynamical systems (deterministic and stochastic)
Stochastic processes with applications to biology, physics and finance
Numerical analysis of dynamical systems and numerical methods for partial differential equations
Atomistic modelling and simulations of Molecules, Materials, and Biological Systems
Development of Scientific Software for High-Performance Computing

We offer a variety of research related courses as well as introductory level courses which help first-year students strengthen their mathematical background. At the beginning of every academic year, in consultation with their tutor, students present a study plan to the doctorate council where they specify what research and training they plan to do in the coming academic year.

In the three years of the program, students are expected to participate in seminars offered by the school and to take part in research internships in institutions both in and outside Italy. At the end of each academic year, with the exception of the final year, the students will then be interviewed on the studies and research they have carried out during the year in front of a committee appointed by the doctorate council. Successfully passing this interview means that the students can keep their post and fellowship, and thus be admitted to the following year. At the interview, the students will present a report on their scholarly activity, their research and its results, seminars, congresses, or other scientific activities they have participated in, and any publications they have produced.

For the admission into the final year, this report will include a section relating to the progress made in their research project. The program is in strict collaboration with the international phd school Gran Sasso Science Institute (GSSI, L'Aquila) and the students will have the possibility to follow all the activities held at the GSSI. Our graduate students also benefit from our close links with The International Research Center of Mathematics and Mechanics of Complex Systems.

Debora AMADORI
Lucio BEDULLI
Matteo COLANGELI
Raffaele D'AMBROSIO
Anna DE MASI
Marco DI FRANCESCO
Donatella DONATELLI
Klaus ENGEL
Lucia FANIA
Davide GABRIELLI
Norberto GAVIOLI
Leonardo GUIDONI
Corrado LATTANZIO

Francesco LEONETTI
Angelo LUONGO
Immacolata MEROLA
Barbara NELLI
Luca PLACIDI
Cristina PIGNOTTI
Vladimir PROTASOV
Bruno RUBINO
Carlo Maria SCOPPOLA
Maurizio SERVA
Stefano SPIRITO
Amabile TATONE
Dimitrios TSAKGAROGIANNIS

Courses for the academic year 2020/21

Courses will be held on MS Teams, channel PhD courses Mathematics and Models 2020/2021, code eknjbzb

Timetable Abstracts

2019/2020 Courses

Algebra

Group theoretical approach for symmetric encryption
Speakers: Aragona-Civino, 10 hours
Program: Gruppi di permutazione ed in particolare gruppi primitivi, accenno alla classificazione dei gruppi primitivi finiti di O’Nan-Scott, crittografia simmetrica e cifrari a blocchi, gruppi generati dalle funzioni di cifratura di un crittosistema simmetrico, attacchi algebrici su cifrari a blocchi, attacchi di imprimitività, studio del gruppo G(SPN) generato dalle funzioni di cifratura di un SPN, condizioni per cui G(SPN) sia primitivo e condizioni per cui G(SPN) sia l’alterno.

Continuum Mechanics

Variational derivation of continuum mechanics equations: 10 hours
Speakers: Dell’Isola-Placidi-Barchiesi
Program: The aim of this short course is to introduce the students to the variational methods that are used in continuum mechanics. Static and dynamic cases will be analyzed. The mathematical derivations will be done on the basis of assumptions that are formulated on the form of the action functional. In the static case, the action is reduced to the total energy functional and a method to derive its form from a discrete model is also sketched.

Arguments: The subjects developed during the course will be a suitable selection of the following ones: 3D continuum elasto-dynamic model derivations for the standard and for the strain gradient case. Euler-Bernoulli beam theory derived from an Action principle as well as from a discrete model. Continuum equations for materials with band gap for the 3D and for the 1D cases. The choice may depend on the interests of the students.

MODELS: Standard and strain gradient elastic materials. Euler beams. Materials with band gap. METHODS: Variational methods. Heuristic homogenization procedure.

Tentative program:

LECTURE 1:
LECTURE 2:
gradient case.
LECTURE 3:
LECTURE 4:
LECTURE 5:
and for the 1D cases will be shown.

It is shown in details the case of 3D continuum elasto-dynamic case for the standard. It is shown in details the case of 3D continuum elasto-dynamic case for the strain

Standard Euler-Bernoulli beam theory will be derived from an action.
Standard Euler-Bernoulli beam theory will be derived from a discrete model.
The derivation of the continuum equations for materials with band gap for the 3D

Bibliography:

[1] dell’Isola Francesco, PLACIDI L (2011). Variational principles are a powerful tool also for formulating field theories. In: F. DELL’ISOLA, S. GAVRILYUK. (a cura di): F. DELL’ISOLA, S. GAVRILYUK, CISM Courses and Lectures. Variational Models and Methods in Solid and Fluid Mechanics. CISM INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES 535, 1–16

[2] Auffray, N.; dell’Isola, F.; Eremeyev, V.; Madeo, A.; Placidi, L.; Rosi, G. Least action principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola’s point of view. The complete works of Gabrio Piola. Vol. I, 1–89, Adv. Struct. Mater. 38

[3] El Sherbiny Mohammed, Placidi L (2018). Discrete and continuous aspects of some metamaterial elastic structures with band gaps. ARCHIVE OF APPLIED MECHANICS, vol. 18, p. 1725-1742

[4] Placidi, L., dell’Isola, F., Barchiesi, E. (2020). Heuristic Homogenization of Euler and Panto- graphic Beams. In Mechanics of Fibrous Materials and Applications (123–155). Springer, Cham.

Dynamical Systems

Geometry

Mathematical Analysis

Mathematical Physics

Selected topics in kinetic theory of gases
Speaker: Colangeli, 10 hours
Program: The course will cover a selection of topics of classical kinetic theory of gases: phase space and Lioville’s theorem; hard spheres and mean free path; BBGKY hierarchy; Boltzmann equation for hard spheres; Non-cutoff potentials and grazing collisions; Fokker-Planck equation; H-theorem and irreversibility; Chapman-Enskog expansion; Kac ring model.

Bibliography: Carlo Cercignani, The Boltzmann Equation and Its Applications, Springer (1988)

Numerical Analysis

Probability

Quantum Computing

Course by Prof. Denis SERRE (INdAM Visiting Professor)

+

Schedule

2018/2019 Courses

Variational Derivation of Continuum Mechanics Equations,

Title: Variational derivation of continuum mechanics equationsProfessor: Francesco dell’Isola and/or Luca Placidi

Aim: The aim of this short course is to introduce the students to the variational methods that are used in continuum mechanics. Static and dynamic cases will be analyzed. The mathematical derivations will be done on the basis of assumptions that are formulated on the form of the action functional. In the static case, the action is reduced to the total energy functional and a method to derive its form from a discrete model is also sketched.

Duration: 10 hours divided into 5 lectures

Topics: The subjects developed during the course will be a suitable selection of the following ones: 3D continuum elasto-dynamic model derivations for the standard and for the strain gradient case. Euler-Bernoulli beam theory derived from an Action principle as well as from a discrete model. Continuum equations for materials with band gap for the 3D and for the 1D cases. The choice may depend on the interests of the students

MODELS: Standard and strain gradient elastic materials. Euler beams. Materials with band gap.METHODS: Variational methods. Heuristic homogenization procedure.
Program: A possible effective program may be the following:

LECTURE 1: It is shown in details the case of 3D continuum elasto-dynamic case for the standard.
LECTURE 2: It is shown in details the case of 3D continuum elasto-dynamic case for the strain gradient case.
LECTURE 3: Standard Euler-Bernoulli beam theory will be derived from an action.
LECTURE 4: Standard Euler-Bernoulli beam theory will be derived from a discrete model.
LECTURE 5: The derivation of the continuum equations for materials with band gap for the 3D and for the 1D cases will be shown.

Bibliography: Some basic reference texts are the following:

dell'Isola Francesco, PLACIDI L (2011). Variational principles are a powerful tool also for formulating field theories. In: FRANCESCO DELL'ISOLA, SERGEY GAVRILYUK. (a cura di): FRANCESCO DELL'ISOLA, SERGEY GAVRILYUK, CISM Courses and Lectures. Variational Models and Methods in Solid and Fluid Mechanics. CISM INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES, vol. 535, p. 1-16, Wien, New York:SpringerWienNewYork, ISBN: 978-3-7091-0982-3, ISSN: 0254-1971

N. Auffray, F. dell'Isola, V. Eremeyev, A. Madeo, PLACIDI L, G. Rosi (2014). Least action principle for second gradient continua and capillary fluids: a Lagrangian approach following Piola's point of view. In: (a cura di): U.Andreaus, F. dell’Isola, R. Esposito, S. Forest, G. Maier, and U. Perego, The complete works of Gabrio Piola, volume I. Advanced Structured Materials. ADVANCED STRUCTURED MATERIALS, vol. 38, Springer Verlag, ISSN: 1869-8433, doi: 10.1007/978-3-319-00263-7_4

El Sherbiny Mohammed, Placidi L (2018). Discrete and continuous aspects of some metamaterial elastic structures with band gaps. ARCHIVE OF APPLIED MECHANICS, vol. 18, p. 1725-1742, ISSN: 0939-1533, doi: 10.1007/s00419-018-1399-1

Turco E., Golaszewski M., Giorgio I., Placidi L (2017). Can a hencky-type model predict the mechanical behaviour of pantographic lattices?. In: (a cura di): Sofonea M., dell'Isola F., Steigmann D., Mathematical Modelling in Solid Mechanics. ADVANCED STRUCTURED MATERIALS, vol. 69, p. 285-311, Singapore:Springer Nature, ISBN: 978-981103763-4, ISSN: 1869-8433, doi: 10.1007/978-981-10-3764-1_18

Perturbation Methods for Nonlinear Dynamical Systems

Quantum Computation and Quantum Information

Control Theory and Hyperbolic Partial Differential Equations

Adriano Festa, PhD in Mathematics, Ricercatore di tipo A, Universit`a dell’Aquila.

The aim of this course is to offer a basic training in nonlinear hyperbolic equations ofHamilton-Jacobi kind, highlighting the connection of this part of theory with the theory of optimal control. The idea is to provide the students with the basic tools necessary for the comprehension of some actual subjects of research in applied mathematics.

Entry requirements/Target: The course is intended for PhD students in Mathematics. Stu- dents are expected to have a basic knowledge of dynamic systems and linear partial differential equations. The short numerical tutorial will be supplied using the programming language MAT- LAB. Some basics in conservation laws, optimization or optimal control are useful but not mandatory for the full comprehension of the more advanced tasks.

If requested, the lectures may be held in English.

Structure of the lecture: The course is composed of 8 hours on 2 weeks and a tutorial of 2 hours in laboratory to be held on January-February 2019.

Reference books: Lecture notes of the course are provided to the students. [1, 3] constitute the main sources of in-dept study, secondary the more generic [2, 4].

Themes: The first 4h of the course is devoted to an operative-oriented exposition of the theory of the Hamilton-Jacobi equations (characterization of weak solutions, fundamental hypotheses that guarantee well-posedness of the problem), its derivation (relation with conservation laws, Dynamic programming principle, relation with optimal control problems). The other 4 hours are devoted to the introduction of numerical techniques for the approximation of the solution of these equations (Finite Differences, Semilagrangian methods) deepening in some more advanced issues of the matter as numerical tools for a fast resolution (Policy iteration algorithm, Fast Marching, Fast Sweeping), some non trivial extensions of the theory (Obstacle problems, Target problems, Differential games).

In both the parts, a special attention is dedicated to the discussion of some applications among image processing, social and biological models, financial mathematics, control systems.

References

[1] Bardi, Martino, and Italo Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton- Jacobi-Bellman equations. Springer Science & Business Media, 2008.
[2] Evans, Lawrence C., Partial differential equations. Graduate Studies in Math., AMS, 1994.
[3] Falcone, Maurizio, and Roberto Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations. Society for Industrial and Applied Mathematics, 2013.
[4] Quarteroni, Alfio, and Alberto Valli. Numerical approximation of partial differential equations. Vol. 23. Springer Science & Business Media, 2008.

Permutation Groups and Applications to Criptography

Lectures on the coupling method

Differential forms and application in geometry and physics

At a first glance, differential forms (also known as co-vectors) might seem as just a technical tool. Locally, 1-dimensional differential forms look just as the dual of vectors, which everybody is familiar with. However, differential forms capture some very deep global geometric- and topologic properties of the space they are attached to. Therefore, differential forms led to the discovery of a vast number of new results in area such as differntial-, symplectic-, contact- and algebraic geometry, algebraic topology, classical- and quantum mechanics, just to name a few. The course will be an introduction into differential forms.

Prerequisites: We will assume knowledge only in calculus in Rn (differentiation and integration). However, it would also be desirable, but not compulsory, some basic knowledge in: smooth manifolds, algebraic topology and differential geometry of surfaces.

References: M. do Carmo, Differential forms and applications, Springer1991 J. M. Lee, Introduction to Smooth Manifolds, Springer 2006 R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Srpinger 1982

Wave Equations on Bounded Domains

Weakly Monotone Functions

1) Prof.ssa Debora Amadori (6 ore, 3 lezioni)
Wave equations on bounded domains

2) Prof.ssa Menita Carozza, Universita' del Sannio (Benevento)
Totale 6 ore (tre giorni consecutivi) a gennaio 2019

Sulle proprietà delle funzioni debolmente monotone
La nozione di funzione debolmente monotona estende la classica definizione delle funzioni monotone secondo Lebesgue. Essa fu introdotta da J. Manfredi nel 1994, in ambito degli spazi di Sobolev, ed è stata studiata in quello più generale degli spazi di Orlicz-Sobolev da diversi autori quali T.Iwaniec, J.Kauhanen, P.Koskela, J.Maly, J.Onninen, X.Zhong.
In recenti lavori sono stati migliorati alcuni risultati, noti in letteratura, riguardanti le proprietà di regolarità delle funzioni debolmente monotone, negli spazi di Orlicz- Sobolev e negli spazi di Lorentz-Sobolev.
Dopo aver introdotto tali spazi di funzioni, le lezioni verteranno, in particolare, sulla proprietà di continuità delle funzioni debolmente monotone in tali spazi , continuità che è stata provata anche al di là di insiemi singolari a Orlicz-capacità nulla, e al di là di insiemi a misura (generalizzata) di Hausdorff nulla, laddove la continuità ovunque fallisca.
Ricordiamo che le funzioni debolmente monotone entrano in gioco, ad esempio, nella teoria della regolarità delle equazioni alle derivate parziali ellittiche: le soluzioni deboli delle equazioni ellittiche tipo p-laplaciano, con ellitticità anche degenere, sono di fatto debolmente monotone.

Weakly monotone functions
The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to H.Lebesgue. It was introduced, in the setting of Sobolev spaces, by J.Manfredi, and thoroughly investigated in the more general framework of Orlicz-Sobolev spaces by diverse authors, including T.Iwaniec, J.Kauhanen, P.Koskela, J.Maly, J.Onninen, X.Zhong. Recent results complement and augment the available theory of pointwise regularity properties of weakly monotone functions in Orlicz-Sobolev spaces and Lorentz- Sobolev spaces. After having introduced these function spaces,I will tell about recent contributions on the continuity properties of weakly monotone functions from these spaces. The continuity outside sets of zero Orlicz capacity, and outside sets of (generalized) zero Hausdorff measure, are also established when everywhere continuity fails.

Remember that weakly monotone functions come into play, for instance, in the regularity theory of elliptic partial differential equations as well: weak solutions to p-Laplacian type elliptic equations, with possibly degenerating ellipticity, turn out to be wea

An Introduction to Optimal Mass Transport

Control of Linear Hyperbolic PDEs

An Introduction to Mathematical Theory of Control

+

Laboratories and Databases

Students have access to a wide variety of computing equipment in the DISIM department. They have access to various scientific laboratories among which we mention the High Performance Parallel Computing (HPPC) with the supercalculator Caliban with a computing power of about 2.5 teraflops. They also have access to databases as ARXIV, ALL, EBSCO, DOAJ, virtual Emeroteca Caspur, JCR, JSTOR, AMS/Mathschinet, Numdam, PUBMET, Science Direct, Scopus, Springer Link, WILEY on line library, ISI web of knolewdge, Web of Science.