PhD Mathematics and Models

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 programme is open to national and international students.

The doctorate council is made up of 25 Professors of the University of L’Aquila.

The coordinator is Prof. Davide Gabriellithe vice-coordinator is Prof. Debora Amadori


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
  • 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

Courses for the academic year 2021/22

Optional course by Prof. Andrea Pascucci (University of Bologna): Stochastic calculus and PDEs in financial applications Abstract & Schedule

Courses for the academic year 2020/21

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


2019/2020 Courses


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.

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)


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

Differential forms and application in geometry and physics

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

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.

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