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ATENEO DI QUALITÀ ACCREDITATO ANVUR - FASCIA A

Courses in Parma

PhD in Mathematics 36th cycle

(PhD Mathematics) Home | Program | Faculty PhD Students

Lectures take place at “Dipartimento di Scienze Matematiche, Fisiche e Informatiche”, Mathematics Building, Parco Area delle Scienze 53/A, 43124 Parma (location). 


Title: Introduction to dynamical systems (3CFU)
Teacher:  Anna Miriam Benini

Syllabus: We will introduce the basic concepts of discrete dynamical systems, that is, systems generated by the iteration of a map on a space with appropriate regularity properties. We will consider circle rotations, symbolic dynamical systems, smooth maps of $R^2$ , and we will conclude with  a brief class in  complex dynamics. In these cases, and in general, we will study invariant sets (equilibrium state), the basics of ergodic theory like invariant probability measures and topological entropy, and if  possibile structural stability and related topics. We will cover selected topics in Chapters 1-6 from A.  Katok, B. Hasselblatt, "Introduction to the Modern Theory of Dynamical systems".

Dates 2020/2021:  The course will take place in the spring semester 2021. There will be 20-30 hours of lectures and the final exam will be a seminar given by the student on a topic inherent to the course.


Title: Intuition, conceptualization and formalization in mathematics teaching and learning (3CFU)
Teacher:  Laura Branchetti

Syllabus: Procedures, concepts, objects, symbols: the complex relationship between epistemic and cognitive meaning in mathematics teaching and learning Intuition in Mathematics and Figural concepts. Concept image and concept definitions.

Textbooks: 

  • Fischbein, E. (2002). Intuition in Science and Mathematics. An educational approach. Kluwer Academic Publishers. doi: 10.1007/0-306-47237-6
  • Tall, D. (Ed.) (1991). Advanced Mathematical Thinking. Springer Netherlands. doi: 10.1007/0-306-47203-1
  • Radford, L. Schubring, G., Seeger, F. (Eds.)(2008). Semiotics in Mathematics Education: Epistemology, History, Classroom, and Culture. Sense Publishers.

Dates 2020/2021: Feb-May 2021 (15 hours).


Title and Credits: Numerical methods for Boundary Integral Equations (6CFU)
Teacher:  Alessandra Aimi

Syllabus: The course is principally focused on Boundary Element Methods (BEMs). Lectures involve: Boundary integral formulation of elliptic, parabolic and hyperbolic problems - Integral operators with weakly singular, strongly singular and hyper-singular kernels - Approximation techniques: collocation and Galerkin BEMs - Quadrature formulas for weakly singular integrals, Cauchy principal value integrals and Hadamard finite part integrals - Convergence results - Numerical schemes for the generation of the linear system coming from Galerkin BEM discretization. Knowledge of basic notions in Numerical Analysis and in particular in numerical approximation of partial differential equations is required. References will be provided directly during the course.

Dates 2020/2021: Lectures will take place in Spring 2021 at the University of Parma for an amount of 24 hours. At the end, an individual project will be assigned. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.


Title: Constraint Satisfaction Problems (4CFU)
Teachers: Federico Bergenti

Syllabus: The course is intended to provide an introduction to the current research on Constraint Satisfaction Problems (CSPs) to students with no specific background in Computer Science or Artificial Intelligence. The course starts with an introduction to CSPs and with an overview of algorithms for constraint satisfaction based on heuristic search. Then, algorithms for constraint propagation are presented (e.g., arc consistency and hyper-arc consistency), and the forms of consistency that they achieve are discussed. Finally, algorithms to treat polynomial constraints are shown (e.g., Buchberger's algorithm and cylindrical algebraic decomposition), with emphasis on polynomial constraints over finite domains (e.g., based on Bernstein polynomials and on Rivlin's bound).

Dates 2020/2021: 8-10 hours in November-December 2020 (flexible).


Title:  Extended kinetic theory and recent applications (4CFU)
Teachers: Marzia Bisi, Maria Groppi

Syllabus: The course is intended to provide an introduction to classical kinetic Boltzmann approach to rare-fied gas dynamics, and some recent advances including the generalization of kinetic models to reactive gas mixtures and to socio-economic problems. Possible list of topics:  
distribution function and Boltzmann equation for a single gas: collision operator, collision invariants, Maxwellian equilibrium distributions; entropy functionals and second law of thermodynamics; hydrodynamic limit, Euler and Navier-Stokes  equations; kinetic theory for gas mixtures: extended Boltzmann equations and BGK models; kinetic models for reacting and/or polyatomic particles; Boltzmann and Fokker-Planck equations for socio-economic phenomena, as wealth distribution or opin-ion formation.

Bibliography:

  • C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988.
  • M. Bisi, M. Groppi, G. Spiga, Kinetic Modelling of Bimolecular Chemical Reactions, in “Kinetic Methods for Nonconservative and Reacting Systems” edited by G. Toscani, Quaderni di Matematica 16, Dip. di Matematica, Seconda Università di Napoli, Aracne Editrice, Roma, 2005.
  • L. Pareschi, G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014.

Dates 2020/2021: About 18 hours in January - February 2021 (flexible). The interested Ph.D. students are asked to contact the teachers in advance to define the calendar.


Title: Fourier and Laplace transforms and some applications (4CFU)
Teacher: Marzia Bisi

Syllabus: Fourier transform: from Fourier series to Fourier transform, definition of inverse transform, transformation properties, convolution theorem, explicit computation of some transforms, applications to ODEs and PDEs of some physical problems.

Laplace transform: definition, region of convergence, transformation properties, Laplace transform of Gaussian distribution, applications to some Cauchy problems.

Definite integrals by means of residue theorem: integrals of real functions, and integrals of Fourier and Laplace useful to evaluate inverse transforms; theorems (with proofs) and examples.

Dates 2020/2021: reading course; pdf slides and videos of all lectures are available on-line, number of expected hours: 24 + individual project.


Title: Introduction to Interpolation Theory (4CFU)
Teacher:  Alessandra Lunardi

Syllabus: The course provides the fundamentals of classical (real and complex) interpolation theory, as well as connections with the theory of powers of operators and semigroup theory, and applications to PDEs. The first part of the course will be devoted to the basic theory, namely

  • Equivalent definitions and basic properties of real interpolation spaces between Banach spaces; Reiteration Theorem.
  • Examples: real interpolation spaces between spaces of continuous functions and spaces of C^k functions, real interpolation spaces between L^p spaces and Sobolev spaces. Connections with trace theory.
  • The Riesz-Thorin Theorem and complex interpolation between Banach spaces.
  • Examples: complex interpolation between Lebesgue spaces.

In the second part of the course the students may choose among the following more specialized topics:

  • Interpolation and domains of unbounded operators in Banach spaces. Applications to regularity theory in PDEs: Schauder theorems for elliptic second order differential equations.
  • Powers of nonnegative operators, relations of their domains with interpolation spaces, operators with bounded imaginary powers.
  • Interpolation and semigroups: real interpolation spaces between Banach spaces and domains of generators of semigroups. Applications to regularity theory in PDEs: Schauder theorems for parabolic second order differential equations.

Reference textbook:
A. Lunardi, "Interpolation Theory. Third edition". Edizioni della Normale, Pisa (2018).

Lecture notes will be provided.

Dates 2020/2021: Spring 2021.


Title: Several complex variables (6CFU)
Teacher:  Alberto Saracco

Syllabus: Theory of several complex variables. Hartogs theorem, Cartan-Thullen theorem, Kontinuitatsatz. Domains of holomorphy, Levi convexity and plurisubharmonic functions. Cauchy-Riemann equation. Sheaves and cohomology (Cech cohomology). The course will be mainly based on Chapters 1-6 of the book by Giuseppe Della Sala, Alberto Saracco, Alexandru Simioniuc and Giuseppe Tomassini: "Lectures on complex analysis and analytic geometry", Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)] 3, Edizioni della Normale, Pisa (2006).

Dates 2020/2021:  Oct. 2020 - Jan. 2021.


Title: Holomorphic isometries (3CFU)
Teacher: Michela Zedda

Syllabus: After recalling basic facts about Kaehler geometry, the course focuses on existence, extension and rigidity results for germs of holomorphic isometries between Kaehler manifolds. In particular, we will introduce Calabi's diastasis function and Calabi's criterium and we will discuss the case of complex space forms and give some explicit examples of holomorphic isometry. The last part of the course focuses on analytic continuation and rigidity results.

Dates 2020/2021: october-november 2020.


Title: Basic theory of the Riemann zeta-function (3 CFU)
Teacher: Alessandro Zaccagnini

Syllabus: Elementary results on prime numbers. The Riemann zeta-function and its basic properties: analytic continuation, functional equation, Euler product  and connection with prime numbers, the Riemann-von Mangoldt formula, the explicit formula, the Prime Number Theorem. Prime numbers in all and "almost all" short intetvals.

Dates 2020/2021: late winter, early spring 2021


Title: Yangians in geometry and representation theory  (4 CFU)
Teacher: Martina Lanini (Roma 2), Francesco Sala (Pisa), Andrea Appel 

Syllabus: The course will be divided in three parts. The first part will describe Yangians and Quantum Loop Algebras as algebraic objects, focusing on their definitions (motivated by mathematical physics), their several presentations, and their category of finite-dimensional representations. The second part will focus on developing the necessary tools to study the geometry of Nakajima quiver varieties and the structure of their equivariant cohomology. Finally, in the last part, these two aspects will come together, showing the natural appearance of Yangians in the context of algebraic geometry, following the approach of Maulik-Okounkov which led to the discovery of a new kind of Yangians.

Dates: January – April, 2021 (30 hours, online)


Title: Introduction to Geometric Measure Theory (6 CFU)
Teacher: Massimiliano Morini (Parma)

Syllabus: The course covers the following topics: review and complements of Measure Theory; covering theorems and their application to the proof of the Lebesgue and Besicovitch Differentiation Theorems; rectifiable sets and rectifiability criteria; the theory of sets of finite perimeter;  applications to geometric variational problems; the isoperimetric problem; the partial  regularity theory for quasi-minimiser of the perimeter.

Dates 2020/2021: reading course.

Hand-written notes of the whole course are available in Italian on the Elly platform.
Further references:

  1. L.C Evans and R.F. Gariepy: "Measure Theory and Fine Properties of Functions"
  2. F. Maggi: "Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory"

Pubblicato Martedì, 19 Maggio, 2020 - 15:17 | ultima modifica Martedì, 17 Novembre, 2020 - 09:54