Title: Numerical methods for Boundary Integral Equations - CFU: 6
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 - In-tegral 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 - Numeri-cal 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: Lectures will take place in Spring 2020 at the University of Parma for an amount of 24 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.

Title: Conformal Maps of the complex plane - CFU: 6
Teacher:  Anna (Miriam) Benini

Syllabus:  In complex analysis a conformal map is a  map which is holomorphic and one-to-one. Conformal maps have many surprising geometric properties. For example,  the image of the unit disk under  a  conformal map always contains a disk of radius at least 1/4 times the modulus of the derivative at the origin. In this course we  plan to examine and prove several distortion theorems which hold for  conformal maps, concluding with the Denjoy-Carleman-Ahlfors Theorem. This is a cross course between complex analysis and complex geometry.

Dates:    February and March 2020  

Title: Constraint Satisfaction Problems - CFU: 4
Teachers: Federico Bergenti, Stefania Monica

Syllabus: The course is intended to provide an introduction to the current re-search on Constraint Satisfaction Problems (CSPs) to students with no specific background in Computer Science or Artificial Intel-ligence. The course starts with an introduction to CSPs and with an overview of algorithms for constraint satisfaction based on heuris-tic search. Then, algorithms for constraint propagation are pre-sented (e.g., arc consistency and hyper-arc consistency), and the forms of consistency that they achieve are discussed. Finally, al-gorithms 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: 8-10 hours in November-December 2019 (flexible).

Title: Group actions on manifolds - CFU: 6
Teacher:  Leonardo Biliotti

Syllabus:  An introduction to Lie group and Riemannian Geometry. Bi-invariant metric on Lie group. Proper action. Fiber bundle, Slice Theorem, stratifcation of the orbit space. Compact Lie group, max-imal torus and Weyl group.

Dates:    to be fixed.

Title:  Extended kinetic theory and recent applications - CFU: 9
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 Ma-tematica, 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: About 20 hours in January - February 2020 (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 - CFU: 9
Teacher: Marzia Bisi
Syllabus: 

•  Fourier transform: from Fourier series to Fourier transform, definition of inverse transform, tran-sformation properties, convolution theorem, explicit computation of some transforms, applica-tions to ODEs and PDEs of some physical problems.
• Laplace transform: definition, region of convergence, transformation properties, Laplace tran-sform of Gaussian distribution, applications to some Cauchy problems.
• Definite integrals by means of residue theorem: integrals of real functions, and integrals of Fou-rier and Laplace useful to evaluate inverse transforms; theorems (with proofs) and examples.

Dates: reading course; pdf slides and videos of all lectures are available on-line.

Title: Pattern formation: nonlinear dynamics and multiscale analysis in reaction-diffusion systems - CFU: 6
Teachers: Gaetana Gambino (Università di Palermo), Maria Carmela Lombardo (Università di Palermo)
Syllabus: here
Details: here

Dates: February 17-21,  2020.

Title:  Partial Differential Equations 
Teacher: Luca Lorenzi

Syllabus:
Depending on the interests of the students, the lectures will cover one of the following arguments:
1) Classical $C^{\alpha}$ theory for parabolic PDEs
2) $L^p$ and $C^{\alpha}$ theory for elliptic equations.

Dates: 20-25 hours in the second semester. Interested students should contact the teacher to agree upon the calendar.

Title: Infinite Dimensional Analysis - CFU: 6+6 
Teachers: Alessandra Lunardi (Parma), Michele Miranda (Ferrara)

Syllabus: This is an introductory course about analysis in abstract Wiener spac-es, namely separable Banach or Hilbert spaces endowed with a non-degenerate Gaussian measure. Sobolev spaces and spaces of continuous functions will be considered. The basic differential operators (gradient and divergence) will be studied, as well as the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup, that are the Gaussian analogues of the Laplacian and the heat semi-group. The most important functional inequalities in this context, such as Poin-caré and logarithmic Sobolev inequalities, will be proved. Hermite polynomials and the Wiener chaos will be described.
The reference book is "Gaussian Measures" by V. Bogachev (Mathematical Sur-veys and Monographs 62, AMS 1998). In addition, lecture notes prepared by the teachers will be available.

Dates: Second semester (may be changed, according to the needs of the students)