Graduate Courses in Mathematics (2019-2022)

Graduate Courses in Mathematics

In addition to the Graduate Courses in Mathematics offered by the Faculty Members of the Program (listed below for the years 2019-2022), graduate students can also attend:

Courses of the Master's Degree in Mathematics held at the University of Parma , University of Ferrara, University of Modena and Reggio Emilia.
Interdisciplinary Courses offered by Centro Linguistico di Ateneo and Doctorate School in Engineering & Architecture at the University of Parma, and similar courses offered at the University of Ferrara and the University of Modena and Reggio Emilia.

2021-2022 (Ciclo XXXVII)

Title and Credits: Numerical methods for Boundary Integral Equations, 6 CFU
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 2021/2022: Lectures will take place in Spring 2022 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 and credits: Introduction to quantum groups, 4CFU
Teacher: Andrea Appel

Syllabus: The course will be a blend of mostly representation theory (quantum groups and Hopf algebras), a bit of basic complex algebraic geometry (blowups), some category theory (braided monoidal categories), and some deformation theory (Hochschild cohomology). The course is intended for a general mathematical audience: I will do everything from scratch, assuming only the basic notions in algebra and geometry.

The first part of the course will provide a parallel between the classical theory of the Lie algebra sl(2) and that of its quantum counterpart Uqsl(2), with a special focus on the role of the universal R-matrix and the Yang-Baxter equation.

The topics discussed in the second part of the course will depend upon the main interests of the audience. Potential topics are: monodromy of the Knizhnick-Zamoldchikov equations and Kohno-Drinfeld theorem; Yangians and quantum loop algebras; Etingof-Kazhdan quantization of Lie bialgebras; categorification and Khovanov-Lauda-Rouquier algebras; Reshetikhin-Turaev invariants; quantum groups at root of unity.

More information here

Dates 2021/22: 24 hours, in the period January-March 2022

Title and credits: Introduction to complex dynamics, 3 CFU
Teacher: Anna (Miriam) Benini

Syllabus: We will give an introduction to complex dynamics following Milnor's book and notes by M. Lyubich. We will investigate the dynamics of polynomials, rational maps and transcendental maps.

Dates: January 10-February 24th, 2022

Title and Credits: Fourier and Laplace transforms and some applications, 4 CFU
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 2021/2022: reading course; pdf slides and videos of all lectures are available on-line, number of expected hours: 24 + individual project.

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

Syllabus: The course is intended to provide an introduction to classical kinetic Boltzmann approach to rarefied 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 opinion 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 2021/2022: About 18 hours in January - February 2022 (flexible). The interested Ph.D. students are asked to contact the teachers in advance to define the calendar.

Title and Credits: Interface and Contact Problems, 3 CFU
Teacher: Heiko Gimperlein, Heriot–Watt University, UK (Visiting Professor)

Syllabus:

Modelling of interface problems between materials: interface conditions and friction laws,
Obstacle problems, friction and contact: (free) time-independent boundary problems as constrained on nonsmooth variational problems, solution using Uzawa and semismooth Newton methods,
Nonlinear analysis of variational inequalities: functional analytic background, classical theorem on wellposedness and abstract approximation, regularity of solutions,
Approximation by finite and boundary elements - from basics to current research: BEM for dummies, variational inequalities and penalty formulations, error analysis, adaptivity, advanced approximation methods, maybe coupling of FEM and BEM,
Towards time-dependent variational inequalities and dynamic contact.

Dates 2021/2022: May 2 - 13, 2022

Title and Credits: Numerical methods for option pricing, 2 CFU (50 ore)
Teacher: Chiara Guardasoni

Syllabus:

Introduction to differential model problems for option pricing in the Black-Scholes framework
Analysis of peculiar troubles and advantages in application of standard numerical methods for partial differential problems: Finite Difference Methods, Finite Element Methods, Boundary Element Method

Dates 2021/2022: reading course always available

Title and Credits: Spectral theory for operators in Banach spaces and applications to semigroups, 6 CFU
Teacher: Luca Lorenzi

Syllabus: The program contains the topics listed here below plus possibly additional material based on the students' interests. To make the course self-contained, an introduction to the theory of semigroups of bounded operators will be provided to students who are not acquainted with this theory.

A survey on operator theory
1.1 Closed operators. Definitions and different characterizations. Closable operators and closure of an operator.
1.2 Spectrum and resolvent of a (bounded or closed) operator. Basic properties. Spectral radius.
1.3 Normal and selfadjoint operators and their basic properties.
Compact operators and Riesz-Schauder theory
2.1 Compact operators: definitions, examples, Schauder theorem.
2.2 Riesz-Schauder theory for compact operators.
2.3 Spectral decomposition theorem for selfadjoint compact operators.
Spectral representation theorem for bounded operators
3.1 Spectral representation theorem for bounded and selfadjoint operators on a separable Hilbert space H.
3.2 Spectral theorem for normal operators
Spectral representation theorem for unbounded operators
4.1. Adjoint of an unbounded operator, selfadjoint and symmetric operators. Definitions and basic properties.
4.2. Dissipative operators. Definitions and main properties.
4.3. Representation theorem for unbounded selfadjoint operators.
4.4. Spectral mapping theorem for the resolvent operator of closed linear operator.
4.5 Spectral representation theorem for selfadjoint operators.
4.6 Positive operators and minimax theorems for their eigenvalues.
Spectral mapping theorems for semigroups
5.1 The spectral mapping theorem for the point and residual spectrum.
5.2 Spectral mapping theorem for eventually norm continuous semigroups and consequences.
5.3 Examples and counterexamples.

Dates 2021/2022: from march to the end of may.

Title and Credits: Operator semigroups and evolution equations, 6CFU.
Teacher: Alessandra Lunardi

Syllabus: The course deals with the basic theory of semigroups of linear operators and evolution equations in Banach spaces. Main topics are

basic spectral theory for linear operators in Banach spaces;
strongly continuous semigroups and the Hille-Yosida theorem;
analytic semigroups;
Cauchy problems for linear evolution equations in Banach spaces: regularity and asymptotic behavior.

Lecture notes (in italian) will be provided. Further bibliography include

K. Engel, R. Nagel: One-parameter Semigroups for Linear Evolution Equations, Spinger Verlag, Berlin, 1999.
K. Engel, R. Nagel: A Short Course on Operator Semigroups, Spinger Verlag, Berlin, 2006.
A. Lunardi: Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser Verlag 1995, 2nd edition 2013.

Dates 2021/2022: reading course.

Title and Credits: Introduction to Geometric Measure Theory, 6 CFU
Teacher: Massimiliano Morini

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.

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

Further references:

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

Dates 2021/2022: reading course.

Title and Credits: Instability and Bifurcation, 6 CFU
Teacher: Paolo Piccione(*), Universidade de Sao Paulo, Brasil

Syllabus: We give an overview of classical results in variational Bifurcation Theory and some geometrical applications, including multiplicity results for Geodesics, Constant Mean Curvature Surfaces, and the Yamabe problem. The detailed program is here.

Dates 2021/2022: november - december 2021.

(*) Supported by INdAM

Title and Credits: 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 2021/2022: reading course.

Title and Credits: 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 2021/2022: late winter, early spring 2022.

Title and Credits: Holomorphic and isometric immersions, 3CFU
Teacher: Michela Zedda

Syllabus: After an introduction of basic instruments of 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.

Dates 2021/2022: 24 hours, December 2021-January 2022 (flexible).

Title and Credits: Colloquia on Mathematical Analysis, 4 CFU
Teachers: Lorenzo Brasco, Michele Miranda

Syllabus: The course consists of a weekly meeting, aimed at presenting and discussing some fundamental research papers on the topics of Mathematical Analysis. Some examples of topics that will be considered are:

variational methods
geometric and regularity properties of solutions to elliptic equations
Gamma-convergence and applications
functional inequalities

The course will consist of approximately 10 meetings of 2 hours each. During each meeting the participants will deliver a talk on a paper (or a series of papers) proposed by the teachers, related to the aforementioned topics.

Dates 2021/2022: November 2021/February 2022. In case of problems due to the pandemics, the course will be shifted to March 2022/May 2022

Title and Credits: Steady Problems in Fluid Dynamics and Elasticity, 3 CFU
Teacher: Vincenzo Coscia

Syllabus: Six lectures of approximately 2.5 hours each.

Steady problems for the Navier-Stokes equations: basic questions and open problems.
The functional spaces of hydrodynamics. Hydrodynamic potentials.
The boundary value problem for the Stokes equations. Existence and uniqueness in bounded and exterior domains.
The boundary value problem for the Navier-Stokes equations in bounded and exterior domains.
The boundary value problem in elastostatics.
Recent results on steady problems in fluid dynamics and elasticity in bounded and exterior domains.

The students will be required to participate in the course solving the assigned exercises. At the end of the course each participant will have to carry out a seminar on a prescribed topic.

Dates 2021/2022: March-April 2022

Title and Credits: BV functions and applications to variational problems; Mumford-Shah, 4 CFU
Teacher: Michele Miranda

Syllabus: This course is an introduction to the theory of functions of bounded variations; we describe fine properties of BV functions and sets with finite perimeter using tools of geometric measure theory. We shall prove the decomposition of the total variation measure defined by a BV function. The notion of special functions with bounded variation, SBV functions, will be introduced and its characterisation via the chain rule will be given; we shall prove closure and compactness results of SBV functions. These properties will be used in the study of the Mumford-Shah functional that has applications in variational problems with free discontinuities (for instance, image reconstruction). Then the notion of Gamma-convergence will be introduced and the Ambrosio-Tortorelli approximation of the Mumford-Shah functional will be described.

Dates 2021/2022: January-February 2022, around 20 hours

Title and Credits: An introduction to uncertainty quantification for PDEs, 4 CFU
Teacher: Lorenzo Pareschi, Giulia Bertaglia

Syllabus: The course aims to provide an introduction to numerical methods for uncertainty quantification with specific reference to PDEs. After defining the main concepts in the field of uncertainty quantification, including some references to probability theory, the course focuses on two main approaches. The Monte Carlo method, in its variants characterized by multi-fidelity techniques, and the methods based on generalized polynomial chaos expansions, both in intrusive and non-intrusive form. Specific applications to the case of hyperbolic systems with relaxation terms and reaction-diffusion equations will be considered. In-depth study by students through specific reading of articles will also be suggested.

Dates 2021/2022: 12h lectures + reading course + home assignments

Title and credits: Computational intelligence and gradient-free optimization, 3 CFU

Teachers: Filippo Poltronieri, Mauro Tortonesi e Lorenzo Pareschi

Syllabus: This course provides an introductory overview of key concepts in computational intelligence with a focus on metaheuristic methods for global optimization. These include Genetic Algorithms (bitstring and integer vector genotype representations) and Particle Swarm Optimization (constrained PSO, quantum-inspired PSO, and a multi-swarm version of quantum-inspired PSO), extended with adaptation mechanisms to provide support for dynamic optimization problems. The main algorithms will be illustrated with the help of simple implementations in Matlab and/or R language. In the last part of the course, using a mean-field approach, rigorous convergence results for some of the methods will be presented.

Dates: Around 12h lectures + 4h assignments, February-June 2022

Title and Credits: (Modal) Symbolic Learning, 2CFU+2CFU (optional, for some research work)
Teacher: Guido Sciavicco

Syllabus: Symbolic learning is the sub-discipline of machine learning that is focused on symbolic (that is, logic-based) methods. As such, it contributes to the foundations of modern Artificial Intelligence. Symbolic learning is usually based on propositional logic, and in part, on first-order logic. Modal symbolic learning is the extension of symbolic learning to modal (and therefore, temporal, spatial, spatio-temporal) logics, and it deals with dimensional data. In this course we shall lay down the logical foundations of symbolic learning, prove some basic properties, and present the modal extensions of classical learning algorithms, highlighting which ones of those properties are preserved, and which ones are not.

Dates 2021/2022: September 2022, 4 lectures, 8 hours

Title and Credits: Introduction to the regularity theory of elliptic PDE's, 3CFU
Teacher: Michela Eleuteri

Syllabus:

Hystorical introduction to the Calculus of Variations
Basic notions about Sobolev Spaces and embeddings
The Hilbert spaces approach to the existence of solutions to nonlinear elliptic systems in divergence form
Classical and Direct methods in the Calculus of Variations (a short introduction)
The Caccioppoli inequality. The hole filling method. Higher integrability
The Nirenberg method of difference quotient: Hilbert space regularity in the interior of a domain
Holder, Morrey and Campanato spaces.
Decay estimates, iteration lemmas
The Schauder theory; constant coefficients, continuous coefficients and Holder coefficients
The XIX Hilbert problem: interior regularity for nonlinear problems
Solution to the XIX problem in 2D
The regularity in the scalar case: De Giorgi's theorem

Dates 2021/2022: second term (March-May 2022)

Title and Credits: Duality Theory of Markov Processes, 3 CFU
Teacher: Cristian Giardinà, Gioia Carinci

Syllabus: The course will present the duality approach to the study of Markov processes. This will combine, in a joint effort, probabilistic and algebraic tools. In particular we will consider several interacting particle systems that are used in (non-equilibrium) statistical mechanics, we will discuss "integrable probability", we will show how (stochastic) PDE arise by taking scaling limits.

Dates 2021/2022: Reading course, beginning of 2022 (the precise schedule will be decided together with the students).

Title and Credits: Designs, Graphs and their Applications, 4 CFU
Teacher: Arrigo Bonisoli, Simona Bonvicini, Giuseppe Mazzuoccolo, Anita Pasotti, G. Rinaldi, T. Traetta

Syllabus: The goal of the course is to introduce the students to ideas and techniques from Design Theory, Graph Theory and Combinatorics. After a general introduction we will present some advanced topics and recent results. They include (but are not limited to) Block Designs, Graph Decompositions, Difference Methods, Signed and Coloured Graphs. We will focus also on open problems and applications.

Dates 2021/2022: Reading course, beginning of 2022 (the precise schedule will be decided together with the students).

Title and credits: Hypoelliptic Partial Differential Equations, 3 CFU
Teachers: Sergio Polidoro, Maria Manfredini

Syllabus: The subject of the course is the regularity theory for linear second order Partial Differential Equations with non-negative characteristic form satisfying the Hormander's hypoellipticity condition. Fundamental solution, maximum principle, local regularity, boundary value problem will be discussed for several examples of equations. Some open research problems will be described. The course will focus on the following topics:

Bony's maximum principle for degenerate second order PDEs, propagation set and Hormander's hypoellipticity condition.
Perron method for the boundary value problem in a bounded open set of the Euclidean space.
Boundary regularity, barrier functions. Boundary measure, Green function.
Fundamental solution. Mean value formulas. Harnack inequalities.
Degenerate Kolmogorov equations. Applications to some financial problems and to kinetic theory.

The program may be modified in accordance with the requirements of the students.

Reference text: A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians (Springer Monographs in Mathematics - 2007). Further references and lecture notes will be given during the course.

Dates: The course will start in March 2022

2020-2021 (Ciclo XXXVI)

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

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

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:

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

Title : Hypercomplex Analysis and Geometry (2 CFU)
Teacher: Cinzia Bisi

Syllabus: We will present the theory of Slice Regularity, analogous of complex holomorphicity, on the main *-alternative algebras which generalize the complex field C: first of all on H, the skew field of quaternions, then on O, the algebra of Octonions , and also on some Clifford Algebras etc....
We will overview the main analytic properties of Slice Regularity and its geometric implications.

Dates 2020/2021: Second Semester a.a. 20-21. 10 hours in presence, around 5 lessons.
I will provide to participants also around 24 recorded lessons with written notes.

Title: Introduction to toric geometry (6CFU)
Teacher: Alex Massarenti

Syllabus: Toric varieties provide an elementary way to see many examples and phenomena in algebraic geometry. The goal of the course is an introduction to toric varieties and to their combinatorial, topological and geometric properties. At the end of the course, the student will be able to recognize and construct examples of toric varieties and to describe their properties.

Dates 2020/2021: 05/10/2020 - 27/11/2020. Around 12 hours + reading course + assigned homework

Title: BV functions and applications to variational problems; Mumford-Shah and Blake-Zissermann (4CFU)
Teachers: Michele Miranda, Elena Benvenuti (1 Lecture), Valeria Ruggiero (1 Lecture)

Dates 2020/2021: January-February 2021, around 20 hours

Title: Symbolic Learning: the point of view of a logician (2+2 CFU)
Teacher: Guido Sciavicco

Syllabus: From the point of view of a logician, the world can be described by formal rules. This notion is at the core of Artificial Intelligence (AI), and AI applications can be classified as deductive or inductive. Deductive AI can be seen as the classical approach, which brings us to classical problems such as (in)computability, (too high) complexity, and fundamental description problems. Inductive AI is better known as Machine Learning, and, in particular, Symbolic Machine Learning. In this course, we want to describe Symbolic Machine Learning as a logical problem, and we do by starting from Propositional Logic, then moving to Modal Logic, to end with Temporal and Spatial Logic. At the end, we shall have a complete picture of what Symbolic Machine Learning is, and how it locates itself in the realm of Machine Learning and AI in general. Two CFUs will be given upon attending classes, and two further CFUs may be earned with additional research work on these topics (to be discussed).

Dates 2020/2021: February 1-4, 2021, 8 hours (2 hours online session for 4 days).

Title: Recent topics in numerical methods for hyperbolic and kinetic equations (6 CFU)
Teachers: Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi

Syllabus: Hyperbolic and kinetic partial differential equations arise in a large number of models in physics and engineering. Prominent examples include the compressible Euler and Navier-Stokes equations, the shallow water equations, the Boltzmann equation, and the Vlasov-Fokker-Planck equation. Examples of the applications area range from classical gas dynamics and plasma physics to semiconductor design and granular gases. Recent studies employ the aforementioned theoretical background in order to describe the collective motion of a large number of particles such as pedestrian and traffic flows, swarming dynamics and other dynamics driven by social forces. These PDEs have been subjected to extensive analytical and numerical studies over the last decades. It is widely known that their solutions can exhibit very complex behavior including the presence of singularities such as shock waves, clustering and aggregation phenomena, sensitive dependence to initial conditions and presence of multiple spatio-temporal scales. This course will cover the mathematical foundations behind some of the most important numerical methods for these types of problems. To this goal, the first part of the course will be devoted to hyperbolic balance laws where we will introduce the notions of finite-difference, finite volume, and semi-Lagrangian schemes. In the second part we will focus on kinetic equations where, due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods requires a careful balance between accuracy and computational complexity. Finally, we will consider some recent developments related to the construction of asymptotic preserving methods, and to the development of efficient methods for optimal control and uncertainty quantification.

Dates 2020/2021: september-october 2021, 18 hours

Title: Consensus based algorithms and machine learning (2 CFU)
Teacher: Lorenzo Pareschi

Syllabus: This course will be focused on some recent results in the development of metaheuristics algorithms for global optimization with application to machine learning. In particular, it will be based on the following article:
J. A. Carrillo, Y-P. Choi, C. Totzeck, O. Tse. An analytical framework for consensus-based global optimization method. Mathematical Models and Methods in Applied Sciences 28, pp. 1037-1066 (2018) .

Dates: reading course, June, July or September 2021

Title: Control and uncertainty in epidemiological modelling (2 CFU)
Teacher: Lorenzo Pareschi

Syllabus: The reading course will be focused on some recent results in the field of epidemiologic modelling. In particular, it will be based on the following article:
G. Albi, L. Pareschi, M. Zanella. Control and uncertainty quantification in socially structured epidemiologic models. Preprint 2020.

Dates: reading course, May or June 2021

Title: Optimization methods for machine learning (6 CFU)
Teacher: Luca Zanni, Valeria Ruggiero, Serena Crisci

Syllabus: Introduction to machine learning. Supervised Learning: loss functions, empirical risk minimization, regularization approaches. Gradient descent approaches: deterministic and stochastic frameworks. Decomposition techniques and gradient projection methods for support vector machines. Stochastic Optimization in learning methodologies: topics and perspectives. Implementation issues for large-scale learning.

Dates: Ferrara, three days- July 2021 (18 hours )

Title: Topics in Discrete Mathematics (4 CFU)
Teachers: A. Bonisoli, Simona Bonvicini, Giuseppe Mazzuoccolo, Anita Pasotti, G. Rinaldi

Syllabus: The goal of this course is to introduce students to ideas and techniques from discrete mathematics. A general introduction to basic concepts in Graph Theory, Design Theory and Combinatorics will be furnished. After that, some advanced topics and recent results will be presented. They include (but are not limited to) Matchings and Colorings in Graphs, Latin Squares, Balanced Block Designs, Decomposition of Graphs and Enumerative Combinatorics.

Dates 2020/2021: From January to March 2021

Title: Duality Theory of Markov Processes (3 CFU)
Teacher: Cristian Giardinà, Gioia Carinci

Dates 2020/2021: Reading course, beginning of 2021 (the precise schedule will be decided together with the students).

Title: Introduction to the regularity theory for elliptic PDE’s (3 CFU)
Teacher: Michela Eleuteri

Syllabus: The aim of the course is to give an introduction to the regularity theory for solutions of elliptic partial differential equations and local minimizers of integral functionals, illustrating some of the classical results available in literature. Some aspects of the regularity of nonlinear elliptic systems will be also given.
The topics treated in the course will be the following:

The Hilbert spaces approach to the existence of solutions of nonlinear elliptic systems in divergence form. The lower semicontinuity of integral functionals by the direct methods of the Calculus of Variations.
The Caccioppoli inequality.
The Nirenberg method of difference quotient to derive Hilbert space regularity in the interior and up to the boundary.
Holder, Morrey and Campanato’s spaces.
The Schauder theory.
The L^p theory.
The regularity in the scalar case: De Giorgi-Nash-Moser Theorem.
De Giorgi’s counterexample to the regularity for systems.
Partial regularity for systems.

References:

L. Ambrosio, Lecture Notes in Partial Differential Equations. https://cvgmt.sns.it/
M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Math. Studies n. 105, Princeton University Press, Princeton 1983.
M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics, Birkhauser, 1993.
D. Gilbarg& N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer Verlag, Heidelberg, New York, 1977.
E. Giusti, Direct Methods in the Calculus of Variations. World Scientific, Singapore, 2003.

Date 2020/2021: Second Semester

Title: Topics in Discrete Morse Theory (3 CFU)
Instructor: Claudia Landi

Syllabus: The goal of this course is to introduce students to Morse Theory from the combinatorial standpoint, as to the study of discrete topological spaces via discrete vector fields. Providing easily computable homotopy-invariant simplifications, discrete Morse theory finds applications at the interface between mathematics and computer science, preeminently in topological data analysis.

Dates 2020/2021: From March to May 2021

Title: Variational methods for imaging (6 CFU)
Teachers: Germana Landi, Federica Porta, Simone Rebegoldi

Syllabus: The image formation model: forward and inverse problems. Regularization techniques. Statistical approach for Gaussian and Poisson data. Forward-backward methods in differentiable and non-differentiable optimization: basic theory, variable metrics, inertial techniques, inexact solution of proximal operator, non-convex problems. Applications in tomography, astronomy and microscopy.

Dates 2020/2021: Course on-line
23/02/2021 9:00-13:00;
24/02/2021 9:00-13:00 & 14:00-16:00;
25/02/2021 9:00-13:00.

Title: Hypoelliptic Partial Differential Equations (3 CFU)
Teachers: Sergio Polidoro, Maria Manfredini

Syllabus: The subject of the course are the linear second order Partial Differential Equations with non-negative characteristic form satisfying the Hormander's hypoellipticity condition. Maximum principle, local regularity, boundary value problem will be discussed for several examples of equations. Some open research problems will be described. The course will focus on the following topics:

Bony's maximum principle for degenerate second order PDEs, propagation set and Hormander's hypoellipticity condition.
Perron method for the boundary value problem in a bounded open set of the Euclidean space.
Boundary regularity, barrier functions. Boundary measure, Green function.
Fundamental solution. Mean value formulas. Harnack inequalities.
Degenerate Kolmogorov equations. Applications to some financial problems.

The program may be modified in accordance with the requirements of the students.

Reference text: A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians (Springer Monographs in Mathematics - 2007). Further references and lecture notes will be given during the course.

Dates 2020/2021: The course will start after March 2021

2019-2020 (Ciclo XXXV)

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

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: h ere

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)

Title: High Order numerical methods for hyperbolic PDE - CFU: 6
Teacher: Walter Boscheri

Syllabus:
- introduction to numerical methods for PDE (finite volume FV and discontinuous Galerkin (DG) methods);
- Godunov's theorem (1959);
- high order FV schemes (reconstruction operators, a priori and a posteriori limiters);
- high order DG schemes;
- high order time discretization: Runge-Kutta and ADER time stepping techniques.

Date: 12 hours (2 days) - March 2020 (to be decided with the students).

Title: Plane Cremona transformations
Teacher: Alberto Calabri

Syllabus: Fundamental points and exceptional curves of a plane Cremona map, examples like quadratic and De Jonquières maps, properties like Noether's equations and inequality, factorization of maps and proofs of Noether-Castelnuovo theorem, Cremona equivalence of plane curves, plane curves of minimal degree with respect to plane Cremona maps.

Dates: February-March 2020 (16-20 hours)

Title: Numerical methods for kinetic equations
Teacher: Giacomo Dimarco

Syllabus: In this course we consider the development and the mathematical anal-ysis of numerical methods for kinetic partial differential equations. Kinet-ic equations represent a way of describing the time evolution of a sys-tem consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful bal-ance between accuracy and computational complexity. We review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the deri-vation of fast algorithms, the notion of asymptotic preserving methods and the construction of hybrid schemes.

Lecture I: Preliminaries on kinetic equations
Lecture II: Semi-Lagrangian schemes
Lecture III: Discrete velocity and spectral methods
Lecture IV: Breaking complexity: fast algorithms
Lecture V: Asymptotic-preserving schemes
Lecture VI: Fluid-kinetic coupling and hybrid methods

References
[1] G. Dimarco, L. Pareschi. Numerical methods for kinetic equations, Acta Numerica, 23 (2014), pp. 369–520.

Dates: to be established, 6/8 hours

Title: High Performance and High Throughput Computing for Data Science

Teachers: Sebastiano Fabio Schifano, Luca Tomassetti

Syllabus: Modern supercomputers are parallel processors, gaining their power from the concurrent execution of thousands of individual CPU-cores, each core in turn able to process vector operations. Developing efficient software to run on these systems requires parallel programming technologies to map at best the computing requirements of the application onto the hardware features of these systems. This course will cover all the fundamental concepts that underpin modern HPC providing hands-on experience, as students will explore these topics through the analysis of real parallel programs. These techniques can also be applied to standard multi-core processors as well as many-core processors, such as recent GP-GPUs and Xeon-Phi systems. Beside the high performance paradigm, high throughput computing is nowadays widely used in virtualized environments, when computation is loosely parallel or embarrassingly parallel. In these cases the work-load can be divided in several independent tasks to be executed on different cpus or cores. The course will cover the architectural aspects and provide practical examples.

As example of learning outcomes we expect students to
    • Understand the key components of HPC architectures.
    • Understand the key components of HTC architectures.
    • Be able to develop parallel and efficient scientific codes for modern computing systems.
    • Be able to use and develop scientific applications on virtualized environments.
    • Measure and comment on the performance of parallel codes.

Dates: 8 lessons of 2 hours each in the 2nd semester to be agreed with students

Title: Advanced Topics in Combinatorics
Teacher: Simona Bonvicini

Syllabus: The cours aims to consolidate some basic knowledge in Graph Theory and Combinatorial Designs, as well as to introduce some arguments that might be investigated by the PhD students in the audience. A series of lectures on some advanced topics related to the following areas will be delivered: Latin squares; block designs; and chromatic parameters for graphs. The following speaker will contribute to the lectures: Bonisoli, Bonvicini, Mazzuoccolo and Rinaldi.

Suggested Textbooks:
T. Beth, D. Jungnickel, H. Lenz, Design Theory, Cambridge
J.A. Bondy, U.S.R. Murty, Graph Theory, Springer
J. Dénes, A.D. Keedwell, Latin squares and their applications, Akadémiai Kiadó, Budapest, 1974.
R. Diestel, Graph Theory, Springer
J. van Lint, R.H. Wilson, A Course in Combinatorics, Cambridge

Dates: The total duration of the lectures will be from 15 to 25 hours, in the period between December 2019 and June 2020.

Title: Introduction to Optimal Transport - CFU: 6
Teacher: Luigi De Pascale

Syllabus:
1. The Rademacher and Alexandrov theorems on differentiability of Lipschitz and convex functions;
2. Rockafellar's characterixation of the sub-differential of convex functions;
3. Formulation of Monge's optimal transport problem;
4. Kantorovich's relaxation and convex duality;
5. Optimality conditions and existence of optimal transport maps;
6. Monge-Ampere equation.

This will be a self-contained introduction to optimal transport theory. The main tools will be introduced and then all the basic theory of existence will be covered. This will allow the students to study autonomously the first book of C. Villani on optimal transport.

Dates: November, 11, 12, 21, 22, 28 and 29

Title: Introduction to the regularity theory for elliptic PDE’s - CFU: 6
Teacher: Michela Eleuteri

Syllabus: The aim of the course is to give an introduction to the regularity theory for solutions of elliptic partial differential equations and local minimizers of integral functionals, illustrating some of the classical results available in literature. Some aspects of the regularity of nonlinear elliptic systems will be also given.
The topics treated in the course will be the following:

The Hilbert spaces approach to the existence of solutions of nonlinear elliptic systems in divergence form. The lower semicontinuity of integral functionals by the direct methods of the Calculus of Variations.
The Caccioppoli inequality.
The Nirenberg method of difference quotient to derive Hilbert space regularity in the interior and up to the boundary.
Holder, Morrey and Campanato’s spaces.
The Schauder theory.
The L^p theory.
The regularity in the scalar case: De Giorgi-Nash-Moser Theorem.
De Giorgi’s counterexample to the regularity for systems.
Partial regularity for systems.

References:
[1] L. Ambrosio, Lecture Notes in Partial Differential Equations. htpp:www.cvgmt.sns.it
[2] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Math. Studies n. 105, Princeton University Press, Princeton 1983.
[3] M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems. Lectures in Mathematics, Birkhauser, 1993.
[4] D. Gilbarg& N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer Verlag, Heidelberg, New York, 1977.
[5] E. Giusti, Direct Methods in the Calculus of Variations. World Scien-tific, Singapore, 2003.

Date: First Semester

Title: Variational methods for imaging and machine learning - CFU: 6
Teachers: Germana Landi, Federica Porta, Valeria Ruggiero, Luca Zanni

Syllabus: The image formation model: forward and inverse problem, regularization, statistical approach for Gaussian and Poisson data. First-order methods in differentiable and non-differentiable optimization. Al-ternating direction multiplier methods.

Introduction to machine learning. Supervised Learning: loss functions, empiri-cal risk minimization, regularization approaches.
Optimization techniques for machine learning. Topics and challenges in Stochastic Optimization.

Dates: February 18-19-20, 2020 (three full days).

Title: Hypoelliptic Partial Differential Equations - CFU: 6
Teachers: Sergio Polidoro, Maria Manfredini

Bony's maximum principle for degenerate second order PDEs, propagation set and Hormander's hypoellipticity condition.
Perron method for the boundary value problem in a bounded open set of the Euclidean space.
Bondary regularity, barrier functions. Boundary measure, Green function.
Fundamental solution. Mean value formulas. Harnack inequali-ties.
Degenerate Kolmogorov equations. Applications to some fi-nancial problems.

The program may be modified in accordance with the requirements of the students.
Reference text: A. Bonfiglioli, E. Lanconelli, F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians (Springer Monographs in Mathematics - 2007). Further references and lecture notes will be given during the course.

Dates: The course will start after March 2020

Modified on 25/11/2023

Graduate Courses in Mathematics (2019-2022)