Courses in Modena-Reggio Emilia
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
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 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.
- 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
24/02/2021 9:00-13:00 & 14:00-16:00;
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