## Modena-Reggio Emilia

**Lectures take place at “Edificio Matematica”, via Campi 213/c, 41125 Modena.**

Title: **Discrete Morse Theory**

Teacher: Bruno Benedetti (University of Miami)

Syllabus: Discrete Morse theory was introduced by Forman in 2000. Together with its ancestor, simple-homotopy theory, it yields a method to simplify any given triangulation, while preserving its homological data. We give an elementary introduction to the theory, using a more geometric and intuitive approach. At the same time, we explain how to find "good" discrete Morse functions quickly, with the help of a computer. If time permits, we sketch also some interesting applications to other areas of geometry, like knot theory or geometric topology.

Dates: from May 29 to June 13, 2019.

Title: **New Directions in Designs and Graphs**

Teacher: Bonisoli Arrigo

Syllabus: In the academic year 2018/2019 the course "New Directions in Designs and Graphs" for the graduate students of the PhD programme in Mathematics will consist of a series of lectures on various topics in Graph Theory and Combinatorics by Bonisoli, Bonvicini, Mazzuoccolo, Rinaldi. Each of them will speak on one ore more topics related to the following areas:

- Matchings and factorizations
- Block designs
- Graph decompositions and graph designs
- Chromatic parameters

The purpose of the activity is to reinforce some basic knowledge of Graph Theory and Combinatorics as well as introduce some advanced topics that may well be pursued by the PhD students in the audience in view of their PhD thesis.

Suggested Textbooks:

- T. Beth, D. Jungnickel, H. Lenz, Design Theory, Cambridge
- J.A. Bondy, U.S.R. Murty, Graph Theory, Springer
- R. Diestel, Graph Theory, Springer
- J. van Lint, R.H. Wilson, A Course in Combinatorics, Cambridge

Title: **Programming parallel architectures: an introduction**

Teacher: Burgio Paolo

Syllabus: In the last fifteen years, tehcnological advances in the design and manufacturing of electronic circuits gave birth to the multi- and many-core era. On one side, multi-core processors are adopted since a decade in "consumer" computers, such as laptops and workstations. On the other side, many-core based accelerators (e.g., GP-GPU-based) are increasingly being used not only in the High-Performance domain but also for embedded computing, and they are used for example to power up the state-of-the-art (and beyond) neural network-based software packages for autonomous driving vehicles.

Unfortunately, exploiting the tremendous potential of these systems requires the programmers of today and of tomorrow to gain first-hand knowledge on the complex hardware platform and software stack (OS, runtime libs..) as well as on both "traditional" and emerging parallel programming patterns for such systems.

- In the first part of the course, the main architectural templates for multi- and many-core computing systems will be described, and their pros and cons will be highlighted;
- in the second part of the course, we will discuss the mostly adopted programming patterns, languages and frameworks that will drive the software development process today and in the next future;
- finally, some words will be spent on how neural network-based techniques are quickly gaining attention as a promising computing paradigm for the next-generation of software, and ultimately will drive the revolution to the internet 5.0 era.

Dates: to be defined (April/May 2019).

Title: **Direct methods in the Calculus of Variations: a short introduction**

Teacher: Eleuteri Michela

Syllabus: The aim of the course is to present a short overview on Direct Methods in the Calculus of Variations. The course will be divided in the following parts:

- Part 1: we discuss conditions under which minimizers of integral functionals are solutions to partial differential equations named Euler-Lagrange equations, showing several examples and counterexamples to the question of existence of minimizers.
- Part 2: we present the basic ideas of the so called Direct Methods in the Calculus of Variations, along the notions of semicontinuity, minimizing sequence and application of a generalization of the Weierstrass theorem.
- Part 3: we prove an existence theorem for minimizers of integral functionals, discussing counterexamples showing why conditions like coercivity, reflexivity and convexity are optimal.
- Part 4: we present some basic facts concerning relaxation theory, explaining a procedure that can be followed once the Direct Methods fail. Then we connect the relaxation theory with the notion of Gamma-convergence, which is the right notion of convergence for a sequence of functionals allowing their minimizers to converge to the minimizer of the limit functional. We introduce the concept of Gamma-convergence through a series of meaningful examples.

L.C. Evans: Partial Differential Equations, AMS 1994.

G. Mingione: Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math, 2006.

Dates: to be defined.

Title: **Phase transitions problems: mathematics and modelling**

Teacher: Eleuteri Michela

Syllabus: The aim of the course is to deal with phase transitions either from the point of view of the modelling and of the mathematical analysis.

After a short introduction recalling the basic notions of thermodynamics, general laws and constitutive relations, the concepts of phase transition of first and second order and the ones of order parameter and free energy, we derive the Allen-Cahn and Cahn-Hilliard equations as examples of phase transitions models with non-conserved and conserved dynamics respectively. These are examples of nonlinear partial differential equations, so we will explain in general what are the main difficulties one can face when dealing with the mathematical treatment of such equations, in particular we will focus on the role of the compactness. We present the so-called Lions’ method, normally used to find existence of solutions to nonlinear PDEs, based on three steps: approximation, a priori estimates, passage to the limit by compactness. We apply this method to the Allen-Cahn equation.

In the second part of the course we want to focus our attention to the classical phase-transition model: the Stefan problem. To this aim, we first present some basic notions of convex analysis, in particular the concept of subdifferential. Then we give some hints about how to solve the weak formulation of the Stefan problem, along the lines of Lions’ method.

We conclude by connecting the notions of phase transitions and hysteresis: so we introduce the concept of hysteresis operator and give a simple example that can be formulated in terms of convex analysis. Then we use it to discuss some generalizations of the Stefan model in the direction of a dissipative behaviour.

References: M. Brokate, J. Sprekels: Hysteresis and phase transitions, Springer, 1996 – Chapter 4

A. Visintin: Introduction to Stefan-Type Problems, Handbook of Differential Equations, Chapter 8, Elsevier, 2008

A. Visintin: Models of Phase Transitions, Birkaeuser, 1996

Dates: to be defined.

Title: **Probability theory and statistical physics**

Teacher: Giardinà Cristian

Syllabus: In the academic year 2018/2019 the course "Probability theory and statistical physics" for the graduate students of the PhD programme in Mathematics will coincide with the Course "Statistical mechanics and dynamical system" which is offered in English for the Master Degree in Mathematics. The purpose of the course is to make the audience acquainted with the following basic topics:

- Large deviations for identical and independent random variable
- Generalization to correlated random variable
- Varadhan lemma
- Curie-Weiss model
- Ising model
- Probabilistic description of phase transitions: breaking of law of large numbers, scaling limits at criticality.

*Large Deviations*, Frank den Hollander, Field Institute Monograph, American Mathematical Society, 2000;

*Entropy, Large Deviations and Statistical Mechanics*, Richard S. Ellis, Springer-Verlag 1985.

Dates: the course consists of 36 lectures, from October 11th to Decembre 20th, Wed and Thu, 9.00-11:00.

Title: **Martingales and Brownian Motion**

Teacher: Giardinà Cristian

Syllabus: In the academic year 2018/2019 the course "Martingales and Brownian motion" for the graduate students of the PhD programme in Mathematics will coincide with the Course "Probabilistic models" which is offered in English for the Master Degree in Mathematics. The purpose of the course is to make the audience acquainted with the following basic topics in stochastic process theory:

Martingales: definitions and properties of discrete time martingales, Azuma-Hoeffding inequality, martingale converge theorem, stopping times, martingale stopping theorem. Applications. Martingales and Markov chains. Continuous time martingales. Brownian Motion: definitions of Brownian motion, Markov and Strong Markov property, heat equation, recurrence, Feynman-Kac formula, arcsine law, harmonic oscillator, exit times from bounded intervals. Stochastic integration: Ito integral, Ito's formula and the martingale representation theorem. Diffusions.

References: S.R.S. Varadhan, *Stochastic processes*, Courant Lecture Notes. Sheldom Ross, *Stochastic processes*, John Wiley and Sons, Inc. Bernt Øksendal, *Stochastic Differential Equations, An Introduction with Applications*, Fifth Edition, Springer-Verlag Heidelberg New Yor

Dates: the course consists of 36 lectures that will be scheduled in the Spring Semester between 1 March and 1 June 2019

Title: **Differential dynamics and non-equilibrium statistical mechanics**

Teacher: Giberti Claudio

Syllabus: Statistical Mechanics consists of two different parts: equilibrium and nonequilibrium. The progress of nonequilibrium statistical mechanics has been much slower and has not reached the spectacular successes of the equilibrium theory. For instance: the well-known phenomenological Fourier's law in transport theory is lacking a rigorous derivation from microscopic dynamics. In these lectures I'll discuss the role of some ideas taken from dynamical system theory in the study of nonequilibrium systems. Moreover I'll discuss also some deterministic models applied in the numerical simulations of such systems (heat baths). Keywords: Statistical mechanics; nonequilibrium; steady states; smooth dynamics; deterministic baths.

Dates: June-July 2019

Title: **Geometric Measure Theory**

Teacher: Leonardi Gian Paolo

Syllabus: This course provides a basic introduction to Geometric Measure Theory, focused on the theory of sets of finite perimeter and BV functions. The topics include:

- Covering, differentiation, and decomposition theorems for measures.
- Hausdorff measure and dimension, rectifiability. Area and coarea formulae.
- BV functions and sets of locally finite perimeter. Compactness, semicontinuity, and smoothing theorems. Isoperimetric and Sobolev-Poincaré inequalities. Reduced boundary of a set of finite perimeter: blow-up, rectifiability, and representation of the perimeter measure. Existence of minimizers of the perimeter. Density estimates and monotonicity formula.
- A glimpse on the regularity theory for minimizers of the perimeter.
- Introduction to the theory of varifolds.

- Maggi. Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory. No. 135. Cambridge University Press, 2012.
- Ambrosio-Fusco-Pallara. Functions of bounded variation and free discontinuity problems. Vol. 254. Oxford: Clarendon Press, 2000.
- Evans-Gariepy. Measure theory and fine properties of functions.
- Simon. Lectures on geometric measure theory. Australian National University, Mathematical Sciences Institute, Centre for Mathematics & its Applications, 1983.Simon.

Title: **Advanced combinatorial designs**

Teacher: Dott. Marbach Trent (Nankai University, China)

Syllabus: This course introduces Latin squares and other related combinatorial designs, and displays current research of these topics. his course is split into four parts, as follows:

- Part 1: Latin squares for design theory- In this lesson, we introduce Latin squares along with their connections to other design theoretic objects. In particular, we show how Latin squares may be used to construct Steiner triple systems of all admissible orders using the Latin square based constructions of Bose and Skolem. The concept of orthogonal it is developed.
- Part 2: Research topics in the theory of Latin squares- Orthogonality is further explored by observing the current research on mutually orthogonal Latin squares and orthogonal arrays. This is used as motivation to introduce transversal of Latin squares. We explore the current state-of-the-art of Latin square transversal research, including topics of existence and enumeration.
- Part 3: Combinatorial computing with dancing links- we introduce the dancing links data structure, and a c++ implementation called exact Cover. This data structure is a tool in modern combinatorial computing that enables existence checking and enumeration of a large variety of combinatorial structures. The method of encoding a problem into exactCover is presented, and examples from the literature using both Latin squares and other combinatorial objects is given.
- Part 4: Applications of Latin squares - In the final lesson, we introduce. The cryptographic topic of distribution schemes. We introduce two Latin square based distribution schemes: one using critical sets of Latin squares, and the other using autotopisms of Latin squares. We show how the later has been implemented to enhance security in a multi-cloud storage scheme.

- Dates: Mercoledì 20 marzo 2019, dalle ore 10 alle ore 13, aula M2.3;
- Giovedì 21 marzo 2019, dalle ore 11 alle ore 13, aula M2.4;

Title: **Hypoelliptic Partial Differential Equations**

Teacher: Polidoro Sergio, 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.
- Bondary regularity, barrier functions. Boundary measure, Green function.
- Fundamental solution. Mean value formulas. Harnack inequalities.
- Degenerate Kolmogorov equations. Applications to some financial problems.

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 2019

Title: **Variational methods for imaging**

Teacher: Zanni Luca (in collaboration with Valeria Ruggiero, Ferrara)

Syllabus: One of the most difficult challenges in scientific computing is the development of algorithms and software for large scale ill-posed inverse problems, such as imaging denoising and deblurring. Such problems are extremely sensitive to perturbations (e.g. noise) in the data. To compute a physically reliable approximation from given noisy data, it is necessary to incorporate appropriate regularization into the mathematical model. Numerical methods to solve the regularized problem require effective numerical optimization strategies and efficient large scale matrix computations. In these lectures we describe first and second-order methods, dual or primal-dual approaches, and Bregman-type schemes and how to efficiently implement the ideas with iterative methods on realistic large scale imaging problems.

Dates: 15 hours, 2019, to be defined.