## Ferrara

**Lectures take place at Dipartimento di Matematica e Informatica, Sede: via Machiavelli, 30 - 44121 Ferrara, Sede distaccata: via Saragat, 1 - 44122 Ferrara**

Title: **Quantum groups and Hopf algebras**

Teacher: Prof. Torrecillas

Syllabus:

1. Definitions and examples of Hopf algebras.

2. Quantum plane.

3. The quantum groups SLq(2) and GLq(2).

4. The quantum enveloping algebra Uq(sl(2)).

5. Monoidal categories.

6. Yang-Bexter equation.

7. Braided monoidal categories.

8. Drinfeld's quantum double.

Dates: 23 June - 6 July 2019, 4 seminars of two hours each. The timetable of the lessons can be established in agreement with the interested students.

Title: **Plane Cremona transformations**

Teacher: Calabri Alberto

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.

Dates: March 2019, 10-15 hours.

Title: **Numerical methods for kinetic equations**

Teacher: Dimarco Giacomo

SyllabusIn this course we consider the development and the mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system 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 balance 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 derivation 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 : G. Dimarco, L. Pareschi. Numerical methods for kinetic equations, Acta Numerica, 23 (2014), pp. 369–520.

Dates: 6/8 hours, to be defined.

Title: **Topics in Continuum Mechanics applied to Biology**

Teacher: Giantesio Giulia

Syllabus: The aim of this course is to describe some mathematical models in Continuum Mechanics which are widely used in Biology.

Starting from a review of kinematics and a short introduction about rheology and hyperelasticity, we present both fluid and elastic models. First, we introduce some constitutive models for fluid flows and show how the Navier-Stokes equations can be generalized in order to include some biological specific behavior.

Second, we study a hyperelastic model of muscle tissue which can describe the ability of the muscle of being activated through a chemical reaction. This process, which is usually called "activation", is now widely discussed in the literature. We present the two most popular approaches: active stress and active strain.

Dates: March/May 2019, 10-12 hours

Title: **Rational surfaces and Sarkisov Program**

Teacher: Mella Massimiliano

Syllabus: Introduction to the study of rational surfaces with special regards to their birational geometry. Introduction to Sarkisov Program, with applications to surfaces and Cremona group of the plane. I will mainly work over an algebraic closed field of characteristic 0, with some digressions over non algebraically closed fields.

Dates: January-February 2019, 10-15 hours

Title: **Monads and their applications**

Teacher: Menini Claudia

Sylabus: Beck's Theorem and its applications. Descent data, symmetries and connections associated to a monad.

Dates: January-February 2019, 10-12 hours.

Title: **Calculus of Variations and Geometric Measure Theory; application to the theory of BV functions.**

Teacher: Miranda Michele

Syllabus: This course is an introduction to the theory of functions of bounded variations; the target is to describe fine properties of BV functions and set with finite perimeter using tools of geometric measure theory. These properties shall be used to prove properties of traces of BV functions on rectifiable codimension one surfaces. We shall also see applications to some PDE equations and minimization of some functional of the calculus of variations.

Dates: Januray – March 2019, 15-20 hours.

Title: **Infinite Dimensional Analysis**

Teacher: Miranda Michele, in collaboration with Lunardi Alessandra

Syllabus: This is an introductory course about analysis in abstract Wiener spaces, namely separable Banach or Hilbert spaces endowed with a nondegenerate 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 semigroup. The most important functional inequalities in this context, such as Poincare' 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 Surveys and Monographs 62, AMS 1998). In addition, lecture notes prepared by the teachers will be available.

References: Lecture notes available at http://dmi.unife.it/it/ricerca-dmi/seminari/isem19 V.Bogachev, “Gaussian Measures”, AMS, 1999

Dates: November 2018/February 2019 (approx. 40 hours, either in Parma or in Ferrara according to the needs of the students)

Title: **Variational methods for imaging**

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

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, to be defined.

Title: **High Performance and High Throughput Computing for Data Science**

Teacher: Schifano Sebastiano Fabio, Tomassetti Luca

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 to agree with students.

Title: **Introduction to Modal and Temporal Logic**

Teacher: Sciavicco Guido

Syllabus: In this course we go over the very basic classic theory of modal logic. We start by some historical notions, syntax, (Kripke) semantics. Then we move to more advanced topics such as axiomatization and semantic tableaux. Finally, we cover some essential aspects of temporal logic that lead to some of the most important current lines of research in this direction.

Dates: 6 hours, second semester. To be decided with the interested students.