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Courses in Ferrara

PhD in Mathematics 36th cycle

(PhD Mathematics) Home | Program | Faculty PhD Students

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

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)

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 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 )

Pubblicato Martedì, 19 Maggio, 2020 - 17:22 | ultima modifica Giovedì, 22 Ottobre, 2020 - 10:41