Modena-Reggio Emilia 2022/23
Lectures take place at “Edificio Matematica”, via Campi 213/c, 41125 Modena.
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 2022/2023: Reading course, beginning of 2023 (precise schedule will be decided together with the students).
Title and credits: Optimization methods for machine learning
Teachers: Federica Porta, Giorgia Franchini, Alessandro Benfenati
Syllabus: Introduction to machine learning. Supervised Learning: loss functions, empirical risk minimization, regularization approaches. Gradient descent approaches: deterministic and stochastic frameworks, also with reduced variance. Hyperparameters choice in the stochastic gradient method. Stochastic Optimization in learning methodologies: topics and perspectives in constrained and
unconstrained fields. Implementation issues for large-scale learning. Link between stochastic gradient and Deep learning. Neural architecture search techniques.
Dates: 13-14-15 February 2023, all day long (9:30-13:00 and 14:00-17:30).
Title and Credits: Geometry of Deep Learning, 4 CFU
Teacher: Rita Fioresi
Syllabus:
- Introduction to Deep Learning, basic steps of the algorithm analogies with the human visual systems and its mathematical models.
- The geometry of the space of data and the space of parameters; KL divergence and its information geometry interpretation.
- Geometric Deep Learning: the algorithm of Deep Learning on Graphs.
- Message passing and GATs: a geometrical modelling via heat equation and laplacian on graphs.
This course will be self-contained as much as possible. The necessary differential geometric concepts (manifolds, Frobenius theorem, Cartan formalism) will be introduced and explained. The necessary programming skills will NOT be assumed, but a part of the course will be "hands on" illustrating key examples on colab.
The exam will consist in a brief exposition of some concepts and the students can choose the part of the
program they like the most and present a focused exposition on one argument.
Six lectures of approximately 2.5 hours each.
Dates 2022/2023: 15h lectures + reading course + home assignments
Title and Credits: New Directions in Designs and Graphs, 6 CFU
Teachers: Simona Bonvicini, Giuseppe Mazzuoccolo, Anita Pasotti, Tommaso Traetta.
Syllabus: The course will present some advanced topics and recent results in Design Theory and in Graph Theory.They include (but are not limited to) decompositions and factorizations of graphs and their symmetries, and the study of several classes of arrays.Open problems and applications will be presented too. Interested students are requested to send an email to: anita.pasotti@unibs.it
Dates 2022/2023: Reading course, beginning of 2023 (the precise schedule will be decided together with the students).
Title and credits: Topological and comparison-type methods for the study of boundary value problems in differential equations, 4CFU
Teacher: Luisa Malaguti
Syllabus: The Course deals with some important methods for the study of boundary value problems to ordinary and partial differential equations. The Leray-Schauder topological degree will be briefly introduced, and its applications discussed in the study of periodic solutions and solutions satisfying Cauchy multi-point conditions in parabolic equations. The upper and lower solutions technique for ordinary differential equations will be then proposed and its application given to the study of traveling wave solutions of reaction-diffusion equations with degenerate diffusivities.
Dates 2022/2023: Lectures will take place from March to June 2023 at the University of Modena and Reggio Emilia for an amount of 18 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
Title and credits: Selected topics on algebraic curves over finite fields (20 hours)
Teacher: Giovanni Zini
The course will consider some selected topics in the theory of algebraic curves over finite fields. Useful previous knowledge: elementary theory of algebraic curves. The topics will be selected among the following ones.
- Maximal curves over finite fields: properties, classical examples (Hermitian, Suzuki and Ree curves), recent families (GK curve, GGS curve, BM curve, Skabelund curves).
- Automorphism groups of curves, and quotient curves: bounds on the size, examples. Automorphism and quotients of the Hermitian curve: classification.
- Rational points of curves over finite fields: criteria and methods for the analysis of absolutely irreducible rational components of curves, in particular for what concerns plane curves.
- Applications of the study of rational points to some remarkable families of polynomials over finite fields which are of interest in cryptography.
Title and credits: Hodge Theory (20hours, 6CFU)
Teacher: Camilla Felisetti
Hodge theory is a powerful tool to understand the topology of Riemannian Manifolds. The aim of this course is to give an introduction to the theory together with applications to complex algebraic geometry. In particular we will treat the following topics:
- Recap on differentiable manifolds, differential forms and vector bundles
- Compact Kähler manifolds
- Hodge theory for Kähler manifolds
- Hodge theory for algebraic projective varieties
Dates 2023: Lectures will take place from September to October 2023 at the University of Modena and Reggio Emilia for an amount of 20 hours. Precise dates will be decided together with the interested PhD students, who are encouraged to contact the teacher in advance.
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