Abstracts of the courses
Algebra and combinatorics
Yasuhide Numata (University of Hokkaido): Basics of Gröbner bases
Ideals and quotient rings of a polynomial ring over a field; term order and Groebner bases; Buchberger Algorithm to obtain a Gröbner basis.
Lea Terracini (Università di Torino): Number theoretical methods for cryptography
A little history. Mono and poly-alphabetic methods. Introduction to. Modular arithmetics and finite fields. Some modern systems of symmetric cryptography (OTP, AES). Asymmetric cryptography, one-way functions. Classical methods in asymmetric cryptography: Diffie-Hellman, El Gamal, RSA. The discrete logarithm problem. Introduction to elliptic curves and cryptography methods based on them. Some notions about post quantum cryptography and Shor algorithm.
Analytic methods in dynamical systems
Paolo Giulietti (Università di Pisa): The transfer operator approach
I will present some basic ideas which have been extremely fruitful in the study of chaotic dynamical systems.
I will introduce the transfer operator and detail how its functional analytical properties are tied to the physical properties of the systems studied, such as the existence of invariant measures and the decay of correlations. I will discuss a wishful spectral picture for such an operator, on suitable functional spaces, for some simple examples which highlight the strengths of such method.
Alfonso Sorrentino (Università di Roma Tor Vergata): Action-minimizing methods
In this part of the course, I will provide an overview of the so-called Aubry-Mather theory, namely a series of variational results related to the so-called action-minimizing principle, that allows one to study the dynamics of twist maps of the annulus and to construct interesting families of invariant sets. Several applications will be discussed.
Harmonic maps
Paolo Caldiroli (Università di Torino): CMC and near-CMC surfaces of the type of the cylinder
In these lectures, we deal with immersed surfaces in $\mathbb{R}^{3}$ with constant mean curvature (CMC, for short) or near-CMC. In particular we focus on surfaces of the type of the cylinder, i.e., immersions of $I\times\mathbb{S}^{1}$ into $\mathbb{R}^{3}$, where $I=[-T,T]$ or $I=\mathbb{R}$. In the compact case one usually prescribes also boundary conditions.
This kind of surfaces appears in gas dynamics, in capillarity phenomena, and also in biology (forms of stems of plants, small unicellular organisms, or individual cells in the simpler aggregates). CMC surfaces well approximate static equilibrium configurations of continua supported only by surface tension. The presence of external forces, like gravity or other fields, may produce pressure differences which locally change, hence, according to the Young-Laplace equation, the mean curvature along phase interfaces is no longer a constant. Hence also the near-CMC problem is meaningful.
In 1841, the astronomer and mathematician C. Delaunay explicitly described a certain class of CMC surfaces with an axis of revolution. Such surfaces are quite important in the construction of a plethora of complete, CMC surfaces, of any genus and any number of ends. They can be obtained by attaching together spheres and pieces of Delaunay surfaces, as first accomplished by N. Kapouleas in the '90s and then by other authors.
In more recent years, also the study of immersions of the compact cylinder $[-T,T]\times\mathbb{S}^{1}$ having fixed boundary raised some interest: the occurrence of bifurcations and symmetry breaking, already observed by J. Plateau in his experimental investigation on the Delaunay surfaces, has been rigorously proved by M. Koiso, B. Palmer and P. Piccione. Even more recently, the corresponding near-CMC problem has been studied and some existence and nonexistence results have been obtained by P. Caldiroli in collaboration with G. Cora and A. Iacopetti.
Shimpei Kobayashi (University of Hokkaido): Harmonic Gauss maps and generalized Weierstrass type representations
The generalized Weierstrass type representation for harmonic maps from a surface into symmetric spaces has been established by J. Dorfmeister, F. Pedit and H. Wu in 1998. A family of flat connections and the loop group Birkhoff and Iwasawa decompositions are the core of the representation, and it has been commonly called the loop group method. Since the unit normal for a constant mean curvature (CMC) surface in the Euclidean 3-space is a harmonic map to the 2-sphere, the loop group method can be applied for a construction of CMC surfaces in the Euclidean 3-space. On the one hand, it has been known that surfaces in in the Euclidean 3-space or 4-space can be formulated by the pair of generating spinors and the nonlinear Dirac equation (it has been also called the Weierstrass representation). More recently, it can be extended to three-dimensional Lie groups with left-invariant metric.
In the first half of lectures, I will explain the basics of generating spinors and the nonlinear Dirac equation, and combining the loop group method.
In the second half of lectures, I will explain more recent work of the construction of minimal surfaces in the three-dimensional Heisenberg group with left invariant metric using these techniques.
Probability and machine learning
Paolo Baldi (Università di Roma Tor Vergata): Large Deviations, rare events, simulation
The course will introduce to Large Deviations and some of their applications. The first part will be about the theory and the tools thereof. The second part will be around applications to the estimation of the probability of rare events and simulation.
Maurizio Parton (Università di Chieti e Pescara): TBA
TBA
Abstracts of the HPRT talks
Luigi Caputi (University of Torino): A gentle introduction to Topological Data Analysis, and applications
Applications of (algebraic) topology to data analysis and sciences – from biology to digit recognition, from neuroscience to finance – have recently led to a new and flourishing interdisciplinary research domain: Topological Data Analysis. At the crossroad between topology, statistics, machine learning and data science, TDA was born from the idea that data have a shape, and that this shape contains relevant information about the data. The aim of the talk is to introduce the basic concepts and ideas in TDA and, in particular, in persistent homology, with an eye towards applications.
Valerio Formichella (INRiM): Mathematics and Statistics in Time Metrology
Time metrology is the science of time measurements, with several applications for scientific research, technology, industry, and even everyday life. Among the major actors in this field, we have atomic clocks, the most precise clocks currently available. The dynamics of their output needs to be accurately modelled from the mathematical point of view, including deterministic and stochastic components. At the same time, statistics and data analysis are necessary when processing time measurement data. An enlightening example on this regard is timekeeping within Global Navigation Satellite Systems (GNSS), requiring the most precise clocks and mathematical tools to provide accurate and trustable Position, Navigation and Timing solutions. The seminar will introduce the audience to the basics of time metrology and to the mathematical and statistical tools used in this field, with particular attention to GNSS applications. Future perspectives including the use of Machine Learning are also briefly discussed.
Mirko Moscatelli (Banca d'Italia): Machine learning for credit scoring
Credit scoring plays a crucial role in the financial industry, helping lenders assess the creditworthiness of individuals and businesses seeking credit. Traditionally, credit scoring has relied on expert judgement and statistical models to evaluate credit applicants. With the advent of machine learning, however, a new way of evaluating credit scoring is starting. In this presentation we will see how machine learning models differ from traditional statistical ones, how they perform in predicting corporate default risk and how it is possible to mitigate their main limitation, the lack of an easily interpretable functional form.
Ottavia Telve (San Paolo): Treatment of COVID-19 recession in Italian default rate projection model
Since the onset of the COVID-19 recession, stress testing models have dramatically overpredicted losses. It is not a surprise: all models are trained on past events, but no pattern exists in current industry training data for an equivalent combination of natural disaster, government assistance and loan forbearance as it has happened in 2020 due to COVID-19. The methodology put in place before the COVID-19 recession is not adequate for describing the link between macro-economic scenarios and the evolution of default rates anymore. The latter applies especially in the Italian system, where the severity of the crisis (in terms of, for example, GDP reduction and unemployment increment) was strongly mitigated with various measures adopted by governments, institutions, and central bank. These actions were adopted to support the economy; hence they did not reveal in an expected increase in company defaults. In this talk, we show an improved version of the classic Error Correction Model used for the Italian default rates projection before 2020. It will be tested to consider the different behaviour between Italian GDP and default rate due to the government measures before and after the pandemic. We illustrate an Italian default rate projection model for Banking Sector, specifically estimated on the 2005Q3-2021Q4 period and with a forecast for the 2022Q1-2023Q4 period, where two different scenarios will be presented.
Additional notes:
Paolo Baldi (Università di Roma Tor Vergata): Large Deviations, rare events, simulation
Paolo Caldiroli (Università di Torino): CMC and near-CMC surfaces of the type of the cylinder
Slides: Lecture 1, Lecture 2, Lecture 3, Lecture 4
Luigi Caputi (Università di Torino): A gentle introduction to Topological Data Analysis, and applications
Valerio Formichella (INRiM): Mathematics and Statistics in Time Metrology
Rosa Gini (ARS Toscana): Using data to assess causality
Shimpei Kobayashi (University of Hokkaido): Harmonic Gauss maps and generalized Weierstrass type representations
Two-dimensional Dirac operator and the theory of surfaces
The Spinor Representation of Surfaces in Space
Mirko Moscatelli (Banca d'Italia): Machine learning for credit scoring
Maurizio Parton (Università di Chieti e Pescara): Machine Learning, Neural Networks, and Geometric Deep Learning for Mathematicians.
Link to the shared folder
Lea Terracini (Università di Torino): Number theoretical methods for cryptography
Slides: Lecture 1, Lecture 2, Lecture 3, Lecture 4