PhD Course: Extreme Value Theory

Nov 262025

Extreme Value Theory

Dr. Amir Khorrami Chokami

Università di Cagliari

Abstract

This doctoral course provides a comprehensive introduction to Extreme Value Theory (EVT), covering both classical probabilistic foundations and modern statistical modeling tools, with particular attention to dependence structures and applications to real data.
We begin with the fundamental limit theorems for maxima and the characterization of max-stable laws (described by the so-called Generalised Extreme-Value family of distributions). We then introduce the Point-Process approach to extremes and the Peaks-over-Threshold (POT) method (leading to the Generalised Pareto Distribution). The probabilistic foundations of extremes are followed by the statistical aspects of the theory, including parameter estimation and diagnostic techniques, all implemented in the R software. A fundamental topic of the course regards the behavior of extremes under dependence. We study how to describe the tail dependence of multiple variables and the connections with the multivariate extreme value distributions. We then address extremes of stationary sequences, including mixing conditions, the extremal index and the clustering of extreme events.

Outline

Fundamentals of Extreme Value Theory and Max-stable Distributions
Point Process Representation of Extremes
Peaks-over-Threshold Method and the Generalized Pareto Distribution
Statistical Inference: Estimation, Diagnostics
Tail Dependence and Multivariate Extremes
Extremes of Stationary Processes: Mixing, Clustering, Extremal Index
Applications and R Implementations

Schedule

TBA

Exam

The final assessment consists of reading and presenting a research article selected from a list provided during the course.

References

Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J., Statistics of Extremes: Theory and Applications, Wiley, 2004
Coles, S., An Introduction to Statistical Modeling of Extreme Values, Springer, 2001
de Haan, L., Ferreira, A., Extreme Value Theory: An Introduction, Springer, 2006
Resnick, S. I., Extreme Values, Regular Variation and Point Processes, Springer, 1987

PhD Course: Riemann Surfaces

Nov 192025

Riemann Surfaces

Prof. Roberto Mossa
University of Cagliari, Italy

Abstract

This 16-hour PhD course is devoted to algebraic functions on Riemann surfaces, following Forster’s Lectures on Riemann Surfaces up to §8 (Algebraic Functions), with emphasis on the construction of the associated Riemann surfaces and on Puiseux expansions near branch points.

In the final part we will explain how these classical tools naturally appear in recent problems in Kähler geometry, notably in the study of Nash–algebraic functions, simultaneous normalization, and rigidity of holomorphic isometries into Kähler manifolds. As a case study we will discuss the paper A. Loi, R. Mossa, Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds, Proc. Amer. Math. Soc. 152 (2024), no. 7.

Schedule

The course will take place in January-February 2026. The course consists of 16 hours divided into 8 lectures. The exact schedule will be announced soon.

Exam

The final exam consists in a seminar on a topic building on the content of the course. The topic for the final exam may be proposed by the students themselves or chosen from a list provided at the end of the lectures.

PhD Course: Insights and Algorithms for Ill-Posed Problems

Set 262025

Insights and Algorithms for Ill-Posed Problems

Prof. Lothar Reichel and Prof. Laura Dykes
Kent State University, USA

Abstract

The aim of this course is to introduce Master’s and Ph.D. students to linear ill-posed problems. Their properties and applications will be discussed. The focus of the lectures will be on solution methods for linear discrete ill-posed problems and on the numerical linear algebra required for the solution of these problems.

While there are no prerequisites, a basic knowledge of numerical linear algebra, least squares, LU, QR and SVD factorizations, and Matlab programming will be helpful for successfully following the lessons.

Outline

Linear discrete ill-posed problems: Definition, properties, applications
Solution methods for small to moderately sized problems: Regularization, Tikhonov regularization, the singular value decomposition, the generalized singular value decomposition, choice of regularization matrix.
Solution methods for large problems: Iterative methods based on the Lanczos process, the Arnoldi process, and Golub-Kahan bidiagonalization. Regularization by Tikhonov’s method and truncated iteration. Iterative methods for general regularization matrices.
lp-lq minimization for image restoration.

Schedule

Monday September 29, 15-18, room B
Thursday October 2, 15-18, room 2
Friday October 3, 15-18, room 2
Monday October 6, 15-18, room B

The first four lectures will be broadcast on Microsoft Teams for students who cannot attend in person.

The final two lectures, each lasting two hours, will be delivered on Teams after the instructors return to their offices. The schedule will be released during the lectures.

Anyone interested in participating in the course should contact the organizers, Alessandro Buccini and Giuseppe Rodriguez.

Exam

TBA

Acknowledgements

The course is partially supported by the INdAM Visiting Professors Program.

PhD Course: Qualitative Properties of Solutions of Uniformly Parabolic Equations

Mag 232025

Qualitative Properties of Solutions of Uniformly Parabolic Equations

Prof. Daniele Castorina
Università di Napoli Federico II

Dott. Simone Ciani
Università di Bologna

Abstract

The course is divided in two sections:

Section 1 – Second Order Parabolic Equations (Ciani)
In the first lectures we will follow Ch. VII of [1], by setting up a definition of solution of parabolic uniformly elliptic equations. In the sequel we will prove existence and uniqueness for the Cauchy-Dirichlet problem, thanks to the method of Galerkin approximations and a priori estimates. Then we will approach regularity theory: first we will prove that the unique solution to the boundary value problem proposed improves its regularity as much as the initial value datum allows, until we reach the smoothness C-infinity by a bootstrap argument. In the last lecture, time permitting, we will comment on the lack of regularity when the coefficients and the data are rough; and give a glimpse of the possible minimal regularity properties affordable, following Chap XI-XII of [2].

Section 2 – The Alexandrov-Bakelman-Pucci method and its applications (Castorina)
The classical Alexandrov-Bakelman-Pucci (or ABP) estimate is a uniform bound for strong solutions of second order uniformly parabolic operators with bounded measurable coefficients written in nondivergence form. Its main feature is being a basic tool in the regularity theory for fully nonlinear parabolic equations. However, the ABP method is fairly general and it can be adapted to a wide variety of different issues such as obtaining a maximum principle in domains of small measure, as well as simplifying the proofs of several isoperimetric and Sobolev inequalitiees. The aim of this second part course is to introduce the ABP method in detail, discuss some of its generalizations and refinements and to give a detailed and complete overview of its applications, explicitly highlighting the improvements of using this technique with respect to previous and more classical tools.

Outline

Existence and Uniqueness
Regularity Theory I – Improvement of Regularity
Minimal Regularity II – Hölder Continuity, Harnack inequality and Applications
The Alexandrov-Bakelman-Pucci estimate
The Maximum Principle in small domains
The Gidas-Ni-Nirenberg Theorem and Isoperimetric and Sobolev inequalities

Schedule

8/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A
9/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A
10/7/2025 11:00 – 13:00 and 15:00 – 17:00 room A

Exam

The final assessment consists of one of the two choices: a list of exercises to solve (during the course) and a seminar; or a written elaborate on selected topics of the course.

References

L. Evans, Partial Differential Equations, Second Edition, AMS, 1998.
E. DiBenedetto, U. Gianazza, Partial Differential Equations, Third Edition, Birkhäuser, 2023.
Berestycki, H., Nirenberg, L. On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22, 1991, 1–37.
Berestycki, H., Nirenberg, L., Varadhan, S. R. S. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math. 47,1994, 47–92.
Cabré, X. On the Alexandrov-Bakelman-Pucci estimate and the reversed Holder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math. 48, 1995, 539–570.
Cabré, X., Ros-Oton, X. Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations 255, 2013, 4312–4336.
Cabré, X., Ros-Oton, X., Serra, J. Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc. 18, 2016, 2971–2998.
Gilbarg, D., Trudinger, N. S. Elliptic Partial Differential Equations of Second Order. 2nd ed., Springer-Verlag, Berlin-New York, 1983.

PhD Course: Introduction to Compositional Data Analysis and Modelling

Mar 232025

Introduction to Compositional Data Analysis and Modelling

Prof. Fabio Divino
Università del Molise & University of Jyväskylä

Abstract

This course introduces the fundamental concepts of compositional data analysis, including the algebraic structure of the simplex and its main properties. It will then cover the basic tools for analysis before presenting the main regression models:
(a) compositional data as a predictor;
(b) compositional data as a response variable;
(c) compositional data as both predictor and response.

All topics will be explored through hands-on lab sessions in R using real data and simulations.

Outline

Lecture 1: Introduction to Compositional Data Analysis. ALR, ILR, and CLR Transformations
Lecture 2: Descriptive Analysis in the Simplex. Introduction to Regression Models
Lecture 3: Regression Models with Compositional Data

Schedule

The course consists of a total of 6 hours, scheduled as follows:

March 17, 15:00-17:00 – Aula A
March 19, 11:00-13:00 – Aula A
March 21, 15:00-17:00 – Aula A

Exam

The final exam consists of a presentation on a specific topic covered in the course.

References

K. Gerald van den Boogaart & Raimon Tolosana-Delgado, Analyzing Compositional Data with R, Springer, 2013.

PhD Course: Introduction to Algorithmic Fairness

Mar 122025

Introduction to Algorithmic Fairness: Principles, Methods and Regulatory Perspectives

Dr. Erasmo Purificato
European Commission, Joint Research Centre (JRC), Italy

Abstract

The course provides a comprehensive introduction to algorithmic fairness, exploring key concepts such as definitions, bias characterisation and the potential sources of unfairness in machine learning models. Initially, we will thoroughly examine fairness criteria, bias detection metrics, and the limitation of fairness evaluation in binary scenarios. Then, we will analyse the emerging multiclass and multigroup approaches, and cover bias mitigation techniques and their practical trade-offs. Finally, the course will examine legal and ethical frameworks governing algorithmic fairness, with a focus on EU regulations such as the GDPR, DSA, and AI Act, as well as global policies.

Outline

Lecture 1: Foundation of Algorithmic Fairness
- Why fairness matters in AI and ML
- Defining fairness and bias
- Potential causes of unfairness in ML
- Fairness criteria
- Conflicts between fairness goals
Lecture 2 – Measuring Bias and Fairness
- Bias detection metrics
- Challenges in binary scenarios
- Extending fairness metrics to multiclass and multigroup scenarios
Lecture 3 – Mitigating Bias
- Bias mitigation strategies
- Choosing the right fairness intervention
- Trade-offs and practical implementations
Lecture 4 – Legal and Ethical Frameworks for Fairness in AI
- Overview of EU Regulations affecting AI and ML (i.e., GDPR, DSA and AI Act)
- Fairness principles in EU Regulations
- Fairness principles in global regulations
- The future of algorithmic fairness and open research challenges

Schedule

The course will have a total duration of 10 hours, scheduled as follows:

May 21, 14:00-18:00 Aula II
May 22, 10:30-12:30 Aula F
May 22, 14:00-16:00 Aula F
May 23, 10:00-12:00 Aula F

Exam

The final exam consists either in a seminar presentation focusing on a specific topic studied during the course or in a test held on the last day of the course. The definitive format will be announced when the schedule is finalized. The course will be held in person. Please contact me if you are interested in joining.

References

The content of the course is based (but not limited to) the following articles:

Simon Caton and Christian Haas. Fairness in Machine Learning: A Survey. ACM Comput. Surv. 56, 7, Article 166 (2024). https://dl.acm.org/doi/10.1145/3616865
Corbett-Davies, Sam, Johann D. Gaebler, Hamed Nilforoshan, Ravi Shroff, and Sharad Goel. The measure and mismeasure of fairness. Journal of Machine Learning Research 24, no. 312 (2023). https://jmlr.org/papers/v24/22-1511.html
Dana Pessach and Erez Shmueli. A Review on Fairness in Machine Learning. ACM Comput. Surv. 55, 3, Article 51 (2023). https://doi.org/10.1145/3494672
Sahil Verma and Julia Rubin. Fairness definitions explained. In Proceedings of the International Workshop on Software Fairness (FairWare 2018). https://doi.org/10.1145/3194770.3194776
Purificato, Erasmo, Ludovico Boratto, and Ernesto William De Luca. Toward a responsible fairness analysis: from binary to multiclass and multigroup assessment in graph neural network-based user modeling tasks. Minds and Machines 34, no. 3 (2024). https://doi.org/10.1007/s11023-024-09685-x
Regulation (EU) 2016/679 of the European Parliament and of the Council of 27 April 2016 on the protection of natural persons with regard to the processing of personal data and on the free movement of such data, and repealing Directive 95/46/EC (General Data Protection Regulation),
2016, OJ L119/1. http://data.europa.eu/eli/reg/2016/679/oj
Regulation (EU) 2022/2065 of the European Parliament and of the Council of 19 October 2022 on a Single Market For Digital Services and amending Directive 2000/31/EC (Digital Services Act), 2022, OJ L277/1. http://data.europa.eu/eli/reg/2022/2065/oj
Regulation (EU) 2024/1689 of the European Parliament and of the Council of 13 June 2024 laying down harmonised rules on artificial intelligence and amending Regulations (EC) No 300/2008, (EU) No 167/2013, (EU) No 168/2013, (EU) 2018/858, (EU) 2018/1139 and (EU) 2019/2144 and
Directives 2014/90/EU, (EU) 2016/797 and (EU) 2020/1828 (Artificial Intelligence Act), 2024, http://data.europa.eu/eli/reg/2024/1689/oj

PhD Course: Interpretable and Explainable Machine Learning Models

Gen 102025

Interpretable and Explainable Machine Learning Models

Dr. Claudio Pomo
Politecnico di Bari

Abstract

The course focuses on methods for interpreting and explaining machine learning (ML) models, including inherently interpretable approaches and post-hoc explanation techniques. Key concepts of interpretability will be introduced, alongside the analysis of interpretable models and the application of explanation methods for complex models. The course critically evaluates existing techniques in terms of fidelity, stability, fairness, and practical utility, while addressing open challenges and future perspectives.

Schedule

The course will have a total duration of 10 hours, scheduled as follows:

March 20, 15:00-17:30 Aula II
March 21, 10:00-12:30 Aula F
March 24, 15:00-17:30 Aula II
March 25, 10:00-12:30 Sala Riunioni II piano

Exam

The final exam consists of a project analyzing a case study using the techniques and tools acquired during the course. The course will be held in person. Please contact me if you are interested in joining.

References

Lundberg, S. M., and Lee, S.-I. A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems, 2017.
Ribeiro, M. T., Singh, S., and Guestrin, C. Why should I trust you? Explaining the predictions of any classifier. Proceedings of the ACM SIGKDD, 2016.
Molnar, C.. Interpretable Machine Learning: A Guide for Making Black Box Models Explainable. 2nd edition, 2022.
Doshi-Velez, F., and Kim, B. Towards a rigorous science of interpretable machine learning. arXiv preprint, 2017.
Agarwal, C., Krishna, S., Saxena, E., Pawelczyk, M., Johnson, N., Puri, I., … & Lakkaraju, H. Openxai: Towards a transparent evaluation of model explanations. Advances in Neural Information Processing Systems, 2022

PhD Course: Introduction to algebraic logic

Ott 242024

Introduction to algebraic logic

Dr. Nicolò Zamperlin
Università degli Studi di Cagliari

Abstract

The course is an introduction to the theory of algebraizability of Blok and Pigozzi. Through an analytic study of the first chapters of Font’s handbook on abstract algebraic logic we will first introduce the elementary notions of universal algebra needed for linking together logic and algebra (closure operators and their lattices, varieties, quasivarieties and equational consequences), then building upon these notions we will consider the case of implicative logics and their algebraic properties, introducing the technique of completeness through the Lindenbaum-Tarski process. Finally we generalize these notions to the class of algebraizable logics (with a glimpse to the larger Leibniz heirarchy), with the ultimate goal of proving the isomorphism theorem and the transfer for the deduction theorem.

Schedule

The course will have a duration of 20 hours, scheduled as follows:

November 7, aula B, h. 15-17
November 14, aula A, h. 15-17
November 22, aula II, h. 9:30-11:30
November 29, aula II, h. 9:30-11:30
December 2, aula II, h. 9:30-11
December 6, aula II, h. 9:30-11:30
December 12, aula B, h. 15-17
January 15, aula B, h. 10-12
January 20, aula B, h. 15-17
February 3, aula B, h. 10-12

Exam

The final exam consists in a seminar presentation. The course will be held in person. Please contact me if you are interested in joining the course

References

Bergman, C., Universal Algebra: Fundamentals and Selected Topics, Chapman & Hall Pure and Applied Mathematics, Chapman and Hall/CRC, 2011.
Blok, W., and Pigozzi, D., Algebraizable logics, vol. 396 of Memoirs of the American Mathematical Society, A.M.S., 1989.
Burris, S., and Sankappanavar, H.P., A course in Universal Algebra, freely available online: https://www.math.uwaterloo.ca/snburris/htdocs/ualg.html, 2012 update.
Czelakowski, J., Protoalgebraic logics, vol. 10 of Trends in Logic: Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2001.
Font, J.M., Abstract Algebraic Logic: An Introductory Textbook, College Publications, 2016

PhD Course: Conic Programming: Theory and applications

Set 232024

Conic Programming: Theory and applications

Prof. Benedetto Manca
Università degli Studi di Cagliari

Abstract

The course covers the theory of conic programming, starting from the simplest case of linear programming and introducing conic quadratic and semi-definite programming. The first part of the course will introduce the theoretical backgrounds needed to define the concept of conic programming. In the second part the case of conic quadratic and semi-definite programming will be addressed together with some applications.

Outline

From Linear to Conic Programming
Conic Quadratic Programming
The quadratic formulation of the Distance Geometry Problem
Semi-definite Programming
The semi-definite relaxation of the Distance Geometry Problem
Diagonally dominant matrices and positive semi-definite matrices
The ellipsoidal separation problem

Schedule

The course consists in 10 hours, two lectures per week. Details will be specified on the occasion of the first lecture, which will be given on October 3, 2024 at 2:30 p.m. in room B of the Department of Mathematics and Computer Science.

Exam

The final exam consists in a presentation on a specific application of conic programming (conic quadratic or semi-definite).

References

Ben-Tal, Aharon, and Arkadi Nemirovski. Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Society for industrial and applied mathematics, 2001.
Liberti, Leo. Distance geometry and data science. Top 28.2 (2020): 271-339
Astorino, Annabella, et al. Ellipsoidal classification via semidefinite programming. Operations Research Letters 51.2 (2023): 197-203.

PhD Course: Introduction to Kähler Geometry

Set 122024

Introduction to Kähler Geometry

Prof. Roberto Mossa, Prof. Giovanni Placini
Università degli Studi di Cagliari

Abstract

This introductory course covers some of the fundamental concepts of Kähler geometry, with particular attention to almost complex and complex manifolds, the properties of Hermitian metrics, and Kähler metrics. Starting from the basics of differential geometry, we will explore the structure of almost complex and complex manifolds. Subsequently, we will delve into the properties of Hermitian metrics, focusing on the definition and characteristics that define Kähler metrics, which play a key role in integrating the complex structure with the Riemannian one. Through concrete examples and applications, students will gain a deep understanding of these concepts, preparing them for advanced studies in Kähler geometry.

Schedule

The course consists of 32 hours divided into 16 lectures. This is the schedule of the lectures:

Martedì 14 Gennaio 15-17 Aula III (Giovanni Placini)
Giovedì 16 Gennaio 11-13 Aula III (Giovanni Placini)
Martedì 21 Gennaio 11-13 Aula III (Giovanni Placini)
Giovedì 23 Gennaio 11-13 Aula III (Giovanni Placini)
Martedì 28 Gennaio 11-13 Aula III (Giovanni Placini)
Giovedì 30 Gennaio 11-13 Aula III (Roberto Mossa)
Martedì 4 Febbraio 11-13 Aula III (Roberto Mossa)
Giovedì 6 Febbraio 11-13 Aula III (Roberto Mossa)
Martedì 11 Febbraio 11-13 Aula III (Roberto Mossa)
Giovedì 13 Febbraio 11-13 Aula III (Roberto Mossa)
Lunedì 17 Febbraio 11-13 Aula III (Roberto Mossa)
Giovedì 20 marzo 11-13 Aula F (Roberto Mossa)
Giovedì 27 marzo 11-13 Aula F (Roberto Mossa)
Giovedì 3 aprile 11-13 Aula F (Giovanni Placini)
Martedì 8 aprile 11-13 Aula F (Giovanni Placini)
Giovedì 10 aprile 11-13 Aula F (Giovanni Placini)

Exam

Precedenti