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

  1. Existence and Uniqueness
  2. Regularity Theory I – Improvement of Regularity
  3. Minimal Regularity II – Hölder Continuity, Harnack inequality and Applications
  4. The Alexandrov-Bakelman-Pucci estimate
  5. The Maximum Principle in small domains
  6. 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

  1. L. Evans, Partial Differential Equations, Second Edition, AMS, 1998.
  2. E. DiBenedetto, U. Gianazza, Partial Differential Equations, Third Edition, Birkhäuser, 2023.
  3. Berestycki, H., Nirenberg, L. On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.) 22, 1991, 1–37.
  4. 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.
  5. 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.
  6. Cabré, X., Ros-Oton, X. Sobolev and isoperimetric inequalities with monomial weights, J. Differential Equations 255, 2013, 4312–4336.
  7. Cabré, X., Ros-Oton, X., Serra, J. Sharp isoperimetric inequalities via the ABP method, J. Eur. Math. Soc. 18, 2016, 2971–2998.
  8. Gilbarg, D., Trudinger, N. S. Elliptic Partial Differential Equations of Second Order. 2nd ed., Springer-Verlag, Berlin-New York, 1983.
 Scritto da in 23 Maggio 2025  Senza categoria  Commenti disabilitati su PhD Course: Qualitative Properties of Solutions of Uniformly Parabolic Equations
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.
 Scritto da in 23 Marzo 2025  Senza categoria  Commenti disabilitati su PhD Course: Introduction to Compositional Data Analysis and Modelling
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:

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 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
  8. 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
Gen 142025
 

MAIN PhD Seminars 2024

Date Speaker(s)
January, 15th Sandro Gabriele Tiddia
January, 29th Andrea Azzarelli
February, 12nd Valentino Artizzu
February, 26th Simone Pusceddu
March, 12nd Nicola Piras
March, 26th Sara Vergallo
Giorgia Nieddu
April, 9th Lejzer Javier Castro Tapia
April, 23rd Matteo Mocci
May, 7th Giuseppe Zecchini
May, 21st Antonio Pio Contrò
June, 4th Matteo Palmieri
June, 18th Michele Faedda
June, 25th Giuseppe Scarpi

All the seminars at 13:00 in Aula Magna di Fisica.

 

Sandro Gabriele Tiddia: LLM Agents: Definitions and Real-World Applications

In this seminar, we explore the concept of ‘agents’ in artificial intelligence (AI), with a particular focus on the role of Large Language Models (LLMs) in powering these systems. The seminar begins by discussing a real-world application where LLMs are used to build a question-answering (QA) system, showing how LLMs can function as ‘agents’ within such a system. We then examine various definitions of ‘agent’ across AI subfields and consider how agents interact with their environment, make decisions, and pursue goals autonomously. Additionally, we revisit earlier works on agency in AI, reflecting on their original, more profound ideas, and connect them to recent developments and applications of LLM-powered agents. The seminar concludes by exploring experiments and use cases from recent literature, highlighting the capabilities and potential of LLM agents across different domains. The goal is to provide a clear introduction to the concept of LLM-powered agents, their role in AI systems, and how this concept has evolved from theoretical foundations to practical applications.

Andrea Azzarelli: Fractional Laplacian and ADMM for glyph extraction

In archaeology it is a common task to extract incisions or glyphs from a surface. This procedure is usually done manually and, therefore, it is prone to errors and it can be extremely time consuming. In this talk we present a variational model to automatically extract these incisions from a smooth surface. We provide a procedure to generate realistic synthetic data and we show the performances of the proposed method on this kind of data.

Valentino Artizzu: End-User Development for Extended Reality: Empowering Users to Create and Understand XR Environments

In this seminar we will see how to enable individuals without prior XR development experience to create and understand XR environments. It focuses on using End-User Development (EUD) techniques to allow users to design, build, and adapt XR systems. The research specifically explores methodologies and tools for non-programmers to construct XR environments. It examines how EUD can facilitate novice developers in comprehending existing VR environments for enhancement purposes. The study further investigates how EUD can empower domain experts to tailor these environments to meet diverse requirements. Additionally, it delves into how EUD can guide domain experts in configuring XR environments to support task learning and demonstration. The goal is to provide a clear overview of the topic using the experiences and the applications developed during the three years of a PhD career.

Nicola Piras: Global and local fit measures for latent class models and extensions

Latent class (LC) analysis is a powerful and flexible statistical tool for model-based clustering with categorical data. An important task in LC analysis is the choice of the number of clusters or classes. The choice of the number of classes is a selection model problem and usually Information Criteria are considered for this purpose. These are measures that weigh model fit (log-likelihood) and model complexity (based on the number of free parameters). The LC models formulation is subject to an assumption of conditional independence between the variables involved. Adherence to this assumption and the correct estimation of the parameters is central to evaluate how well the model fits to the data. While global selection and the goodness of fit is verified through Information Criteria, model conditions must also be checked. Specific statistics can be defined that allow to verify the local independence assumption. In the literature of LC analysis the statistics considered are the Bivariate residuals. The standard LC model can be modified to handle more complex data structures, and fit measures must be adapted to the new formulations. In this talk, after having briefly discussed the results in the standard formulation, I will also present the extension to the case in which data have a multilevel cross-classified structure. This structure is present when observations are simultaneously nested within two groups, for example, children nested within both schools and neighborhoods. An application is illustrated using an Italian dataset on the evaluation of students of their degree programmes, where degree programmes are nested in both universities and fields of study.

Sara Vergallo: Mathematics as a foundation for learning Machine Learning from primary school onward

The proliferation of artificial intelligence (AI) in many people’s daily lives has led the education sector to recognize the importance of teaching elements of AI—and in particular Machine Learning (ML)—from the earliest stages of schooling (Karalekas et al., 2023). Consequently, there is a need for resources, guidelines, and studies on the feasibility and methods for integrating ML into lower‐school settings, beginning in kindergarten (Sanusi et al., 2023), so that children can understand how the machines they interact with every day work (Lin & Brummelen, 2021). Teaching machine learning in primary and secondary school is very challenging, also due to students’ deficiencies in data analysis—especially in classification (sets) and data representation (trees, two‐way tables) (Grillenberger & Romeike, 2019)—an essential competency included in mathematics curricula from primary school onward and in secondary‐school computer science curricula, yet insufficiently promoted or stimulated (Grillenberger & Romeike, 2019). We investigated the level of these mathematical skills in a fifth‐grade class at a primary school and proposed a didactic pathway composed almost entirely of unplugged activities for learning the basics of ML. The activity fostered an improvement in the children’s classification and representation skills and received a high level of enthusiasm; this latter point is significant, as one of the critical issues identified in the literature is the lack of engaging activities within the school context (Grillenberger & Romeike, 2019). The results from these in‐class research activities will then be considered within a broader research overview concerning the use of non‐standard approaches (such as game‐based learning) to enhance mathematical skills, which are also necessary for a better understanding of computer‐science content.

Giorgia Nieddu: Use of GenAI to Support Learning and Teaching Mathematics

This seminar will present our recent experiments on the use of large multimodal models for learning and teaching mathematics. The first, conducted in North Macedonia, explored how students interact with GenAI in electronics problem-solving activities with mathematical content; the second, carried out in collaboration with the University of Turin, investigated how GenAI can support teachers in preparing educational materials. In this latter study, conducted within teacher training courses, we observed teachers’ behaviors and attitudes, aiming to understand whether AI could be useful in their lesson planning and how.

Lejzer Javier Castro Tapia: Mod-2 Cohomological Classification of Orbit Spaces of Free Involutions on the 2-Fold Projective Product Space

The action of a compact Lie group on a topological space describes the symmetries of our space, in that sense the properties of the orbit space of these actions has attracted many mathematicians over the world since the beginning of the twentieth century. In this talk we present in an informative way a cohomological classification via spectral sequences of orbit spaces of free involutions on the two-fold projective product space, a manifold that generalizes the usual projective spaces and wich was introduced for the first time by Donald Davis in 2010.

Matteo Mocci: Automatic Walkability Assessment using AI and multi-input image classification

Walkability is a key element of sustainable and livable cities, influencing public health, environmental impact, and social connectivity. Traditional methods for assessing walkability, such as surveys and audits, are often time-consuming and limited in scale. Recent advancements in artificial intelligence and computer vision offer new opportunities to automate and enhance these assessments using street-level and aerial imagery. This seminar explores how deep learning and multi-perspective image analysis can provide more comprehensive and scalable walkability evaluations. By integrating insights from urban planning, AI, and geospatial analysis, we will discuss the potential and challenges of these emerging technologies in shaping more pedestrian-friendly cities.

Giuseppe Zecchini: On the algebraic study of substructural logics by means of Płonka sums

Logic can be intuitively defined as the science of correct reasoning: given a certain set of premises, we want to be able to establish their consequences. Algebraic Logic can naively be defined as the study of Logic through the methods of Algebra. In the first part of this talk, we will explain in detail what a logic formally is and what it means to study it algebraically. In the second part, we will introduce and motivate substructural logics, which are traditionally defined by means of Gentzen-style sequent calculi in which one or more of the structural rules (exchange, weakening, contraction) of Gentzen’s LK calculus for classical logic are restricted or eliminated. Finally, in the third part, we will present the algebraic counterpart of substructural logics, residuated lattices, and make some remarks on the study of their structure through the construction of Płonka sums, a construction introduced in Universal Algebra in the 1960s by the eponymous Polish mathematician.

 Scritto da in 14 Gennaio 2025  Senza categoria  Commenti disabilitati su MAIN PhD Seminars 2025
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

  1. Lundberg, S. M., and Lee, S.-I. A unified approach to interpreting model predictions. Advances in Neural Information Processing Systems, 2017.
  2. 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.
  3. Molnar, C.. Interpretable Machine Learning: A Guide for Making Black Box Models Explainable. 2nd edition, 2022.
  4. Doshi-Velez, F., and Kim, B. Towards a rigorous science of interpretable machine learning. arXiv preprint, 2017.
  5. 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
 Scritto da in 10 Gennaio 2025  Senza categoria  Commenti disabilitati su PhD Course: Interpretable and Explainable Machine Learning Models
Giu 172024
 

Foundations of Digital Systems and Applications

Dr. Zsolt Ercsey
University of Pécs, Hungary

Outline

  1. Introduction to internet technology
  2. Comparison of the business model of mobile operating systems (Android and iOS)
  3. Introduction to a mobile tv recommendation system
  4. Introduction to a sport shooting scheduling system
  5. Solution of a human resource allocation problem

Schedule

Please register to the course through this form.

June 24th, 9.30-11.30 Aula F
June 25th, 9.30-12.30 Aula F
June 26th, 9.30-12.30 Aula F
June 28th, 9.30-11.30 Aula F (final test)

Exam

The final exam consists in a test, which will be taken on the last day of the course.

Mar 072024
 

Quantum computing: foundations and state of the art

Prof. Michele Marchesi
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The seminar is about the current status of Quantum Computing. It will present the math fundamentals of quantum computing, and then will cover Quantum Annealing, which is the current most consolidated application field, featuring working machines with 5000 qbits. The seminar will then cover the current status and possible applications of Quantum Computing, including Shor’s algorithm and the quest for quantum resistant cryptographic algorithms.

Schedule

March, Monday, 11st and Tuesday, 12nd – 17:30-19.00 (Palazzo delle Scienze, Room B) 

 Scritto da in 7 Marzo 2024  Senza categoria  Commenti disabilitati su Seminar: Quantum computing: foundations and state of the art
Giu 232023
 

Boundary value problems with nonsmooth and multivalued terms

Prof. Vasile Staicu
University of Aveiro, Portugal

Abstract

The aim of this course is to present some methods and tools used in the study of nonlinear boundary value problems involving multivalued maps and nonsmooth functions and illustrate those methods with some recent results concerning existence and multiplicity of solutions. The course will be given during the period April 26 – May 16, 2023, under the support of Visiting Professor Program of INdAM.

Outline

We start by introducing some notions of multivalued analysis: multivalued maps and their continuity. Then we introduce some elements of nonsmooth analysis that we use to develop nonsmooth critical point theory succeeded in extending a big part of the smooth theory to nonsmooth functionals: the main tool in the variational method for solving boundary value problems, which consists of trying to find solutions for a given boundary value problem by looking for stationary points of a real functional defined on a space of functions in which the solution of the boundary value problem is assumed to lie.

Degree theory is a basic tool of nonlinear analysis and produces powerful existence and multiplicity results for nonlinear boundary value problems. We present a generalization of Brouwer degree theory to multivalued perturbations of monotone type maps, developed in a joint paper with Aizicovici and Papageorgiou (Mem. Amer. Math. Soc., 196, 2008). Then we consider a nonlinear Dirichlet problem with nonsmooth potential (hemivariational inequality) and use variational method to prove two constant sign solutions (one positive and another negative) and then by using degree theoretical approach we prove the existence of a third one, nontrivial solution, distinct from the previous two.

We present several existence and multiplicity results, with sign information for the solutions of nonlinear boundary value problems driven by the Laplacian, the p-Laplacian and then try to extend such results to nonlinear boundary value problems driven by fractional order differential operators and to nonlocal pseudo-differential inclusions, possibly including obstacles or constraints. It is reasonable to expect that the pairing of nonlocal diffusion operators and set-valued reaction will provide a more realistic model for applications to such problems as quantum mechanics, image restoration, and financial mathematics, which typically present a high degree of uncertainty, rather than elliptic equations with smooth, single-valued reactions.

Schedule

27/4 11-13 28/4 9-11 2/5 11-13 5/5 9-11 9/5 11-13 11/5 11-13 12/5 9-11 16/5 11-13 (room F)

References

Vasile Staicu, Lecture Notes, 2023

Open questions will be considered and analyzed with the Ph. D. students.

 

 Scritto da in 23 Giugno 2023  Senza categoria  Commenti disabilitati su PhD Course: Boundary value problems with nonsmooth and multivalued terms
Gen 042023
 

Matrix functions: Computation and application to Complex Networks analysis

Dott. Caterina Fenu
Dipartimento di Matematica e Informatica
Università degli Studi di Cagliari

Abstract

The aim of the course is to introduce basic concepts related to the computation of matrix functions and their use in the analysis of complex networks. Half of the course will be dedicated to the implementation of some algorithms by using the MATLAB software.

Outline 

– Linear algebra: eigenvalue decomposition, Jordan decomposition, singular value decomposition;
– Introduction to matrix functions and bilinear/quadratic forms: orthogonal polynomials, Lanczos algorithms, quadrature formulas;
– Introduction to complex networks theory: graphs, centrality/communicability indices, network indices;
– Use of the MATLAB software.

Exam

The oral exam will consist of a discussion of a research paper concerning the topics of the course.

References

1. G. H. Golub and G. Meurant, Matrices, Moments and Quadrature with Applications, Princeton University Press, Princeton, 2010.
2. N. J. Higham, Functions of matrices: theory and computation, SIAM, 2008.
3. W. Gautschi, Orthogonal polynomials: computation and approximation, Oxford University Press, 2004.

 

 Scritto da in 4 Gennaio 2023  Senza categoria  Commenti disabilitati su PhD Course: Matrix functions: Computation and application to Complex Networks analysis
Nov 292022
 

Factorizations in monoids and rings

Laura Cossu
University of Graz

Outline

Many problems in algebra involve the decomposition of certain elements of a ring (or more generally of a monoid) into a product of certain other elements (hereinafter generically referred to as building blocks) that are in some sense minimal. The classical theory of factorization investigates factorizations in which the building blocks are atoms, i.e., non-unit elements of a monoid that are not products of two non-units. For example, it is well known that every non-zero non-unit of a Dedekind domain (more generally, of a Noetherian domain) can be written as a finite product of atoms and that in general such decompositions are not unique. On the other hand, examples of factorizations that lie beyond the scope of the classical theory include additive decompositions into multiplicative units in rings; cyclic decompositions of permutations in the symmetric group of degree n; idempotent factorizations of the “singular elements” of a monoid; and so on.

After an introductory overview of the history and main results in the classical framework, in this course, we combine the language of monoids and preorders (partial orders that are not necessarily antisymmetric) to make first steps towards the construction of a “unified theory of factorization”. In particular, we prove an abstract existence theorem that recover, among others, a classical theorem of Cohn on atomic factorizations in cancellative monoids, a classical result by Anderson and Valdes-Leon on “irreducible factorizations” in commutative monoids, but also a theorem by Erdos on idempotent factorizations of square singular matrices over fields. We also introduce a notion of “minimal factorization”, suitable for the new abstract setting, and we qualify and quantify the related non-uniqueness properties in specific cases but also in some generality. Examples will help to motivate and illustrate the theory.

Prerequisites: Standard knowledge of (basic) algebraic structures and (binary) relations.

Duration of the course: 10 hours.

Schedule

– November 28, 16:00-18:00 (Aula B)
– November 29, 16:00-18:00 (Aula A)
– November 30, 16:00-18:00 (Aula A)
– December 1, 16:00-18:00 (Aula A)
– December 2, 16:00-18:00 (Aula A)

Exam

Written. Writing of a short essay.

References

1. L. Cossu and S. Tringali, Abstract Factorization Theorems with Applications to Idempotent Factorizations, e-print (Aug. 2021), arXiv:2108.12379.
2. L. Cossu and S. Tringali, Factorization under Local Finiteness Conditions, e-print (Aug. 2022), arXiv:2208.05869.
3. A. Geroldinger and F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure Appl. Math. 278, Chapman & Hall/CRC, Boca Raton (FL), 2006.
4. S. Tringali, An Abstract Factorization Theorem and Some Applications, J. Algebra 602 (2022), 352–380.

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