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:ˆ

• 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

1. Bergman, C., Universal Algebra: Fundamentals and Selected Topics, Chapman & Hall Pure and Applied Mathematics, Chapman and Hall/CRC, 2011.
2. Blok, W., and Pigozzi, D., Algebraizable logics, vol. 396 of Memoirs of the American Mathematical Society, A.M.S., 1989.
3. 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.
4. Czelakowski, J., Protoalgebraic logics, vol. 10 of Trends in Logic: Studia Logica Library, Kluwer Academic Publishers, Dordrecht, 2001.
5. Font, J.M., Abstract Algebraic Logic: An Introductory Textbook, College Publications, 2016

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

January-Febrauary 2025 (TBA)

Exam

The final exam consists in a seminar.

QUBO and Quantum Annealing

Prof. Michele Marchesi
University of Cagliari and NetService spa

Abstract

The seminar, lasting about an hour, presents the problems of unconstrained binary quadratic optimization, where the variables assume binary values (0/1 or -1/1), and the function to be optimized is a quadratic form with real coefficients. It will discuss various real problems that can be represented as QUBO and how to incorporate constraints using penalty coefficients. Exact classical solvers, which can only be used for small problems due to the NP-complete complexity of QUBO problems, and the main heuristic solvers: Tabu Search and Simulated Annealing, will then be presented. Finally, the Quantum Annealing approach for solving this type of problem on specialized quantum computers will be presented.

Schedule

July 25th, 9:30-10:30 (Palazzo delle Scienze, Aula B)

From Theory to Practice: Derivative-Free Optimization, Bilevel Problems, and Real-World Applications

Prof. Remigijus Paulavičius
Vilnius University, Lithuania

Abstract

This talk explores the journey from theory to practice in derivative-free optimization (DFO), primarily focusing on the impact of DIRECT-type algorithms and their application to practical problems. While focusing on this key aspect, the presentation also delves into several other pertinent areas of mathematical optimization, such as bilevel optimization, offering a broader perspective on the field’s advancements and challenges. The presentation begins with an overview of the DIRECT algorithm’s role in DFO, highlighting its strengths and limitations. The subsequent discussion delves into advancements in DIRECT-type algorithms and their integration into software tools, facilitating practical applications. The talk presents a comparative analysis of deterministic and stochastic DFO methods through benchmarking studies, evaluating their performance and suitability for various problem domains. Then, the author showcases his contributions to the bilevel optimization field, where he worked on a general bilevel algorithm and underscored the critical need for such algorithms and software tools in practical applications, including ML. Finally, the talk showcases successful collaborations between academia and industry, highlighting the practical implementation of DFO techniques in real-world scenarios with examples from GlobeTrott Travel and Girteka Logistics, demonstrating the impact of DFO in solving complex optimization problems in business settings.

Schedule

June 19th, 10:00-12:00 (Palazzo delle Scienze, Aula Magna Matematica)