2nd Semester 2024/25: Topics in Algebraic Logic and Duality

Instructors

Rodrigo Nicolau Almeida, Simon Lemal

ECTS

6

Description

This project is aimed at students with an interest in algebraic logic and duality theory, seeking to further their knowledge in the field and gain some research experience. This year the focus will be on a classical topic of research which is still quite active: amalgamation, and its interconnections with interpolation properties, as well as other related interpolation style properties (e.g. uniform interpolation).

 

The project will consist of three components:

• Core theory lectures: There will be four lectures. These will be on the (1) the basic theory of interpolation and amalgamation; (2) Maksimova style classification results of systems with amalgamation; (3) definability theorems and epimorphism surjectivity; (4) uniform interpolation and coherence properties. The lectures will only cover the basics of the material, and students are expected to study independently. 
• Study case presentations: four classical settings will be studied to exemplify the methods developed: CPC, positive logic, IPC and S4. These present different challenges and will give students a taste for the variety of problems that may arise in the process of studying amalgamation properties. These will be presented by students in seminar style.
• Research students: students will be asked to write a report on their work. This can consist of a detailed write-up of some of the study cases under analysis. Ambitious students are invited to pick from a selection of small open research problems: amalgamation in non-distributive settings, coherence properties in tense logic, epimorphism surjectivity in fragments of modal logic, etc.

Organisation

Week 1: First two lectures + seminar. The lectures will be given by the instructor, and the seminars will be done in a collaborative setting between students and instructor.

Week 2: Second two lectures + seminar. Project consultations.

Week 3: Two to three seminar presentations.

Week 4: Final presentations.

Reports should be submitted within two months of the ending date of the project in order to get credits. 

Prerequisites

This is a research project on algebraic logic, so it presumes a good level of mathematical maturity and willingness to experience research in mathematics. The kind of background knowledge that will be presumed can be obtained, for example from taking Mathematical Structures in Logic (especially the basic algebraic theory of Boolean and Heyting algebras, and the associated duality theory) or Model theory (especially with a focus on Universal Algebra). Some basic concepts of category theory and of modal logic will be touched on at points, but are not crucial.

Assessment

(At least)two oral presentations, and a final report.

References

Alexander Chagrov, Michael Zacharyaschev. (2007) Modal Logic.

Dov Gabbay, Larisa Maksimova. (2007) Interpolation and Definability in Modal and Intuitionistic Logic.

Sam van Gool, George Metcalfe, Constantine Tsinakis. (2017). Uniform interpolation and Compact Congruences.

 

More references will be made available during the project.