1st Semester 2023/24: Topology in and via Logic
- Amity Aharoni, Rodrigo N. Almeida, Nick Bezhanishvili (Responsible), Soren B. Knudstorp
Topology is one of the basic areas of contemporary mathematics, and finds applications in all areas of logic: from traditionally mathematical subjects (model theory, set theory, category theory and algebraic logic) to areas of philosophy (epistemic logic, formal epistemology), formal semantics or computation (domain theory, learning theory). Its key idea is that one can understand space through very simple units - so called "open" and "closed" sets, and their interaction -- in a way that can capture both the intuitive properties of physical space, and also more abstract notions of "space": spaces of ideas, information spaces, or even spaces of actions, for example, in computation.
In this project we will familiarise students with the basic concepts of topology as they are used in logical practice. This will consist in a series of introductory 2-hour lectures (including time for questions and discussion) covering the basic core concepts one tends to encounter in these settings (continuity, neighbourhood filters, compactness, connectedness, separation); tutorial sessions, guiding students through practical applications of the concepts, as well as guiding them in finding an appropriate presentation topic; as well as forays into more advanced topics, pursued by the students. Importantly, we will emphasise how topology appears naturally in many logical contexts.
The lectures will be hybrid, although we strongly encourage in-person attendance. The (tentative) schedule is as follows:
2 lectures (Continuity, Neighbourhoods, Beginning Separation axioms) + 1 Tutorial.
2 lectures (Separation Axioms, Beginning of Compactness, Basic topological constructions) + 1 Tutorial.
2 lectures (Compactness, Connectedness) + 1 Tutorial.
Individual consultation (to help develop presentations, and further research interests)
Presentations, to be scheduled at the end of week 2.
The topics of presentation should focus on a more advanced concept or idea relating topology to logic. These can include (but are not limited to) specific aspects of the following broad topics: topics in point-free topology; Polish spaces and the spaces of Cantor and Baire; Baire category theorems; Alexandroff and Boolean spaces; density and scatteredness. Students can propose topics, so long as they fit in the scope of the project, and suggestions will also be made available by the instructors.
Basic mathematical maturity as can be expected from having taken a course heavily reliant on proofs, e.g. Philosophical Logic, Dynamic Epistemic Logic or Introduction to Modal Logic. Attendance of these courses can be useful to appreciate some of the examples, but is not necessary.
Basic familiarity with topics of mathematical or philosophical logic will be useful in choosing topics.
Homework Assignments, Oral Presentation (week 4).
Ryszard Engelking, (1968) General Topology.
Steven Vickers, (1996) Topology via Logic.
Jorge Picado, Aleš Pultr, (2012) Frames and Locales: Topology without Points.
Lecture notes for the project will be made available and updated throughout the project.