Topology in and via Logic (2024)

Coordinated project, January 2024, ILLC.
Coordination: Rodrigo N. Almeida , Søren B. Knudstorp, Amity Aharoni.
Powered by: Nick Bezhanishvili.

Project Description

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).<br>

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 first in a series of introductory 1.5-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), as well as forays into more advanced topics and relationships, pursued by students.
Importantly, we will emphasise how topology appears naturally in many logical contexts, and use them to develop intuition about the crucial concepts of topology.

References

Ryszard Engelking, (1968) General Topology.
Steven Vickers, (1996) Topology via Logic.
Jorge Picado, Aleš Pultr, (2012) Frames and Locales: Topology without Points.
Tai-Danae Bradley, Tyler Bryson, John Terilla, (2020) Topology: A Categorical Approach.

Lecture notes for the project will be updated throughout the project. You can find the most recent version here. We thank you for any typos you might notice!

Tutorials

There will be three tutorials. Amity Aharoni will be in charge of designing the tutorial sheets and running the tutorials. You can find the tutorials here:

• Tutorial 1
• Tutorial 2
• Tutorial 3
• Tutorial 4

Assignments

There will be three Homework Assignments, which will be posted in this page. Solutions can be submitted until one week after the end of the block (i.e., 9th of February). You can submit your solutions here. You can find the assignments here:

• Assignment 1
• Assignment 2
• Assignment 3

Registration and Presentations

Students are expected to form a team of between two and three students. Please indicate the team, and in due time, the topic, in this sheet: Google Doc.
You can find some suggested topics for presentations here.

Lecture Schedule

There will be six lectures and a guest lecture throughout the project. These will take place in the following schedule, and will be broadcast online via this link:

You will find the recordings of the lectures here. These recordings will be deleted in April 2024.

Student Presentations

You can find below the schedule for the presentations: