Welcome to the course Duality Theory: Connecting Logic, Algebra, and Topology! It is given during the winter semester 2024/5 at LMU Munich as part of the Master in Logic and Philosophy of Science.
Motivation
This course is an introduction to duality theory, which is an exciting area of logic and neighboring subjects like math and computer science. The fundamental theorem is Stone’s duality theorem stating that certain algebras (Boolean algebras) are in a precise sense equivalent to certain topological spaces (totally disconnected compact Hausdorff spaces). The underlying idea is that the two seemingly different perspectives—the algebraic one and the spatial one—are really two sides of the same coin:
- formulas/propositions vs. models/possible worlds,
- open sets of a space vs. points of the space,
- properties of a computational process vs. denotation of the computational process.
In terms of content, the focus of the course will be to introduce the mathematical theory, after a philosophical motivation. In terms of skills, the aim is to learn how to apply the tools of duality theory. We will illustrate this with applications—especially to philosophical phenomena—that make use of dualities by combining the often opposing advantages of the two perspectives.
General information
We have two sessions per week during the semester:
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The lecture given by me, Levin Hornischer, on Wednesdays from 12:00 to 14:00 in room 021 (Ludwigstr. 31). Note the sharp (aka “s.t.”) times! Here the main content of the course is introduced. Below you find a schedule of when we cover which topic.
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The exercise class given by Marta Esteves, on Thursdays from 18:15 to 19:45 in room 028 (Ludwigstr. 31). Here the solutions to the homework exercises are discussed.
Lecture Notes
The lecture notes are written as the course progresses. You find the latest edition here: duality.pdf.
They are a rewritten version of last year’s notes, which you can find here.
Formalities
All the organizational details for the course are described in this file: formalities.pdf.
Schedule
The schedule below describes in which week we will cover which material in, respectively, the lecture and the exercise class. The ‘Date’ is the Wednesday of the week, i.e., when the lecture takes place; the exercise class is a day later.
| Week | Date | Chapter | Lecture | Exercise Class |
|---|---|---|---|---|
| 1 | 16.10.2024 | 1 | Intro to the course and 1.1.1. | Intro to partial orders (ex 2.a) |
| 2 | 23.10.2024 | 1 | Finishing sec. 1.1 | More on partial orders (ex 2.b) |
| 3 | 30.10.2024 | 1-2 | Finishing ch. 1 and until beginning of 2.1.2 | Ex 2.c & 2.d |
| 4 | 06.11.2024 | - | cancelled | Bonus session on order-adjunctions: ex. 2.e & 2.f |
| 5 | 13.11.2024 | 2 | Finishing ch. 2 | Ex. 2.g-i |
| 6 | 20.11.2024 | 3 | Sec. 3.1.1 | Ex. 2.a-b |
| 7 | 27.11.2024 | TBA | TBA | TBA |
| 8 | 04.12.2024 | TBA | TBA | TBA |
| 9 | 11.12.2024 | TBA | TBA | TBA |
| 10 | 18.12.2024 | TBA | TBA | TBA |
| 11 | 08.01.2025 | TBA | TBA | TBA |
| 12 | 15.01.2025 | TBA | TBA | TBA |
| 13 | 22.01.2025 | TBA | TBA | TBA |
| 14 | 29.01.2025 | TBA | TBA | TBA |
| 15 | 05.02.2025 | TBA | TBA | TBA |
Essay topic
You can find a list of potential essay topics here.