## Café #1 - January 20

For "homework" read chapters 1 and 2 of Topology: A Categorical Approach and attempt the following:

- 1.1 draw examples
- 1.3 check Zariski Topology property
- 1.10 prove quotient topology characterized by given universal prop
- 2.2 prove or disprove a local consistency property
- 2.9 proof about locally compact Hausdorff spaces
- 2.15 prove or disprove a compactness property
- Optionally, exercises: 1.11, 2.21

## Café #2 - February 10th at 1:30-2:30pm EST

We'll start to follow along with the Albin lectures (found here).
For the 10th, aim for listening to the first two lectures + whatever supplemental reading you think might be relevant / helpful from Hatcher (link).

Instead of assigning problems, look at Albin's homework #1 (link) and choose one you think might be tractable to work on. As usual we can discuss on slack or via ad-hoc meetings in the upcoming weeks.

## Café #3 - Selected Topics in Kleene Algebra

Cheng hasn't assigned any homework.

## Café #4 - Prep for Jean-Eric Pin's Lecture

In preperation for Dr. Jean-Eric Pin's lecture on the Generalized Star Height Problem, we will discuss the following concepts:

- first order logic
- words and languages
- finite deterministic automata
- regular expressions
- monoids
- free monoids
- monoid morphism

- metric spaces

Dr. Pin has provided us with

some hand-written nodes which we can use as a resource.

## Café #5 - The Collatz Conjecture, in Ivy

No need to prep for this one. Max will use the Collatz Conjecture (which he surely will not resolve) as a motivating example for a tutorial on Ivy, and more generally, on formal methods and inductive invariants.

## Café #6 - Infinite Analysis & Nets

Read chapter 2 through section 2.6 (Nets) of Infinite Dimensional Analysis: A Hitchhiker's Guide. We will discuss the covered topics.

## Café #7 & #8 - More Infinite Analysis

Read through section 2.8 (Compactness) of Infinite Dimensional Analysis: A Hitchhiker's Guide. We will discuss the covered topics.