# Thursday January 6th

## Beach, Bamboo and Topology

## Happy New Year!

- I am still here for another ten days or so, and I am still doing sightseeing and hanging out with friends tomorrow!
- It’s crazy here; I ended up pretty much cancelling all my meetings. Sorry! It’s just really hard to plan focused stuff here, although I did end up getting my hair done and getting health checkups here. It’s so nice to go to your regular people for that kind of stuff!
- I don’t think most of my friends here know what I do, but that’s okay. I grew up in Theatre and the Arts (everyone here does, to some extent), and went to school with a bunch of
people who do that for a living here (are on radio, work in tv, paint, play musical instruments, dj, teaching Arts at the local university etc), which is pretty refreshing! I get to see
shows for free every year and hear about what they’re working on, which is cool! So a lot of musicians, actors and that sort of vibe, or starting small
businesses, working freelance, and that sort of thing. It’s a nice break!
A smaller economy also means that people do
*lots*of things, so being skilled at one thing or working a job in one capacity doesn’t define you in the way it does in some of the larger, mega-corporate environments, and people are a lot more flexible. I find that to be refreshing, but I understand the tradeoffs of each environment. - Flexibility is key here; you can plan, but not really. You get used to it; things are very fluid, so that thing you planned to do on the hour sharp, if you really planned on doing that then, is probably going to disappoint you because things just come up. Also, people may (or not) show up, or someone might bring a friend or two to hang out. You don’t take it personally, and later on you’ll probably get into an hour long conversation with them about how they met some person along the way you both know and that derailed their plan or something, but it’s a story. It’s kind of a thing you get used to I think.
- I’ve also been doing yoga with my parents in the morning, which has been really fun. It’s been a while, although I used to do that and pilates pretty regularly. I might try the videos once I get back and make them a morning routine thing during my semester.
- My Algebra 2 class was annoyingly cancelled; it makes me sad because our Pure Maths department has the most awesome classes that I don’t think some students realize the value of until they graduate (it’s easy to go for shiny hype classes), so anyways; now I’m doing Algebra 4 (Rings, Galois Theory, Topology), Elliptic Curves and Random Probabilistic Graphs, and sitting in on Number Theory I think. I also laughed at the “horseshoe maths” video this week.

## Notes from This week’s Real Analysis

- I’ve been working on Chapters 5 and 6, which are on Topology and Continuity. These notes are in progress.
- This was the first chapter I think I’d really like to give another go at and slow down.
- Chapter 5: The Topology of R:
- Open Sets
- Closed Sets
- (On my own, read up on Open sets, Zariski topology, Hausdorff spaces and Scheme Theory)
- Finite Subcover
- Open Covers
- compact sets
- Heine-Borel Theorem
- A Pedagogical History of Compactness which talks about compactness, the Intermediate Value Function and Bolzano, Heine, Borel, Lebesgue, Weierstrass, Cantor’s work, Poincare, Hausdorff and Dedekind
- Frechet’s definition of sequential compactness, Cousin’s Theorem
- Cousin’s Theorem

Note: (I’d like to spend some time going over this chapter*)

- Chapter 6: Continuity
- Continuous functions
- Dirichlet’s function
- Dirichlet distribution
- Dirichlet function
- “The Dirichlet function is discontinuous everywhere, and you can use it to construct functions that are only continuous at a single point.
- It is also a standard example of a function that is not Riemann integrable (although it is Lebesgue integrable).”
- Quora Dirichlet function
- Thomae’s function (super dope!!!) aka The Popcorn Function
- Very entertaining video on The Popcorn Function
- Hurwitz’s Theorem

- Topological definitions of continuity
- We start by looking at an arbitrary function; can we define its inverse without ambiguity? We cannot, but we can do this for a set.
- The Extreme Value Theorem

## Here are some photos as promised…as you can tell, I’m having a blast!

## It’s so green here someone carved out the island’s shape

## Driving

## More Driving

## In primary school

- I would have lunch with my grandparents, walk to this beach and then go back in the afternoon to school. There was a hole in the fence that you could take a shortcut through to get back to school, too.

## The beaches are restricted because of COVID

- The hours are something like 6am to noon, but we just drove by. You’re on an island; the beach is always there. Some people were fishing.
- This beach is not one of the more scenic ones, but it was nice to see the ocean air, and of course, visit since I lived in this area very briefly before I moved.
- The town itself has just 20,000 people and is an oil and natural gas town. It used to be populated by expats and had a gated community for those working in the oil and gas industry, which led to some tension during the Black Power movement era.

## My friend in Vermont

- I have a friend who loves the mango fruit, but didn’t know there were different kinds. I grew up with at least 20 kinds.
- They are so prevalent, people just show up with bags and give them away if they have trees in their yard.
- My dad used to sit
*in*the tree with a bowl, pepper and salt and eat them, too (also pretty common).

## They’re called five-fingers here, but also known as Carambola

## This is a Mountainous area

- There’s a waterfall nearby that we hiked growing up, too.

## Yes, people have pools here

## The food is also pretty colourful, and tasty

- I’m going to miss the food the most! (yes people and animals too lol).
- My cousin has this amazing dog that catches the ball, brings it back and drops it again for you to throw. Love him!
- I think this was a mix of plantains, cassava / yucca, salad, fish , a roasted potato with barbecue and garlic sauce, and a pie from a bakery.

## And that’s it!

Written on January 6, 2022