More Real Analysis, Sequences, Series

My phone is acting up

• I have some more pretty pictures, and if I can get my phone to do right things, I’ll post them.
• I’ve been to many places this week (pretty much every day or every other day), but I’ve also been briefly productive in the evenings. It’s really tough to have the same level of focus and productivity here that I do in the US. Here there are a lot of things that just pop up. It’s the place of spontaneous people dropping food off, random visits and things pretty much just spiralling out of control so planning work things is kind of tough. Some of my friends are also business owners or work freelance, so I can only grab them in windows of time, because there are windows of time they’re literally unreachable (somtimes their phones die, or they’re out for months on tour and that sort of thing). I’ve gotten used to seeing some people some years, and well, if they’re randomly somewhere else, I’ll catch them another time. You learn to be flexible because there are limits to what you can plan. It’s the nature of how things are here. One of the things that has been crazy to compare is even for where I am, how much space there is here. There’s just so much space to build large houses, and people do. I think my dad made the observation that we’re not really a tourist country, in the sense that our “tourists” are usually immigrants with ties to home. Some of my relatives are also randomly in sporadic places of the world (Malaysia? Switzerland? New York? London?), or have a window of time before leaving or coming back, so it’s tricky. So yeah…
• I can’t get to my presentation this week, because of the nature of how things are here, but I figured I would try over the weekend and next week to complete that.
• I have some meetings next week, so we’ll see how that goes. I’ve tried to push the meetings for the New Year, because of some things that are out of my control that make it better to have them then. And even then, there are constraints on that and I don’t know how it will work, but we’ll see (shrug). I mean, I am on vacation, so, if it doesn’t work out, oh well?

This week

• Last week, I went through chapters 1 and 2, and this week, I am working through chapters 3 and 4.
• I’m not letting the work overtake my vacation; I plug away a little bit each day, usually late in the evening. The great part is that I’m doing it because I want to.
• So far I’ve looked at:
  • isoradial graphs
  • winding bijections
  • Dedekind cuts
  • Cantor’s diagonalization argument
  • Russell’s Paradox
  • I started a repo on Measure Theory and Real Analysis resources
  • The Squeeze Theorem
  • The Monotone Convergence Theorem
  • Bolzano-Weierstrass Theorem aka sequential compactness theorem
  • Cauchy Criterion
  • convergence and divergence test
  • geometric series
  • Harmonic Series and Series p-test
  • Basel Problem: wiki: proofs / solution using L’Hopital’s Rule, Parseval’s Identity and Cauchy)
  • 3Blue1Brown video using ideas from light and inverse square rule for intuition for Basel Problem. This is funny to me because I did do lighting design and know the Inverse Square Law concept well. It’s usually how you would calculate the beam spread of a fixture in your inventory using photometrics. Random, useless information is always great.
  • I also came across some great notes from a Maths grad seminar at CMU on the Basel Problem

Interesting

• I’ve enjoyed chapters 3 and 4 immensely; sequences in particular are fascinating. In Chapter 4, they talk about the Basel Problem.
• The Basel problem is this Challenge by Pietro Mengoli in 1644 that was solved by Euler in 1735. The 3Blue1Brown video uses circles and geometry, and cited that anything involving Pi usually involves circles and geometry. Interestingly, I think of roots of unity when I think of circles usually, thanks to my post-quantum cryptography and Number Theory in classes.
• For Measure Theory, I found a neat Math Overflow question with some resources and Terry even has a pdf too. For now, I’m just going through the book, and trying to take my time and understand things, while on vacation (think of it as a book I’m reading while on vacation).
• A friend from my Maths class even offered to study with me, so I may take them up on it in the New Year for Real Analysis. I’m taking a break and doing sightseeing and stuff during the day, though. This is just enjoyable, so I’ve found time to do it, too.

Anyways, that’s it