SIMO Retiree Blog

(Dylan here) In January, Zhao Yu wrote a piece about elliptic curves. To prioritise the various remarkable consequences that follow from the theory, many things were black-boxed. Today, I would like to open the box. I expect to make a mess and not clean it up; I hope you are okay with that!  But first, some personal tales. Qn. Is (olympiad) maths actually useful ? I've wrestled with this question for a long long time; it would be comforting to know that what I'm passionate about and invest a lot of time and effort into is also objectively meaningful. While I don't yet have a complete answer, here are my thoughts thus far.  First, I acknowledge the open-endedness and vagueness of the question as posed. What is (or isn't) mathematics? Does the answer vary between different types of mathematics (e.g. pure/applied, high school/undergraduate/research)? What does useful mean, and is there an objective aspect? Of course, fuzzy terms and open interpretations doesn't stop u...

(David here.) In this post I want to share a really interesting factoring algorithm that I learnt while taking an advanced cryptography class , and it will serve as a good springboard into talking about the mathematics and algorithms of lattices.

(Wee Kean here.) I'd like to give a brief intro to linear algebra and wrap up with some personal experience with learning math.

(Yu Peng here) Sometimes, you feel like you should be on the right track in solving a problem, but your solution falls short a little, due to issues like having a suboptimal bound, or having a condition you need but isn't true. At other times, you could just be struggling to find the optimal construction. At such times, it would often do you good to take a step back, and think about how you can optimise your solution.

(This is Yan Sheng.) Exercise : What are the greatest results in math, ever? List 5 of them (or 10, or 20, or...) that you think should be on any reasonable compilation of all-time greatest theorems.

(This is Glen.) Some time ago, Etienne posted about  this problem on rectangle tilings , to which this was my reaction in our blog Discord channel: Glen declares this to be his favourite combi problem. I, too, think that this problem deserves an article of its own, and so this is my attempt.

 (Jit here). There is a famous theorem of Lagrange that says every natural number $n$ is a sum of four non-negative squares (so that $0$ is allowed). Let's try to prove this theorem.