Summer Self-Study

This is a (kind of a) diary of my individual summer self-study. First of all, let me state the reason I want to write this piece of text. Along the summer break, there were activities that I did. (And I pretty much enjoyed all of them) However, one concern I have at the end of the break is that "Am I really improving myself this summer?", or "What is my progress, if any?". Among the community around me, I see everyone improving themselves everyday. Everyday, I wake up and see someone going forward while I'm the one who stays still. The frustration builds up and become depression. To push myself forward, I must be confident enough to say "It was not a useless dream!", and therefore, I decided to write this report, to (at least) tell myself that my progress is non-zero. It is positive. Yes, $\varepsilon > 0$. For one day, in the future, where I come to see what I have done during this summer, here is the summary.

Before the "Summer Self-Study"

In June 2022, I was finishing my first-year finals. It was cool and I'm glad I did electromagnetism in the semester. I went back to Thailand and yes, relax! (of course!) I met my friends and family. We traveled on trips, in vacations. In fact, I almost did not really stay at home!

Back in July 2022, when I was at home, I watched VNL (2022 FIVB Volleyball Women's Nations League) with my family, which inspired me the mindset of sports-playing, once again. I played a little bit of badminton, a little bit of volleyball (even though I cannot really play). This also reminded me of the anime Haikyuu, which is my favorite sport anime.

With that mindset active, along with the ไม่แก่ตาย, Imagination, and เสี้ยววินาที all looping inside my head, I'm ready to get myself back into competitive programming.

Competitive Programming

I started practicing the same old stuffs: BOIs, IOIs, JOIs, CEOIs, etc. Same old 5-hour contest format. But now, along with the IOI Thailand contestant team. (Yes, at first I was unable to do it, but with enough emotional force and a bit of warmup trainings, I'm back with the strength of being able to do a 5-hour contest.) The camp was fun (at least for me it is, not sure about others though lol).

After that there's a trip to France and Germany. Yes, my first time to Germany. Overall, it was nice. I was challenged with a problem:

Let $p$ be a prime. Let $\{a_1, \dots, a_p\}$ and $\{b_1, \dots, b_p\}$ be sets of all residues modulo $p$. Can $\{a_1b_1, \dots, a_p b_p\}$ be a set of all residues modulo $p$?

which took me some time to think about it. And finally, I'm not able to solve it; Got a hint and ah-ha! (try it yourself if you want)

Ok, done with the trip, I got back and tried the IOI2022 day 1 and day 2 (yes, timed 5-hour with normal contest environment). The result shook my heart.

With this score in the computer, I can finally say that I've improved through the practices I've done. This gave me a lot of motivation; it signaled me as if I'm better at problem solving, as if I have really improved, compared to the IOI2019 and IOI2020. I spent a relatively high amount of free time in the week after just for competitive programming practices.

As time went by, finally it came to an end. I lose my purpose. "Why am I even doing competitive programming?", I thought. At that time, I started to shift between reading books in the library, designing new problems, and doing competitive programming (but less often). Not so long, I began to feel tired of competitive programming. (Yes, I have no "competitive programming friends" anymore. I have no one to talk to, and I have completed one of my goals, so it doesn't matter anymore now.)

This begins the summer self-study. To optimize my free time, trying not to waste it on some useless stuffs, I try to study mathematics on my own.

Summer Self-Study

One of the reasons I tried to self-study more is because I think I've practiced on specific problems for an enough amount for the summer. I had been a nice frog for an appropriate amount of time already, why not try to fly up. At first, maybe not too high, but just try to see what's around. That's when I looked into problems that I was interested in, for example:

• About $\textsf{PRIMES}$ and its relationship with $\textsf{NP}$-completeness. (or even $\textsf{NP}$-intermediate, if exists)
• Analytic number theory: Proof of Dirichlet's theorem on arithmetic progressions.
• Transcendental number theory: Inspired by a video on whether $\pi^{\pi^{\pi^\pi}}$ is an integer? (or rather, a tweet, which I have forgotten, which links to this video)
• Collatz conjecture, as you can see. It is one of the easiest-looking problems.
• Goldbach conjecture, as it also looks easy. However, I'm quite dissatisfied with the theory of partitions of integers. Not in a position to say much, but I think additive number theory has relatively low progress compared to multiplicative number theory.
• Basic algebra, as I have not learned anything in year one. (Yes, this is a huge point of frustration for me. I studied year one and I still have no knowledge of undergraduate abstract algebra? What a joke!?)

Also, I plan to study

• Algebraic number theory, but well, time runs out and I still have no knowledge of basic algebra. I saw something called "class field theory" in some books, but I have no idea what that is...
• Something like algebraic variety? Homology? Cohomology? I really have no idea about all of these. I randomly found a result called Grothendieck-Riemann-Roch theorem and I have absolutely zero idea about what it is.
• Information theory, as it looks interesting after pondering about what actually is "information". And it looks promising after I did a tiny bit of quantum mechanics.
• Differential equations. lol I'm gonna be joking on myself. I hate differential equations anyway. However, it would be nice if I know what I'm doing. It would be nice if I can apply those into physics that I'm interested in.
• Galois theory: What did he do about degree-5 polynomial equations?

Success

Here goes the list of things I've succeeded in the summer self-study.

• Understanding basic properties of groups.
• Proving that a proper non-trivial subgroup of a cyclic group is also cyclic.
• Understanding the Dirichlet characters, and the orthogonality relation. (Only understood the proofs, still not getting the big picture but that's ok)
• Understanding the key points in the proof of Dirichlet's theorem on arithmetic progressions. i.e. the nonvanishing of $L(1, \chi)$ for nonprincipal character $\chi$, the ingredients (which are related to $\sum\limits_{\substack{p \leq n\\p \text{ prime}}} \frac{\log{p}}{p} = \log n + \mathcal{O}(1)$)
• Enjoying Codeforces (and increased the rating back to orange :) yay)
• Understanding the machinery of quantum mechanics (the model of wave function) and its "popular" interpretations (i.e. Copenhagen interpretation and realist interpretation).
• Making a set of videos on basic algorithm design, analysis, and proof.

Failure

Here goes the list of things I've failed in the summer self-study. (Failure is a bit pessimistic... Hmm... I'd prefer to say "not yet succeeded" instead)

• Cannot solve any problems of IMO2022 (tried P1, P2, P5, got partial result on P1).
• Tried to come up with a new theory on "biarrownomials" which is defined by $B(x) = \sum\limits_{i=1}^N P_i(x)(x \uparrow\uparrow i)$ where $(P_i)_{i=1}^N$ is a finite sequence of polynomials with integer coefficients. The set $\mathcal{B} \subseteq \R$ of numbers that is a zero of some biarrownomial is countable. Hence, biarrow-transcendental numbers exist, and there are infinitely many. The question of whether $\pi^{\pi^{\pi^\pi}}$ is an integer or not has its answer false if $\pi$ is biarrow-transcendental.
• Tried to prove that alternating groups $A_n$ with $n \geq 5$ is simple but got stuck.
• On Goldbach's conjecture, I present another conjecture that would imply Goldbach's conjecture. If the statement "For all even number $n$, if $n$ can be prime-partitioned into $p_1 + p_2 + \dots + p_k$ then $n$ can also be prime-partitioned into $p+q$" is true for some $k = 2^l$ then we can prove Goldbach's conjecture by induction.
• On Collatz conjecture, I don't think the busy beaver formulation will work because finding the busy beaver number is even stronger than proving some hard theorems. Also, on the other hand, trying to formulate busy beaver in terms of lambda calculus gave me headaches.