I’d forgotten to upload this week’s finds, so it’s been rebranded ‘Week 0’. Unfortunately, there are at least 90 days of backlog on this project - each week’s worth possibly as large as this page - so I will have to git commit to memory the absolute mountain of information I’ve learnt until now. After some consideration, I’ve decided to employ the $\rightarrow$ symbol as shorthand for ‘this led me to discover/learn’ You’ll notice that the format is a little different from weeks >1, I hadn’t organised the first week very well.
Mathematics
- If you really want an intuitive explanation of something, I would recommend seeing Quora answers - They tend to have fantastic resources for building up intuition, contributed by what seems like a tightly-bound community of mathematicians who really know how to write. Not as rigorous as on the stackexchange.com sites, but far less rigid and formal (once I found someone using category theory to explain a trivial group property - talk about overkill).
- Category theory is like an upgrade in elegance to the already superb group theory that I’m learning. Math3ma and John Baez’s blogs really got me hooked (especially John Baez’s elucidation on their connection to spin networks) - so that in turn led me to find more about topological spaces (one of the categories where I haven’t ventured at all really). I enjoyed the Kolmogorov classification and pathological counter-examples to Hausdorff spaces. This apparently is used quite a bit in algebraic geometry (Grothendieck!), which leads me back to the modularity theorem and FLT! I actually understand a bit of Wiles’ first page. This goes on to elliptic curves (FLT, yet again) and Diffie-Hellman key exchange, and the general discrete-logarithm problem (along with man in the middle attacks).
- Applied group theory, finally! I found two really nice PDFs from Harvard about Rubik’s cube and sudoku groups. Sadly they didn’t contain the derivation of their solution. Speaking of Rubik’s cubes, I thought they might be useful for my IB Maths IA (proving successive lower and upper bounds, Superflip, group presentations, etc.). I have to find out what else GAP can do (besides finding the group order). And on the topic of sudoku (suprisingly a relatively recent invention), I found about about Knuth’s X algorithm (need to find more about Dancing Links though)
- …and back to number theory - I rediscovered the baby-step giant-step algorithm (and recalled my poor Python implementation of it); Lagrange’s group theorem is super cool, because it proves Euler’s (totient) Theorem. Carmichael numbers seem interesting - I wonder how Erdos derived the bound on their density
- Schafli symbols are fun for the dual polyhedron (Honeycomb lattices and Coxeter groups probably need some work on my part though)
- Just touched up on some group theory - normal subgroups, commutator groups, direct sum/product/tensor product - nothing too fancy, but terms which have the unfortunate habit of slipping from my mind. Today also marks the end of an apparently intensive abstract maths unit - I’ll aim to shift towards theoretical physics and quant a bit more now. Still dont know what a Dynkin diagram is though (probably need to consult a uni PDF).
- Coxeter and exceptional groups could be the concluding subtopic in this unit - E8 and F4 in particular (I need to get finished with the Standard Model so I can savour all of these elegant symmetry theories). I think operator theory and elliptic equations could come in handy, since I’m defecting to the physics side now.
- John Conway seems to pop up in all sorts of esoteric number theory (Nimbers were cool - didn’t see any applications though; reminded me of surreal numbers).
- The Abel-Ruffini Theorem and Galois and automorphism groups were fun while they lasted, but I think my interest in them is slowly fizzling out $\rightarrow$ Review quartic formula derivation.
- Russell’s paradox looks nice when formalised in Brilliant (reminds me of when I tried to use first order set theory rigorously in routine school maths exercises - I didn’t know the ‘equivalent’ symbol though), and is vaguely reminiscent of my approach to programming, “concretise then generalise” (I’ll explain that some other time) $\rightarrow$ Godel’s incompleteness theorems should be fun to see too. (Whenever I’m bored, I can just consult this list!).
- I did grasp root systems a little, but I’m still uncertain as to how they relate to Dynkin diagrams. On the other hand, the source that piqued my interest in them in the first place, John Baez’s excellent (but unfortunately, developed before CSS) blog is wholeheartedly bookmark-worthy.
- I only just realised the link between gradient line integrals and complex integrals (both evalute to zero for a closed loop (no poles, of course)).
- Hopefully, now that I know a bit about Sp and Spin groups, I can advance into the modular group unimpeded.
- I surprisingly hadn’t seen the method for finding the quadratic residues mod a prime (how do you generalise to composite?).
- Pareto efficiency seems to be a useless metric, but I disagree that Nash equilibria are similarly impractical (the max-row max-column shortcut will prove to be useful if I ever do behavioral microeconomics).
- The semidirect product as the ‘opposite’ of quotient groups is a good perspective (plus this applies to exact sequences) - [Nihar from the future: “You fool! You fell victim to one of the most common blunders!”]
- I chanced upon an old friend, the volume of an n-ball (courtesy of Zach Star).
- I did end up seeing the quartic formula (it’s the first PDF in my library!), and it wasn’t nearly as hard as I remember it to be (respect for Ferrari and Cardano for working all of that out without having variable names).
- I hopped back to Galois theory (Rational root theorem, Alon’s great answer on solvable groups; extensions like differential Galois groups - looks promising if I can find a source).
- To get a better idea of algebraic geometry, (hail Grothendieck!) I found a MIT pdf on Bezout’s theorem $\rightarrow$ projective representation of vector and Hilbert spaces. I learnt about the two alternative forms of the fundamental theorem of algebra (besides the degree n - n roots one) along with the observation that its neither fundamental nor using algebra!
- Looked at Helmholtz decomposition again (some minus signs looked suspect).
- Short exact sequences are completely clear now.
Set Theory, Logic and Linear Algebra (don’t ask)
- The hierarchy of set grammars is somewhat reminiscent of Turing machine hierarchies (nothing beyond superficial, I doubt) - still need to see what ‘TT’ is.
- I’d consider Brilliant to be the de facto source on proof and technique-related pages - Fermat’s method of infinite descent, the well ordering principle were well layed out. I had to go back to Vieta root jumping (the Legend of Six!) - seems like a potential blog post to me.
- I couldn’t quite grasp Godel’s first incompleteness theorem beyond Godel numbers (foremostly due to laziness).
- How do you define numbers in set theory (is it cardinality? In that case, surreal numbers almost feel like the natural step upwards).
- Also, I should find out some corrolaries to the Axiom of Choice and Brouwer’s fixed point theorem (beyond the trivial and mildly amusing).
- Obligatory xkcd: The principle of explosion (seemed a bit ‘suspicious’ at first, but on closer inspection, it turned out to be painfully obvious).
- I looked at the construction of natural numbers in set theory (it’s what I expected) - Richard’s paradox was better than Russell’s, if you ask me $\rightarrow$ the existence of non-definable real numbers really surprised me.
- Induction on real numbers was unexpected, even if impractical.
- The question of identifying whether a vector is a linear combination of other $\rightarrow$ 3b1b’s rank/nullity video (which I think is superseded by Zach Star’s more modern video - whose videos I should watch more of).
- The FangCheng method for Gaussian elimination is nifty (especially considering it predates it by a good 1000 years).
- Canonical vs standard isomorphisms seem clear to me now, after seeing the isomorphisms between a vector space, its dual and its double dual.
Category Theory
- Math3ma’s posts on the Yoneda Lemma were great (Category Theory just keeps on getting better and better!) $\rightarrow$ Topology, whose basics I finally understand (and now continuity and convergence are so much cooler).
- A Berkeley paper managed to link cobordisms (I think) to Hilbert spaces, so that’s another reason to push on with Category Theory.
- I found a master’s thesis linking Category Theory with QFT - but the sprawling cheat sheet I unearthed on a GitHub repo is going to be even more useful.
- Once again Math3ma doesn’t disappoint - just got started on the limits and colimits series $\rightarrow$ Once over-abstract notions like ‘dual’ and product are ironically intuitive in Category Theory (the arrow reversal for dual may well be the coolest thing I’ve seen in a long time) - it might be worth taking some university maths courses just for this.
- I enjoyed reading about categorical limits on math3ma (although the colimit part hasn’t been released yet, I can invoke the sacred arrow-reversal and try and derive what they actually are - by pure thought!).
- I didn’t find any sources on CT for physics to be all that captivating (hint: scroll to the end to see what punchline they’re building up towards - maths is said to be anti-comedy, after all)
- I’m really curious about the smash product - in fact, come to that, seeing the semidirect/wedge/smash products defined categorically should remove the final barriers in my understanding of them.
- The categorical construction of functional languages was a serendipitous find.
- While “application-based category theory” may seem like an oxymoron, Tatsuyo’s thesis on categorical functional languages does just that
[Nihar from the future: I ended up not doing any category theory (at least explicitly) for the next month :( ]
Physics
- Finally got back round to the Tennis Racket Theorem, Euler’s Theorem and in continuation, Euler Angles, gimbal lock and 4-d rotations. Oh, and the method finding the continued fraction for a number was hiding in plain sight (at least, the greedy algorithm)
- The spin-statistics theorem is much easier than Srednicki’s book makes it out to be; Weinberg’s book is sadly not well typesetted, and nLab is unreasonably convoluted, but a (still incomplete) source that I found is physicstravelguide.com - incidentally, one of the contributors is Jakob Schwichtenberg - his chapter on representation theory for physics is effectively what started this unit.
- I still admire Lubos Motl’s SE answers, but man, if that guy doesn’t have an appetite for stoking controversy, I don’t know who does (his blog posts also happen to be a string of cynical but entertaining rants on matters from Jordan algebras to the EU).
- I managed to get a copy of Landau’s CM (which weirdly has the same font as H.C. Verma)
- I asked a fairly mundane question about the physicists’ Lie algebra on SE, but it still garnered a couple of upvotes, with a handful scattered between two (again, straightforward) answers.
- I’m diving back into the Standard Model - from the deep end. I’m decidedly lukewarm to the approach in Schwartz and Srednicki (make unmotivated propositions first, introduce representation theory MUCH later) - Peskin and Schroeder (except for the mathematical dubiousness in the first chapter) seems to really pick up from chapter 3, supplemented of course by the Columbia QM book (may have to skip to a practical-oriented section though).
- So I just touched up on helicity, chirality (Physics from Symmetry really helped here), conservation laws, boost-vs-translation $\rightarrow$ Euclidean group.
- Clifford algebra should be really useful too, but I haven’t looked up anything on that yet.
- “Nature favours simplicity” - works for Lagrangians (just make it gauge-invariant and a Lorentz scalar!) - and alternative Lagrangians (e.g. with the Levi-Civita tensor) should uncover a web of links.
- I couldn’t resist looking up renormalisablity $\rightarrow$ IR and UV cutoffs and the renormalisation group will be coming up soon $\rightarrow$ Wien’s displacement law is far more accurate than people give it credit for (I mean, seriously, Rayleigh-Jeans fails at like 5 Hz).
- I’ve come to the conclusion that prior to QCD, I should definitely get a solid grounding in QED and renormalisation (I’ve thrown in supersymmetry too, just for fun).
- I should explore the Ising model further (it’s also a potential Python project) $\rightarrow$ I had a thermodynamics intro (with beta and micro-states) - still on the search for a quality cheatsheet though.
- Renormalisation group explains renormalisation better than the wiki page proper, funnily enough.
- Noether has a second theorem? $\rightarrow$ if gauge symmetries correspond to redundancies in the description of a physical system, why do they have a corresponding charge?
- I’d forgotten that the Dirac spinor conjugate had a gamma matrix attached to it! $\rightarrow \gamma_0^2=1$
- I did end up skipping Columbia QM chapter 8 (but now, the only thing worth reading is Clebsch-Gordan coefficients and the symplectic group - there just aren’t enough concrete examples in it).
- A treatise that I found on Clifford algebra might prove to be, more than anything, a fantastic refresher on group, representation and character theory (I say ‘refresher’, but I’ve only barely pored over the last of those).
- The Dirac adjoint is used, of course, for Lorentz invariant transformation. The Clifford algebra pdf was truly awesome and super dense.
- A hunt for a QED document led me back to one of the first I had ever seen - David Tong’s lecture notes! I had no idea that all 6 sections were there on the Cambridge website, but I am very thankful $\rightarrow$ I also found a colourful Comic Sans PPT too - informative and succinct (and Comic Sans!)
Chemistry
- I couldn’t find a good source on molecule symmetry character tables - all the LibreTexts thus far appear to be garbled and badly formatted
- In relation to molecular symmetry groups, crystal symmetry groups look even more intimidating (I should get my Bravais lattice up to speed)
Computer Science and App Stuff
- Lambda calculus to Turing machine is interesting, but the Oracle computability hierarchy is even more so.
- A music-to-score application would be really nice. I was inspired by a Two Minute Paper where some guys from IIIT effectively used ML to perform lip reading.
- Not going to restart Statistics and Machine Learning though, perhaps my previous burst was a tad too intensive (feels a bit like burnout).
- I did learn what a melspectogram is, so that took me to DSP: filters, window functions and oscillators at a medium-to-high level.
- Finally finished the news aggregator app, after just a week of learning Flutter. Ridiculously intuitive and with far less mess than fiddling around with XML in Android Studio. Found out about this app with 10m+ downloads - they just started using AI to summarise news articles to no more than 60 words.
- I am going to get back into ML - NLP, Transformers and Normalising Flows kicks off right where I left (Two Minute Papers will continue to be a riveting source) - sadly I may have to be confined to the theoretical subspace only, given the obscene training times.
- I did revise Hamiltonian MC sampling (because of NUTS)
- I thought of making a Pomodoro timer app $\rightarrow$ sprite animation in Flutter could reignite my interest in app dev.
- Game Jam expositions are also entertaining and thought-provoking, but I feel Unity is a bit clunky for my taste. Anyway, I think I’ve reached the point where my stream of app ideas has run dry, so I may have to take a break here.
- Surprisingly, Spotify uses P2P to stream music without overloading its servers (!). Shazam uses a bit of DSP $\rightarrow$ 3b1b’s generalised Fourier Transform video (the Doppler effect example was a great example).
- I was thinking about voting systems (it spontaneously occurred to me while browsing an otherwise mundane list of revolutionary app ideas), so naturally I trundled off to find out why the (notoriously overhyped, imo) blockchain doesn’t fulfill the voting criteria $\rightarrow$ I should review Arrow’s paradox.
- It’s a close call between Cassandra and MongoDB, but Cassandra takes the win here with CQL (very similar to SQL) $\rightarrow$ Microservices seem like a much more natural approach than a hulking monolith (of course, I can consult a resident Software Architect!), popularised by Netflix’s blog on Medium $\rightarrow$ neural diff tools (how does regular hash-based diff tools work?).
- Dad taught me about =VLOOKUP, uncovering the power of spreadsheet editors (and saving me from an agonising 40 minutes). Pivot tables seem to tie in nicely with my previous topic of databases, essential for my new app idea, a content-sharing platform, similar to Medium (with features from Reddit and Quora), but just for Science! Incremental, of course, and using Firebase $\rightarrow$ How can I use BaaS?
- I was taken aback by C++ “concepts” - they’re nothing revolutionary, but they highlighted to me the limitations of templates - but the fact that they’re not structural (cf. Rust’s traits) provides a much more relaxed third-party experience.
- It’s curious that “synchronous” tasks means the opposite of what it does in colloquial usage - asynchronous is obviously more efficient, but the problem of optimising a program at compile-time to its ‘maximally-asynchronous’ flow is equivalent to the Halting Problem, and so is undecidable.
- Software architecture is quite interesting (as it turns out, ‘agile’ and ‘efficient’ aren’t just meaningless buzzwords). I stand corrected on the DB-War though; Mongo looks very appealing now. I looked at a couple of examples of chat app and streaming architectures, but they weren’t very informative.
- Most sources on n-SAT don’t seem to explain what ‘n’ actually is (turns out it’s n booleans per term in disjunctive normal form) $\rightarrow$ Soon going to verify the Cook-Levin theorem. Angular looks interesting (evidently I’m endorsing Google), but having to do JS, CSS and HTML seems like a backache compared to Flutter’s elegant integrated system.
- The UI of my chat app looks polished (no CustomPaint) $\rightarrow$ Firestore vs Realtime Database? - Google’s docs are kind of misleading - seemed to imply that Firestore had a delay of up to 3 seconds (in reality, it’s almost seamless, but in hindsight, 3 seconds isn’t really a problem either).
- Found out how the various definitions of ND Turing Machines are equivalent (I like the self-replicating one the most).
- I touched up on some factorisation algorithms (Wiki has it nicely categorised at the bottom) $\rightarrow$ refresher of the baby step-giant-step algorithm $\rightarrow$ group element inverse powers was SO obvious all along (hint: g^|G| = 1 - reminds me of Lagrange’s theorem)