Aaaand I forgot this week too.
Maths
- I can’t wait for the Hodge star-cross product duality from Michael Penn - in the meanwhile, I contented myself with a newfound blog series on exterior algebra by Alex Kritchevsky (quite nice actually) $\rightarrow$ A proper book on Geometric Algebra is a must.
- I think I’m now in a position to tackle the relation between the gamma matrices and Clifford Algebra (Geometric Algebra apparently doesn’t have the Hodge Star, which I’ve been dying to see used in a practical problem besided cleaning up Maxwell’s Equations).
- Cool notation for matrix norms - $1/p \rightarrow q/p \rightarrow 1/q$, rows first (e.g., frobenius norm is [2, 2]). The L-infinity norm is hilariously just the supremum (which I found out means the limit of the terms, but might not be contained in the set itself, contrary to the maximum.
- Socratic is actually a great surfing alternative to Quora maths. I should really find out about the more exotic finite simple groups (monstrous moonshine! what a name - and a very deep, difficult connection to the j-invariant which I hardly understand
- I found a cool page on the E8 group, detailing some kind of calculation, but more importantly, its properties and signifance (maybe String Theory will prove to be useful for maths!). I think I understand root systems too (talking about them from the lattice perspective is way more understandable), and the E8 Dynkin Diagram makes sense (thank you John Baez!) - but I’m not sure why the leg has to stick out in the middle - the relation of the icosahedron to E8 was honestly mind-boggling.
- I like the visualisation of the real projective plane with the infinite circle surrounding it (although I’m still puzzled by the complex projective space).
- I was trying to find the proof of the $\R^n \rightarrow \R^{2n}$ embedding theorem that I read in Carroll, but I couldn’t remember the name.
- What’s the Fano plane? - its symmetries form a finite simple group, according to Alon.
- Cool integral formula from Michael Penn (I find that x1.5 - x1.75 is a little more my speed)
Logic and Set Theory
- Reading about logical fallacies is a fun pastime of mine.
- The By::Analogy Quora space has a collection of nice proof perspectives (mostly populated by the great Roman Andronov - he ranks fourth on my list after Brian, Senia and Alon)
- I was completely wrong in thinking that necessity was a subset of sufficiency (using arrows in such a detailing is a strict no-no, which is why math3ma’s post on this confused me).
- The Wug test (and unrelated Lex/Yacc and parser research that I did when doing SMILES parsing for Molecular VAEs) has convinces me to maybe (?) take a look at Chomsky’s linguistics work (some claim that it’s just airy and back-fitted - if that’s the case, I’m sure there’s something on universal grammars in some other context)
Physics
- The Kerr effect seems interesting (I should read about negative refractive indices!) $\rightarrow$ Dispersion relation and group velocity (I had visited these in April, I believe, but the first signs of Chrome history decay are showing - the links weren’t purple).
- Don’t anyons have (anti)commutation relations? I think I should find the answer in its representation theory - speaking of which, I really enjoyed the Harvard RT book by Noah Miller, but I think a break from QFT and QM is long overdue.
- Over the last week, I’ve been reading QFT for the Gifted Amateur. It’s EXACTLY what I needed at this stage. I picked up from Chapter 18 and just like that I cruised through Feynman Diagrams (and I’m obsessing over their elegance) and a bit of Spinors. It’s laid out in the perfect manner for me (Wick’s theorem, contractions, interaction picture) - all of it made perfect sense and brought intuitive insights. One thing though - Feynman’s propagator is kind of pulled out of a hat (but IIRC, Schwartz has a complete derivation without resorting to the something-Plemelj theorem)
Computer Science
- ‘Dynamic Programming’ is a strangely fancy way of saying that you need memoization and data structures.
- I read the closest pair of points problem (personally, I like brute forcing all problems because I’m lazy) $\rightarrow$ the element distinctness problem is best left to randomness (thank you, birthday problem) - even smashing everything together in a hash table works.
- The difference between Quantum Computers and NDTMs reminds me of the common misconception that the measurement/decoherence process (which I should touch up on) is not ‘random’, as they say.
- SAT makes sense (exponential time in disjunctive/random $\rightarrow$ conjuctive). Incidentally, I found a Haskell SAT solver on Andrew Gibiansky’s blog. It’s really well designed and although I couldn’t finish it, it got me thinking: could I do this in Rust? (that might be stretching its Enum capabilities but it should be worthwhile)
- Transposition tables are worth researching even if I don’t have plans for a chess engine - it’s a good exercise in data management (hey, caches!). In fact, I have to find out more improvements to the MCTS method in general (not just RAVE and other technical ones - I mean bridging alpha-beta and MCTS).
- Fortune’s Algorithm looked super difficult (a bit more research into Bezier curves MIGHT be useful to this end) - again, I’d just brute force each pixel.
- I found Nicky Case’s website! The Game Theory connection to social relationships was seriously cool - I admire their exposition and art style.
- I understood how Voronoi diagrams are the dual of a Delaunay triangulation (how do you generate dual diagram screen coverings? I’ll try and use SFML to have another crack at the physics engine without having to deal with OpenGL (SFML actually looks a lot like Piston in Rust - but it has audio and networking too).
Stats and Machine Learning
- Back to Li’l log! Reinforcement learning and Transformers are on the menu today $\rightarrow$ from Brian Keng’s excellent (and much easier to read) blog, I sampled log-likelihood for generative models, the one neuron approximator (seeing the proof should be a good section on Lebesgue integration) and VAEs with Normalising Flows (sorry Eric Jang).
- Hyperparameter optimisation is going to link nicely with operations research (I’ll try and throw genetic algorithms in there too). Unfortunately I’d forgotten exactly what priors and posteriors were, so a quick trip to Stats SE sorted that out
- I picked up Bayesian Data Analysis by Gelman (looks comprehensive, but a bit too long in the current climate)
- spinningtop’s docs are a fantastic intro to RL (they don;t throw a lot at once unlike Lilian’s blog - although hers too is great - and RL seems to be her main focus). In fact, the state transitions (or Markov Decision Processes in general) reminded me a lot of Turing Machines (obviously). Q-Learning apparently converges faster than SARSA.
- Following Andrew Gibiansky, I should try and use k-means (k-means++ probably, unless I find a simple but more powerful alternative) to do MNIST classification, just to see how high I can get it.
- I also looked at symmetric matrix decomposition, just for fun
- Lilian has a gripping tale on Evolutionary Algorithms (goodbye, GA!). Finally, I should try and make a project using OpenAI GPT-3