I’ll just call it ‘Maths’ from next time
Number Theory, Abstract Algebra, Analysis, Topology
- The Wikipedia page on dessin’s d’enfants is sadly not very illuminating.
- I looked at how Liouville numbers were ‘too rationally-approximable’ to be algebraic on proofwiki.
- I found out what a regular graph is -> the Cayley graph for groups! The fact that group characters induce eigenvectors for the Cayley graph adjacency matrix blew my mind!
- I didn’t know there was a bijection between N and algebraic numbers (in Cantor’s non-constructive proof of transcendental numbers: this also finds their density).
- I found a site called mathphysicsbook which provides a shallow dip into mathematical applications to physics - that’s where I was introduced to fibre bundles.
- Peter Cameron also has an excellent blog - I read about integrals of groups -> touched up on the commutator subgroup, abelianisation, subnormal series, prefect groups - now it’s all making sense! It’s wonderful realising that it must be a subgroup - so you can identify it quickly!
- I had forgotten than Spin(3) was isomorphic to SU(2) - that was confusing for a while.
- ‘Torsion group’ is a really weird name, I thought it was related to rotations.
- As part of my beginnings in analysis, I read about set closure, interior, nowhere dense sets, etc.
- I’d forgotten the Perlin noise algorithm, it’s actually pretty cool - sighack.com made crafted some beautiful visual art using it.
- The free object seems reasonably simple.
- Turns out the inverse Galois problem is hard.
- I was searching for some concrete automorphisms of Steiner systems, but it didn’t yield much useful. I suppose I’ll be taking a break from mathematical group theory (maths in general, in fact, after a few days) and focus on applied stuff.
- The stars and bars method on Brilliant had a great set of associated problems. I was trying, without success, to find a proof of the upper bound on Skewes’ number, or even its existence (Littlewood’s proof). Now I’m interested in topological fibrations (inspired by the Hopf fibration).
- I’d like to see a scenario where it pays to see the power set as functions to {0, 1}.
- St. Andrews had a cool page on factor rings and isomorphism theorems - nothing too radical.
- I’d really like to understand Hilbert’s basis theorem - a prerequisite to that is Noetherian rings, so I turned to math3ma - she pointed back to Keith Conrad! His ‘blurbs’ on tensor products, finite fields and Noetherian rings are also very handy.
- You can construct a 65537-gon but not a heptagon :) - I was looking at Galois-based proofs of the impossibility of trisection and non-Fermat prime+1 polygon construction.
- Peter Cameron had a paper relating Steiner systems to the S6 outermorphim!
- I went on a math3ma binge today, looking at the hierarchy of integral domains, tricks using Eisenstein’s criterion, functors (the Jacobian really cleared things up), automorphisms of the disk, the Yoneda embedding, an incredible proof of Brouwer’s fixed point theorem, the fundamental group of the circle (one-line proof) that really kicked off an interest in topology, quantum entanglement and SVD (one of my absolute favourites).
- I randomly remembered ‘Iwasawa theory’ in Wiles’ proof and incredibly, it related the absolute Galois group, p-adic analysis AND ring theory - a good incentive to learn all three (it’s truly remarkable how often I serendipitously chance upon new ideas that relate concepts I’m currently interested in!). Sadly the AMS article that I found on it became very hard very quickly (some background reading is evidently in order, but I wish I could come across a Keith Conrad-style introduction to it).
- Again, I remembered Heegner numbers, so I’ll need to read about class numbers before I can find out more.
- The general fibration looks rather complex, so I’ll focus on Hopf fibrations, for which I found a friendly intro relating to quaternions. Because of the incredible one-line proof of the fundamental group of the circle, I was inspired to read more about pointed spaces, reduced suspension, topological join - even the smash product (quotients are ubiquitous, it seems, so my abstract algebra prep turned out to be useful in this regard); parallelizable manifolds - now fibre bundles look interesting too: I’ll begin with the vector bundle, since it seems pertinent to general relativity too.
- What are Galois reps?
- Finally, I looked at Marsaglia’s ziggurat algorithm.
- All this got me pretty interested in geometric topology. Why is the smash product of S_m and S_n homeomorphic to S_m+n?
- math3ma had lovely posts on operads, quite a bit clearer than the n-Category Cafe one.
- I suppose vector bundles make sense, but I’d still like to see them in the context of GR, or a phase space.
- Tai-Danae Bradley has a paper on Applied Category Theory - in chemical reaction networks and NLP - it’ll be a fantastic read!
- I was looking at parallelizable manifolds, which reminds me - I should see the proof of the Hairy Ball theorem.
- The intermediate and mean value theorem wiki pages were worth the read.
- I need to read more on monoidal categories - I feel they underpin a trove of info.
- Quadratic fields and integers make sense now, as does $\mathbb{PR^2}$ vs. $\mathbb{CP}$ (the trick is in the quotient).
- I don’t know why a fancy ‘Dedekind cut’ is needed in the completeness of the reals.
- On math3ma, I also took a look at the formalisation of the Yoneda lemma (very useful in category theory proofs), maximum vs maximal (again, posets vs. tosets).
- Cornell has a decent book on vector bundles and K-theory: not sure how long my interest will last though. I’m still looking for an intuitive explanation of the Hopf fibration.
- I’d love to be able to understand the exotic R4 manifolds (the great Michael Freedman worked on those)
- Bott periodicity is a potential object of interest - it connects Clifford algebras and vector bundles.
- I saw a fun generalisation of Fibonacci numbers to $\mathbb{C}$.
- Brahmagupta’s solution generator for Pell’s equations was fairly simple, courtesy of Brilliant (no, they’re not sponsoring me).
- As part of a school topic, I was reading about chaos theory (which seems to be rather vague to me) - but I was astounded to find its relation to topological symmetry breaking!
- I finally found a simple explanation of Hopf fibration by David Lyons, quite visual - this special case is far, far easier than fibrations in general then.
- I’m still trying to find the reason behind why the Jordan curve theorem is so hard - I’ll stick with the polygon curve proof for now.
- math3ma’s automorphisms of Riemann surfaces are good practice for analysis.
- I appear to have scrawled “covering space is fibre bundle w/ discrete fibres’, but a week later, I am struggling to decipher this.
- Homotopy, homology and fundamental groups were rather confusing first, but I think I’ve got it now.
- Tom Leinster’s ‘Doing without diagrams’ is also a great guide on how to generalise and apply category theory in proofs.
- I kind of got bored of ring theory, so I’ll leave that aside for now - geometric topology too, most likely - I don’t like to aimlessly dwell on a topic for too long, it narrows my exploration.
- Differential Galois theory sounded amazing, but unfortunately, it’s not used very much since the algebraic world is hard to translate into a form suitable for Lie theory. Ah, only separable field extensions have [L:K] = |Gal| (forgive the shorthand)
- I hadn’t even looked at the difference between total and partial ordering facepalm. I looked at some tutorials on using Coq too - I’d like to understand it’s type checking functionality better.
- Is there a systematic way to construct larger cardinals that is reasonably intuitive?
I’ve decided to add a separate subsection for distinguished reads:
- Svante Janson - Roots of Polynomials of Degrees 3 and 4 - cool use of Galois theory for cubics and quartics (plus some Fourier analysis of finite groups!)
- Couple of PDFs on Steiner systems - but nothing overly deep: I was looking for more on their automorphisms
- Nigel Boston, UMadison, The Proof of Fermat’s Last Theorem - I’d consider this the next stage of Simon Singh’s excellent book; it’s an excellent mixture of history and the more technical aspects, including profinite groups and deformations of elliptic curves. I didn’t get an opportunity to read much of it, but I’ll definitely be coming back to it
- Keith Conrad’s p-adic Expansion of Rational Numbers is very well explained, plenty of enlightening examples. His ‘Factoring in Quadratic Fields’ is also peppered with super-useful examples, but overall it was fairly easy.
- ‘Olympiad Number Theory Through Challenging Problems’ looks really impressive, it’s just what I wanted (I doubt I’ll learn any new techniques, but who doesn’t love tons of solved problems)
- ‘Iwasawa Theory: A Climb up the Tower’ got real hard, real fast (it is highly concentrated after all) - but looking back, I probably should have been more comfortable with PIDs and UFDs beforehand. Jim L. Brown has a much longer, but much more comprehensive introduction on the same. They could both be useful in conjugate with the Nigel Boston paper (*sigh* when will I ever get time to read all of these?)
Physics
Wow, I hardly did anything in Physics this week.
- Non-linear optics and phonons (Fourier decoupling!) look really promising - I’ll have to go through Griffiths chapter 3 again though.
- Allen Chou’s 3D Game Math series is also very useful - very cool method for quaternion renormalisation and implementations, a fun use of Taylor polynomials in sine and cosine approximations, and numeric springing for animations.
- I was trying to find the significance of the Heisenberg group - but what better place than Peter Woit’s book? I had abandoned it a while back, but I’ll persist now - the tensor product of representations was great.
- I just realised, the electric field equations in 2D look very similar to the Cauchy-Riemann equations.
- I returned to Peskin and Schroeder, just for fun, picking out interesting bits from chapter 7.
- I was looking at the Kepler problem since John Baez had a nice presentation on the SO(4) symmetry of hydrogen, and I came to the impressive Bertrand’s theorem. I’m looking for some kind of intro to string theory, inspired by the SO(32) embedding - maybe Joe Polchinski’s one? Even if only for the sake of learning about Calabi-Yau manifolds, M-theory looks like it’s worth a shot. I sorely miss being a kid, where time isn’t at a premium and I could just sit down with a pencil, paper and pure thought.
- I reopened QFTGA, picked up right where I left off - statistical physics, generating functionals, the path integral, Wick rotations, symmetry breaking (I loved this chapter) and the introduction to renormalisation. This book is severely underrated: it covers such a vast array of topics, explains everything incredibly lucidly and has cute diagrams to boot - in fact, its explanation of QED processes (not computation) is superior to P&S’s. I’m also really excited to get a taster of Non-Abelian gauge theories AND QFT for condensed matter.
- I’d also like to read the epic 5-part Division Algebras and Supersymmetry series by John Huerta, John Baez’ student
Computer Science and App Development
- A YouTuber named Dani has an excellent tutorial on grappling physics - good use of quaternions, plus I put into practice line renderers and joints for the first time in Unity.
- On a side note, I cranked out a Python script to add the history to the Chrome extension … until my epoch time was off by a magnitude of 3 (a harmless mistake)
- Sebastian Lague’s Marching Cubes and Ray Marching videos are rather impressive - but Hydraulic Erosion was my favourite, I feel like doing something involving shaders in Unity now.
- I remembered the oddly-named ‘Wavefunction Collapse Algorithm’ from a couple of years ago - no new improvements, but I’m trying to explore the procedural generation universe.
- I was just browsing methods to remove jitter and stack-sinking from my physics engine. While I did find some interesting propositions, like applying variable positional corrections, force-based dynamics on pybullet and Gamedev SE, I’ll only be implementing stuff like warm starting and sleeping in the future.
- Miraculously, and I can’t stress how lucky I was, I sorted out the physics engine. I was mournfully doing some routine code cleanup (adding persistent contacts in preparation for warm starting) when I flipped two minus signs to make it more symmetric. All of a sudden it was super-stable, and I sped it up a fair bit by caching a Cholesky factorisation and then using cho_solve. Of course! I’d forgotten my training - floating-point subtraction kills off precision! It’s now working excellently with just 8 rounds of joint sequential impulse - all I need to do now is improve the speed, clock out more FPS. Later, I reverted to multiple sequential impulse, which wasn’t jittery for the collisions with the new update! All right, straight to GitHub!
- In other news, I also decided to get to work on my Milk Delivery app, designed to connect local grocery outlets with my neighbourhood during the pandemic.
Robotics and Machine Learning
Robotics, eh? Who’d have thought it?
- I randomly remembered Andrew Gibiansky’s quadcopter post - I hadn’t really gone through it properly previously. It was incredible! Robotics is a perfect mix of physics, programming and linear algebra - much like a physics engine, but more practical and more diverse. His paper was probably the best, but I decided to look at a few other quadcopter dynamics papers. It gave me a great overview of what to expect in a robotics problem, but it’s quite complex, so I’ll start with some simple robotics examples - hopefully I can find a course.
- In a stroke of incredible luck, I found a course on Underactuated Robotics by Russ Tedrake, MIT - it is OUTSTANDING. Incredibly intuitive, with visual explanations of phase portraits, applications like Acrobots and cart-pole (just like in RL! In fact, robotics even uses Bellman’s equations in cost analysis!) - this definitely spurned a lifelong interest in robotics. One qualm - a lot of potentially useful information is relegated to the appendices and later chapters (I’m not sure whether the first few chapters were just an exposition, but I wasn’t aware of, say, the manipulator equations).
- I also found a paper on Sparsity in Robot Dynamics - it’s always fun to apply abstract theoretical concepts.
- OpenAI’s blog looks very cool - today I read ‘Deep Double Descent’ and ‘Activation Atlases’.
- Naive Bayes’ classifiers are nice and simple - its unfortunate that my knowledge of classification methods began with ANNs (I’m sure many others’ too) because for simple applications, there are far sturdier tools for the job.