Good Reads
I don’t want this getting lost at the bottom, so:
- Of course Keith Conrad has to make a cameo here - I enjoyed his proof of the simplicity of PSL_n(F) using Iwasawa’s criterion
- Oxford University’s Prof. Steven H. Simon’s Lecture Notes on Solid State Physics was an excellent, humorous introduction to a subject which I never thought I’d enjoy - solid state physics (in fact I used to have a Gell-Mann-like attitude that the only type of physics worth doing is particle physics!). It builds on the bits that I did enjoy in Griffiths Ch.5 - and even the friendliness of QFTGA pales in comparison to this - it could do with a few more real-life examples though. On a related note, I’ve heard that QFT somehow arises out of graphene physics, that could be interesting.
- There are actually so many good papers that I’ve got on my phone - unfortunately, the file organisation system isn’t very ergonomic at all. One of the greats is Claire Frechette’s introduction to dessins d’enfants - originally I was mystified as to why there was so much graph embedding and combinatorics - but the final build up to algebraic geometry showed me just how diverse dessins are
- Jonatan Lindell’s ‘Profinite Groups and Infinite Galois Extensions’ is a very terse, compact account, building up right from the axioms of set theory. It should serve as my go-to reference for such matters, but it’s not the most pedagogical of tools - I for one don’t vacuum in lemmas and suddenly start speaking Galois.
- An unnamed PDF (possibly a section from David Tong’s classical mechanics notes) on Green’s functions for PDEs was a great survey on Fourier transforms, fundamental solutions and Green’s identities
- Columbia’s GR Lecture Notes look very good from a mathematical perspective, but I prefer the physical-oriented style, like David Tong’s
Algebra and Analysis
- I revised a bit of Riemannian geometry - fairly vanilla at this point, except the Lie bracket of vector fields which I had forgotten (or misunderstood, having not been thorough in vector vs vector field).
- I looked at MIT’s conformal mapping, useful for GR which I’ll be restarting.
- I rounded off my Galois theory research with some specific inverse Galois problems, general methods for finding Galois groups (an excellent compilation on SE), symmetry reduction criteria (with Keith Conrad’s S_n and A_n - still not sure how to factor polynomials modulo p) and explicit bases. I wasn’t aware of cycle types, but I see now that they’re useful in conjugacy classes (looks like a partition problem to me?). Also, a site called GroupNames is a great summary of the properties and relations of small groups.
- I also revised some complex analysis - homolomorphic functions, an excellent proof of their analyticity, Parseval’s theorem, hramonic functions, the Weierstrass M-theorem as the analogue of the direct comparison test.
- sporadic.stanford.edu has a fairly good site on Fourier transforms of groups, and I found a great overview of p-adic analysis by Cambridge.
- rreusser.github.io’s visualisation of the sphere eversion was beautiful!
- I was introduced to zeta functions of linear operators (I believe in the mathematical context of causal structure?) so naturally I was led to Green’s function methods for differential operators (as I am returing to physics), and functional analysis is probably a good mix between them - I can start with the functional trace and determinant
- In my research on operator zeta functions, I came across the nLab page which, contrary to my previous experience (see 22 July), I found quite enlightening!
- I suppose it’s because I’m more familiar with category theory and topology now, so I decided to explore related topics - [moved to the Physics section]
- *-algebras are SO straightforward now (four months ago when I’d searched this, I hardly knew what a ring was, much less an algebra)
- Emily Riehl’s article on configuration spaces and cohomology groups looks really interesting, I’d love to understand it properly.
- Brilliant’s page on analytic continuation was illuminating: I can now easily visualise it, after reading on essential singularities, removal singularities, poles and zeroes. Descartes’ rule of signs was quite unexpected.
- Representations and isomorphisms of the p-adic groups should be next on my list - it should help in clearing things up.
- The Gaussian trick for the volume of the n-ball was a really neat find.
- sporadic.stanford.edu has a great paper on Hecke algebras (my motivation was the modularity theorem) - I only read the first few pages where they introduced Coxeter groups - my first thought was that they look like generalised braid groups, at least in the group presentation.
- I read an introduction to infinite-dimensional Lie algebras (ties in with vertex algebras), but I think the Witt algebra is a prerequisite.
- Continuing with complex analysis, I found the unbelievable Picard’s theorems, Cauchy’s integral formula and Liouville’s theorem too - using which jeremykun had a great proof of the Fundamental Theorem of Algebra (finally!)
- Lifting maps to covering spaces was so obvious in hindsight, just compose with the covering map.
- The uniformization theorem was good to know (no proof) - I realise I’d seen it before on math3ma.
- To compose Mobius transforms, multiply the matrices (this is like a reverse representation).
- The Fat Cantor set was a great pathological example (I had just arrived from ‘How to be Small’ on math3ma) - it came from the same source as the long line (‘A Few of my Favourite Spaces’).
- Another great application of 2-adics: Monsky’s theorem.
- I’d forgotten that nilpotent groups had a terminating subnormal series - but why are they useful? Also, what’s the $W_{1, 2}$ condition (some kind of generalised C_1).
- I guessed right - there’s no formalisation of proper classes in the ZF axioms
Category Theory and Linear Algebra
- I had to relook at math3ma’s adjunctions series - I’m more interested in them after hearing the maxim, ‘adjunctions are everywhere’, and the free-forgetful adjunctions. I think a good exercise would be to take a look at the ‘big’ categories - Grp, Ring, Top, etc., and functors, adjunctions, nats, etc. between them - I think the big picture will also be useful in progressing to 2-categories.
- Tai-Danae Bradley’s book, “What is Applied Category Theory?” is quite good, it really picks up in the later chapters - it goes hand in hand with ‘Seven Sketches in Compositionality’ (which she recommends) which could serve as the QFTGA of category to me. I also looked at Grp and Functor categories.
- The illustrious Emily Riehl has another great paper called ‘Infinity Catgeories from Scratch’ - I’ll read it when I return to catgory theory
- I hadn’t realised the connection between SVD and the Moore-Penrose psuedoinverse - John Cook had a great post on that, as well as on surprises in numerical linear algebra (I was mostly aware of these)
- Apparently only normal matrices are unitarily diagonalisable (I’m not sure whether Griffiths mentions this at the end)
Physics
- I was wondering what Noether’s theorem does to gauge fields (turns out you just make $\alpha(x)$ constant and treat it as global).
- I was mulling over whether to look at the use of structure constants in non-Abelian gauge theories, but I’ll cross that bridge when I come to it.
- WHY do Feymnman Parameters work? Also, why does the c-limit exist at all? - the usual rapidity argument is ridiculous, it just shifts the blame to the arctan(v/c) equation.
- On nlab, I read about the Feynman propagator (this one really had a wealth of ideas), Mellin transform for the Schwinger parameterisation (I believe that’s equivalent to Feynman parameters, but I can’t remember how), causal structure, hyperbolic differential operators, advanced and retarded Green’s functions and the rather difficult Hadamard distribution.
- I wasn’t able to find an example for a null basis in Minkowski space - I’d seen it on the causal structure Wiki page, which served as a great refresher for the kinds of paths and cones (I recall skipping that in Caroll) - although I still didn’t understand the term ‘chronological future relative to T’.
- I really want to learn the principal U(1) bundle formulation of electrodynamics (I think it’s level four in difficulty - with geometric algebra at 3).
- Speaking of causal structure, I’d like to see some common conformal maps for solving problems in GR (I actually found one, by UFJF).
- So the Wightman propagator is to normal ordering what the Feynman propagator is to time ordering, right?
- I finally found a decent line bundle connection on phenomenologica.com - while it only hinted at the relation between the two, it was interesting nonetheless.
- I’m drifting into PDE theory now, with the Green’s function of the d’Alembertian.
- Nakahara’s book, ‘Geometry, Topology and Physics’ looks really solid - not even in the context of physics, but because physics tends to be pedagogically more straightforward, so I’ll use it for homology instead :)
- David Tong’s lecture notes on kinetic theory (?) were a great tool for linear response functions (this errs on the side of engineering, I feel).
- Because I am absolutely in love with quotient spaces, I really want to see a description of gauge invariance as a modulo.
- The moving magnet and conductor problem was a good exercise in special relativity.
- Does the Levi-Civita connection form some sort of pseudo-Lie algebra? [Nihar from the future: what]
- I needed some reminding of Weyl spinors and Noether’s theorem worked out (nothing David Tong’s lecture notes couldn’t handle)
- The difference between propagators, Green’s functions and kernels is fairly clear now.
- I suppose Cartan’s theorem on closed subgroups will come in handy for the Lorentz group subgroups.
- I was trying to link graph theory and Feynman diagrams, nothing conclusive though.
- I took a look at the screened Poisson equation - the 1/(r-r’) trick always seems to work. Also I still don’t get why the Yukawa potential dies off (it looks infinite to me)
- The photon sphere was a great application of the Schwarzschild metric -> (in Kerr vs. Reissner-Nordstrom, I think I’ll tackle the former first, it seems more interesting despite its difficulty).
- The Geodesics wiki page is rife with different interpretations, including very exotic ones regarding the ‘double tangent bundle’ and ‘geodesic spray’ (no idea what they do or anything).
- I saw Dirac’s belt trick in action!
- Today I returned ot Schwartz QFT again. Best decision ever - it trumps P&S in almost all regards. That’s why I read almost 200 pages in a single day. I had previously avoided this book since chapters 5 and 6 looked forbidding and unmotivated (maybe they are), but having done it elsewhere, I picked up from chapter 7 and realised just what I was missing out on. He emphasises group theory and Lorentz invariance as a means of postulating Lagrangians and elegantly attacking equations. He works through the calculations as much as P&S, and the calculations in chapter 13 may not be as detailed, but they are still solid - he also motivates spinors far, far better than P&S. He brings in the Ward identity early, and uses gauge invariance to good effect too. I also liked how he used invertibility for linear operators (I’m going to continue functional analysis because of him) and showed several methods to prove the spin-statistics theorem (at least for spins $0, \frac12$ and $1$), along with a stress on building up from phi-fourth to phi-third to scalar QED to spinor QED for new ideas (this really helped me). The group theory could have been a bit more pronounced, but that’s just my John Baez-influenced opinion right there.
Computer Science and Robotics
Technically Scientific Computing
- Since my physics engine is pretty much complete (feature-wise), I had a little fun setting up some scenes - ramps, motors, a paddle, which was my first use of combined rigidbodies (it worked first time surprisingly), even some user look-around. All I need to do is haul in the distance constraints - although this’ll require some reworking since in its current form, ‘constraints’ is a hobbled together UUID-indexed dict. Once I’m done, though, I’ll have a physics engine supporting multi-body, springs, strings, pendulums, ramps, collisions and motors - awesome. I used cProfile to churn out some optimisations (AABB and local-to-global transforms were taking a long time - again, I’d love to port this to C++) + Pyglet has a cool FPS visualiser
- The Robotics SE is surprisingly good - looking at the top questions might help. I researched a bit about inverse kinematics.
- I managed to find a great overview of robotic control and dynamics - it alluded to the SE(3) group often, which was quite interesting
- I thought a good example of linearization control would be the unicycle dynamical system, but I couldn’t find a good treatment of it (hopefully it includes the Udwadia-Kalaba method too)
- I looked at numerical methods to solve PDEs, but engineering is SO BORING (disclaimer: this opinion is subject to change) - I might as well find out the difference between finite difference, element and volume (none of the sources so far have been useful)
- In my search for ‘warm starting’ in physics engines, I came across an epynomous technique in Bayesian optimisation!
- Jeremy Kun had two outstanding posts on cryptanalysis and word segmentation on n-grams - I knew that a naive Bayes classifier was in order!