I haven’t added in the PDFs in yet, they’ll be here soon.
Maths
- I’ll try and understand the proof of why there are no finite dimensional unitary representations of the Lorentz group.
- I looked at geodesics on an ellipsoid, the Cauchy problem makes sense and I suppose the closed subgroup theorem is good to be aware of.
- Although I didn’t get past the first chapter, the Diagrammatic Calculus of Coxeter and Braid Groups by N. Gowravaram and Uma Roy (MIT PRIMES) looked really beautiful (Dynkin diagrams and Weyl groups are quite straightforward after all, perhaps my aversion to papers back then hindered my progress - I’d still like to look at a few examples, e.g. of $\mathfrak{sl}_n$, but I believe I’ll need the Cartan matrix). To add to this, I found a ‘Graphical Introduction to classical Lie algebras’, which uses category theory too!
- I encountered Lipschitz continuity and the cascading inclusion chain (continuous vs. uniformly continuous: roughly, use the same delta for all x).
- Regularisation involved the expansion of the gamma function at the poles, which naturally led to the Weierstrass factorisation theorem and a short proof for the Euler-Mascheroni constant integral.
- I looked at Banach’s fixed point theorem applied to cosine and polynomials in general.
- Now I have a good example of a bundle section - vector fields and the tangent bundle (it just looks like an inclusion map backwards).
- I also found a great paper covering rings and modules (the obvious next step), and my quest to undestand the Calabi-Yau manifold led me to a summary of differential geometry.
- The homotopy theory on wiki is actually a pretty good summary of cofibrations and related concepts too.
- There was a useful Warwick PDF on elementray p-adic results (I’m quite comfortable with them now), also Keith Conrad (of course!) lecture notes on p-adic infinite series (which I haven’t got round to reading yet).
- There’s actually been a full-cycle algorithm for the 3x3 Rubik’s cube, speaking of which, I’d like to understand Thistlethwaite’s algorithm.
- The Peter-Weyl theorem looks like a useful sledgehammer (indeed, I alluded to it in my first Maths SE answer regarding the completeness of he spherical harmonics - although I am yet to see the Stone-Weierstrass proof).
- Retractions and deformation retractions in topology make sense now (and its use in Brouwer’s fixed point theorem).
- I tried to find something interesting on $\text{O}(\infty)$ or $\text{D}_\infty$, but it’s not quite clear.
- Box vs product topology looks a lot like direct sum vs product to me (I’m guessing they’re categorical limits/colimits resp. somehow).
- How do different free groups (and different vector spaces) differ among themselves?
- As usual, I forgot the difference between Hilbert spaces, metric spaces and Banach spaces. I also found something called ‘Finite element exterior calculus’ which looks wildly interesting, but seems quite difficult.
- I liked Gauss’ heptadecagon constructibility proof with the Gaussian periods (now I see why Fermat primes work - because the roots ‘halve all the way down’) - how do I find the general cos(2pi/n)?
- I looked at distribution on nLab (+ bump functions, which emerge as the Fourier transform of continuous functions -> Schwarz space. That’s why fields go to zero at infinity.) as well as symmetric group representations - Young tableau find use here!, and I grokked compact spaces once and for all.
- The Nilsen-Schreier theorem is cool and all, but I feel like it has some elegant applications which I can’t seem to find.
- I missed this - L’Hopital’s rule states that g’(x) cannot be zero anywhere on the open interval containing c. On the whole, analysis seems forbidding because of its rigor and many definitions, but it’s extremely versatile, and can be used in a myriad of ways, even in the most unexpected of places.
- The FEEC paper inspired me to read about Galerkin methods - the best resource was a UToronto Lecture. In fact, this could pave the way for something new (and something that I’ve been getting at for ages) - functional analysis. It’s slightly more advanecd than regular analysis, but I feel it’s more memorable and practical - I looked at Lp completeness, weak derivative, Hilbert spaces again. Robin vs. mixed boundary conditions is pretty simple.
- It’s important to remember that diffeomorphism is weaker than Lie isomorphism (obviously, but sometimes it makes a difference).
- The subset toplogy (and hence ‘continuity of inclusion’) make sense now.
- The $W_{1,2}$ thing I was puzzled about earlier are actually just Sobolev spaces.
- I found a great PDF as well as a website (with code) on the Galerkin method (which really mirrors my chemistry Hartree-Fock program in the sense of ‘basis-pursuit’).
- The FEEC paper is a little clearer than the presentation (which was a little too concise).
- I checked out the fundamental theorem of abelian groups (and how it’s different from the generalised Chinese remainder theorem).
- Also, chain complexes are like a generalisation of exact sequences. Even de Rham cohomology is starting to make sense (especially in the context of differential forms - Carroll was a tad hasty in his exposition in the GR book) - but first I had to revise differential forms, and then simplicial complexes from none other than Jeremy Kun.
- The norm topology is obvious in hindsight, but again, it’s something that I often take for granted.
- UChicago has a pretty good introduction to schemes, but I’m not that interested in algebraic geometry anymore (much like number theory, its too abstract.
- Integration of k-forms is confusing - I understand the pullback, but how come the wedge product vanishes under the integral?
- I’m beginning to view the Fourier transform as a Schauder basis now. Also, the Baire Category Theorem has changed my perspective on ‘nowhere dense’ sets.
Linear Algebra and Category Theory
- ‘Thirty-Three Miniatures in Linear Algebra’ by Matoušek is exactly the kind of thing I was looking for - a short set of useful tricks that could prove handy in problem solving.
- The determinant can be calculated from the trace (I hadn’t seen the relation with Newton’s symmetric polynomials).
- The one-by-one substitution proof that all bases of a vector space are the same cardinality was cool - its something that I usually just take for granted.
- I’ll try to formalise my knowledge of linear algebra (‘the only thing mathematicians really understand’!) - so I looked at the critical distinction between Hamel vs Schauder bases (how can the Baire Catgory Theorem be used to prove that Hamel bases are finite or uncountable for Banach spaces? Also, how does the axiom of choice come into play here?), and the incompleteness of the space of polynomials
- I looked at cokernels as the measure of failure of surjectivity and the constraints on a linear equation -> zero morphisms, universal properties (again, it seems to be a catch-all term)
Physics
- I looked at some analytical methods for finding the Klein-Gordon propagator on SE - I wonder if there’s an easier method for the light-cone integral?
- I started regularisation and renormalisation (and importantly, their difference, aided by an SE answer. I thought that all it took was a ‘hard cutoff’, but that breaks almost all the symmetries) in Schwartz with Appendix B and Chapter 16 (in this regard it’s similar to, but more involved than, P&S chapters 6 and 7). I find it to be peppered with some cool motivating examples (Casimir effect, the Lamb shift especially, bringing me back to 2S1/2-2P1/2 splitting with first-order perturbation theory!) which puts the theory into perspective nicely.
- I’m guessing that Schwartz’ dLIPS is just the invariant four-volume element?
- Also I was correct, Schwinger and Feynman parameters are equivalent. Two annoying things that I just cleared up: Lorentz invariance vs. covariance and dimensions of the metric tensor (convention).
- I continued with symplectic geometry for Hamiltonian mechanics (classical, mind you. I want to be able to look at Hamiltonian mechanics with the Poisson algebra + symplectic manifold formalism in a classical context, which also finds use in my newfound hobby, robotics!). So, I read a bit about symplectic manifolds and their relation to phase spaces (this part isn’t too strong), the interior product (when you expand the exterior algebra equivalence class, it’s a bit like applying the product to the first term only)
- I couldn’t resist getting a taste of IR divergences (looks similar to the one in P&S), Yang-Mills theory and QCD Feynman diagrams - it seems to mirror QED, just with some color generators attaches; consequently, the pace is rather fast. To complement Schwartz, I need to find a good overview on practical calculations in QED and QCD (Schwartz often leaves us ‘hanging’ as to what to do with the result).
- John Baez actually has lecture notes on Hamiltonian mechanics - unfortunately the later ones haven’t been transcribed to LaTeX yet. MIT Open Courseware might be really handy for this.
- Again, I’m continuing with formal QFT, so I read about n-point functions, the Wightman axioms, Wick rotations (which just ‘wraps’ to periodic Euclidean spacetime), operator-valued distributions, adiabatic switching (this was there in QFTGA). Fock spaces are just the tensor algebra of single-particle states, as I intuited correctly. Although I didn’t delve too deeply into Wick algebras (they are hardly mentioned outside nLab), the equivalence between matrix and wave mechanics is just due to them being different representations of it.
- Interstellar encouraged me to revisit General Relativity - wormholes and time dilation seems like a good place to start.
- The Casimir negative energy sounded suspicious, and it is (negative relative to the zero-point energy).
- I revised some degenerate perturbation theory (I was looking for something formulated in a more matrix-y way) -> quadratic and linear Stark effects, presented by Richard Fitzpatrick were great
- Rigged Hilbert spaces just look like they add Dirac orthonormality
- ER=EPR sounds interesting, speaking of which, Lenny Susskind’s lectures could complement David Tong’s lecture notes very nicely.
- Now I know what Kahler manifolds are (it’s hard to keep track of all these definitions, but each one always seems to satisfy about 3 properties) and states on C* algebras. Unfortunately, I’ve found out that while Algebraic QFT may be mathematically rewarding, it was formulated at a time when the physical underpinnings weren’t entirely fleshed out, and consequently, they fail to describe several phenomena that were found later. I credit it with introducing me to functional analysis though.
- I read a bit about the weak interaction in Schwartz (I was under the impression that it would be taught before QCD, but it seems quite challenging).
- I also finished the first lecture in ‘Mastering Quantum Mechanics’ by Prof. Barton Zweibach, MIT. Nothing too difficult, it seems to mirror Griffiths Chapter 1.
Computer Science, Robotics, App Dev
- I finished outlining the booking and payment screens for the app, creating it won’t be easy though.
- gieseanw.wordpress.com is a fantastic blog - he covers expression templates in C++ vs. the new coroutines (now I’m excited for them); double dispatch, forward declarations - I hadn’t seen a C++ blogger before this.
- Julia actually has SFML bindings, now I’m leaning towards it (it is garbage collected, but it shouldn’t matter too much).
- I even browsed through some of Jeremy Kun’s other topics, simulating a fair coin with a biased one, hybrid images, 3d Game of Life
- I managed to get the client-side prototype working in Flutter, just a bit of UI - I think I’ve developed a slick minimalistic style (perhaps not the most appropriate for milk delivery). I used animations and collapsible appbars for the first time - Flutter makes the workflow super smooth. I found a useful date picker by Syncfusion, but they need licensing so it might not make the final cut. In fact, I’d originally over-engineered it a little, so I imporved the ease-of-use front. Overall it was a thorough review of Dart and Flutter.
- Zero-knowledge proofs were quite interesting (I think I’ve seen then before albeit in a different form - discrete log or blockhain).
- I also know what mimaps are now (they avoid the Moire effect too!). For some reason, I decided to read a bit more about computer graphics -> back-face culling, alpha compositing, normal mapping, gamma correction, z-buffers, normal mapping, then onto shading models - Lambert and Phong (a review by Cornell).
- An important point on the discrete log problem - it seems O(n) in terms of the group order, but its exponential in terms of element exponentiation - what the actual computation involves
- The MIT presentation on forward and inverse kinematics (which, turns out, is just a frame choice - effector vs base) often referenced ‘grasping’, so I’ll take a look at that (I believe it’ll be more physics-oriented, using, e.g. friction). Anyway, a UMass PDF cleared some things up on linearization of control (its just like physicist’s linearization; I don’t know what I was anticipating)
- Much in the spirit of Andrew Gibiansky’s posts, Jeremy Kun also has a set on KNN for digit classification (his repertoire is astonishingly comprehensive)
- I’m trying to learn the meaning behind the expectation-maximisation algorithm - I’m also going to be looking at interesting machine learning algorithms, e.g. KNN for local outlier detection, and hopefully something related to structured data (even for NLP - reminds me of Chomsky’s hierarchy, which amazingly implies that regex cannot be used to parse HTML)