Weekly and monthly summaries of interesting finds and what I’m working on. Many of the later posts can be found on my Physics page.
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The Good Stuff (21/9/20 - 27/9/20) 21 Sep 2020
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....
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The Good Stuff (14/9/20 - 20/9/20) 14 Sep 2020
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...
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The Good Stuff (7/9/20 - 13/9/20) 07 Sep 2020
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...
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The Good Stuff (31/8/20 - 6/9/20) 31 Aug 2020
Number Theory and Algebraic Geometry
I found a short article on lattices, by John Baez (importantly, it goes into the non-elliptic curve formulation of the j-invariant through the eta function). St. Mary’s College, California has an enlightening paper on the Lindemann-Weierstrass Theorem I didn’t see a method to calculate the q-series of the j-invariant without a deep detour into L-functions (apparently this is one of Noam Elkies’ areas of expertise). One method involved the Hardy-Littlewood...
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The Good Stuff (24/8/20 - 30/8/20) 24 Aug 2020
I’ll probably be renaming (and/or splitting up) ‘Group Theory, Calculus and Fun Stuff’, probably into more manageable sections like Number Theory, Group Theory, Abstract Algebra - Category Theory, Linear Algebra and Logic+Set Theory already exist.
Group Theory, Calculus and Fun Stuff
The series differential equation for Bernoulli polynomial might come in handy. MathCounterexamples.net is very, very useful - particularly for topology and vector space concepts. The supremum’s like a categorical limit on posets. I saw...
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The Good Stuff (17/8/20 - 23/8/20) 17 Aug 2020
If anybody comes across an error, please feel free to correct me!
Maths
Out of all the subjects, I think Brilliant excels most at Number Theory (although Stanford’s Cryptography website comes close) - I tried my hand at some problems using Euler’s totient theorem, the rather obvious rational root theorem and the Extended Euclidean Algorithm (super important with Bezout’s theorem for RMO) with the Chinese Remainder Theorem (which, on the surface, looks extremely fancy, but...
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The Good Stuff (10/8/20 - 16/8/20) 10 Aug 2020
Here’s where I’ll be noting fun reads, interesting finds, cross-topic connections, useful sources and unresolved questions that I’ve had over the week. This weekly recap is in part inspired by the illustrious John Baez and his weekly finds in mathematical physics. This is the first post of a long series, so I wouldn’t be too surprised if I bring in some major overhauls.
I’ve decided to split it up by category, so you’ll have some...
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The Good Stuff (4/8/20 - 10/8/20) 04 Aug 2020
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...
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The Good Stuff (20/7/20 - 27/7/20) 20 Jul 2020
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...