I draw mathematical worlds. These ideas don't fit on the page: They're too abstract, too far reaching, or too high dimensional. To draw them, I cheat.

Here are some shortcuts and shorthands I use while illustrating math

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There are a handful of easy to draw high dimensional spaces. Learn them, and they will serve you well. I spend half my time drawing tori, like I'm running a psychedelic donut shop.

Math lives atop a scaffolding of physics. Take this arboreal arrangement of quantum field theories, teeming w/ mathematical critters. A mysterious duality lurks beneath, hinting that critters of different realms share an unexpected kinship.

Welcome to the mirror symme-tree.
🧵 physicists describe the universe as a quantum field theory, but we have few exact results. This is part of a strained relationship with math -- You get a million dollars if you can define these things (). Instead, we build a toy model to poke and prod.
2/en.wikipedia.org/wiki/Yang%E2%8…

Symplectic geometry starts with a "closed, non-degenerate two form", and turns that into its whole personality. But what does symplectic taste like? IMO, one of these nerd clusters. Crunchy tangy nerds w/ a gummy center.

Here's how symplectic's flavor changed over the years
1/🧵 Here's a picture of the symplectic 2-form, hard at work. It takes in two vectors and turns it into a sum of areas in a non-degenerate way. 2-forms are familiar to geometry, but this one's special because its closed -- Its exterior derivative is 0.

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In Zelda Tears of the kingdom, you can only rotate vertically and horizontally by 45°. Here's a tip for rotating around the third axis: Rotate all four directions, in order, for a 45° rotation

→↓←↑ = ↺
→↑←↓ = ↻

This happens because of some pretty neat math

1/🧵 ↑ and ↓ rotate the object in opposite directions, as do → and ←. So you might think the opposite arrows would cancel each-other out in →↓←↑, ending up back where you started. But for rotations, _order matters_. The difference between →↑ and ↑→ is a twist ↻.

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When I close my eyes, all I see are strings.

I've become somewhat obsessed with the patterns made from stretching straight lines across a circle, better known as string art. Simple rules can yield beautiful patterns and reveal hidden math.

Join me on my explorations:
1/🧶 It all started in seventh grade math class, during our modular arithmetic unit. We drew a clock, and imagined how the hands move after waiting three hours. Addition by 3, mod 12. Connecting the starting and ending numbers with string, we get a satisfying pattern

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My first paper is out! Last thread I gave background about hyperbolic crystals, and how they turn quantum to algebraic geometry (AG). In the paper, we continue this story with the help of a curious critter from AG, the Higgs bundle…

(this part will be a bit more technical)
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https://twitter.com/chessapigbay/status/1488375642391470081

Quick recap: Quantum mechanics sort electrons by representations of a hyperbolic crystal’s symmetry. But after representing the crystal structure as a Riemann surface, these representations become holomorphic vector bundles! And hence, complex algebraic geometry.

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Today I experemented with giving a very narritive heavy talk, like a full on storybook. The formula definitly needs some refinement, and I shouldn't have done it w/ only a week notice, but I think the aesthetic is delightful! A couple more slides

Exciting day today: My first paper is up on Arxiv!! I’m a big boy now

arxiv.org/abs/2201.12689

We’re extending a bridge between condensed matter and algebraic geometry, built out of quantum matter on *non-euclidean crystals*

Let me tell you about it, in a friendly 🧵!

1/ The world runs on crystals. Your Twitter machine is an array of crystalline semiconductors, with 1s and 0s encoded in the energetics of their electrons. The orchestrator is Bloch’s theorem, which sorts electrons by their interaction with the underlying periodic lattice.

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Color theory really has bottomless subtly. Today I learned the space of colors has geometry, which lead me down a rabbit hole connecting colors with special relativity?? and Riemann surfaces? This stuff is so cool!

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https://twitter.com/akivaw/status/1482883705479778307

Apparently there's a long tradition of Physicists shoving their grubby hands into color theory, like with Weinberg or Ashtekar on this geometry. We can thank Schrodinger for axiomatized color theory. Imagine fitting the perceivable colors into an abstract space:

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Imagine a string streaming in a current. In something like a whirlpool, the string would form a loop. A vortex ring might make a knotted loop.

Wild question: Is there a single current that can tie ~every~ knot?

🧵/ More precisely: the current is a vector field describing a point’s velocity. The point’s path can form a loop, or “periodic orbit”, which may be knotted. Is there a vector field which realizes every knot as a periodic orbit?

The key lies in chaos.

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What’s this Riemann-Roch thing I keep hearing about, and why do people care? In a way, it says “singularities of complex functions can hide inside wormholes”

Hopefully if I explain, I’ll understand it myself…

(I promise this is cool! Prereqs: just some complex analysis)
1/n Complex geometry is so interesting because it’s inherently nonlocal. It studies holo/meromorphic functions, whose value on any tiny portion of ℂ practically determines the whole thing (think analytic continuation). So, meromorphic functions can “see the whole space”
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Lie groups, or manifolds w/ group structure, are very cool. Their groupiness lends some out-of-the-blue oddities, such as:

Every sphere in a lie group can be collapsed to a point

The proof is really neat, as you’ll hopefully see in this visceral and very pictographic 🧵

1/ By “a sphere in” a space, I mean a continuous map from the 2D sphere to that space. This can be “collapsed to a point” if there’s a continuous family of maps taking the original to one which sends everything to a point.

But what do group structures have to do with spheres?2/