Warping geometric structures and abelianizing SL2(R) local systems
“The abelianization process of Gaiotto, Hollands, Moore, and Neitzke parameterizes SLK(C) local systems on a punctured surface by turning them into C× local systems, which have a much simpler moduli space. When applied to an SL2(R) local system describing a hyperbolic structure, abelianization produces an R× local system whose holonomies encode the shear parameters of the hyperbolic structure.
This dissertation extends abelianization to SL2(R) local systems on a compact surface, using tools from dynamics to overcome the technical challenges that arise in the compact setting. Thurston's shear parameterization of hyperbolic structures, which has its own technical subtleties on a compact surface, once again emerges as a special case.”
Limitations on cloning in classical mechanics
“In this paper, we show that a result precisely analogous to the traditional quantum no-cloning theorem holds in classical mechanics. This classical no-cloning theorem does not prohibit classical cloning, we argue, because it is based on a too-restrictive definition of cloning. Using a less popular, more inclusive definition of cloning, we give examples of classical cloning processes. We also prove that a cloning machine must be at least as complicated as the object it is supposed to clone.”
Potentially cluster-like coordinates from dense spectral networks
The space of GLK(C) local systems on a punctured surface has a natural cluster coordinate system, which was first described by Fock and Goncharov (2006), and can be computed through an elegant geometric construction introduced by Gaiotto, Hollands, Moore, and Neitzke (2013, 2013). This construction, in at least some cases, can be extended to surfaces without punctures, yielding an algebra of functions on the space of local systems that behaves like a “cluster algebra without clusters.” A very sketchy overview can be found on this poster, presented at a CRM workshop on the cluster-algebraic and integrable systems aspects of positive Grassmannians in July of 2015.
The Geometry of The Night Sky (or, An Ape Pointing at The Stars)
The night sky looks like a sphere (this is more subtle than it sounds), and the Lorentz group acts on it by Möbius transformations. This action gives an isomorphism between the Lorentz group and the Möbius group, so if you’re piloting a starship, you can work out your heading in spacetime from the positions of the stars. The slides, from a talk I gave at JMM 2015, mostly follow the text, although you’ll have to figure out for yourself how they line up. Either by chance or by necessity, the exposition here is very similar to the one in Penrose and Rindler's Spinors and space-time, so you can look there if you want to see a similar argument in a different style.
Where do alternating multilinear maps come from?
“Alternating multilinear maps, as the pointwise constituents of differential forms, play a fundamental role in differential geometry, while other kinds of multilinear maps—for example, symmetric ones—hardly show up at all. In these notes, I’ll try to illuminate the geometric nature of alternating maps by linking them to our intuitive concepts of volume and area.”
Farey Sets in Rn
If you take the lattice Z2, scale it by 1/n for each n from 1 to 40, and plot all the results on top of each other, you’ll see the picture to the left. It may look quite startling, but all its intricacy can be understood using just two basic ideas from number theory: Farey sequences and Bézout’s identity.
An illustrated proof of the Perron-Frobenius theorem
You may not know the Perron-Frobenius theorem’s name, but you probably know its deeds. It can be used to reason about the stability of Markov chains, the growth and decay of populations and economies, and the convergence of ranking schemes.* It also has a lovely geometric proof, an application of the Brouwer fixed-point theorem and the properties of projective transformations. These slides will walk you through the argument.
* To learn more, check out MacCluer’s “The Many Proofs and Applications of Perron’s Theorem” and Keener’s “The Perron-Frobenius Theorem and the Ranking of Football Teams.”
Matrix Algebra and Error-Correcting Codes
These notes started off as an enrichment piece for computer science and electrical engineering majors studying matrix algebra at UT Austin. They're meant to show how the tools you pick up in a first matrix algebra course—things like matrix multiplication, linear equations, column spaces, null spaces, bases, pivots, column operations, and inversion—can be used to design and implement error-correcting codes.
Relativity and Quantization
Quantum mechanics is full of relativistically invariant wave equations, like the Schrödinger and Klein-Gordon equations. One way to see why these equations are so useful is to look at their solution spaces as representations of the symmetry groups and observable algebras of free particles. If you’d like an introduction to the Klein-Gordon equation that skips the traditional hand-wringing about negative probabilities, you might like this one.
Classification of two-dimensional Frobenius and H*-algebras
“After a quick review of Frobenius and H*-algebras, I produce explicit constructions of all the two-dimensional algebras of these kinds.”