“The Hamiltonian operator describing a quantum particle on a path often extends holomorphically to a complex neighborhood of the path. When it does, it can be seen as the local expression of a complex projective structure, and its perturbations become deformations of that geometric structure. We’ll describe the Hamiltonian of a free particle as a complex projective surface, and we’ll use tools from quasiconformal geometry to study its perturbations. Our main results are loosely modeled on the algebraic ‘transformation theory’ results that underpin the exact WKB method. They’re meant to serve as a foundation for efforts to gain a more geometric understanding of the exact WKB method.”

“Twisted SL₂(C) local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's ‘shear-bend coordinates’ parameterize these local systems of charts. On a surface with punctures, Gaiotto, Hollands, Moore and Neitzke's "abelianization" process computes the shear-bend coordinates of a twisted SL₂(C) local system without reference to its hyperbolic geometry. Using tools from dynamics, we'll generalize abelianization to compact surfaces, leading to a dynamical recipe for the shear-bend parameterization. This recipe lends itself well to numerical approximation, and it may clarify the changes of coordinates that relate different shear parameterizations.”

“The abelianization process of Gaiotto, Hollands, Moore, and Neitzke parameterizes SL_K(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.”

“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.”

One way to build a quantum curve on a Riemann surface is to choose an atlas of local Schrödinger operators related by Liouville transformations. This method produces a convenient space of quantum curves with a beautiful geometric interpretation, but its motivation has never been clear to me. I show here that it can be motivated using an operator-ordering rule, which comes with conceptual questions of its own.

An electron hopping along a chain of atoms is a basic system in physics. When the sequence of atoms is determined by the shape of a flat surface, the motion of the electron becomes related to the shape of a pleated hyperbolic surface. This relationship is mediated by a motley crew of mathematical objects, including cutting sequences, difference equations, flat SL₂(C) bundles, and a weak version of the Anosov property. I show how they fit together into an unlikely bridge from geometry to physics.

This introduction to homology was presented in a data science discussion group. It’s intended as background for discussions of persistent homology, cohomology, and other related topics. I expanded the slides and cleaned up the whiteboard before posting them here. It’s a work in progress, and I’m hoping to develop it more the next time I have a chance to present it.

I describe an old-fashioned way to quantize an energy level of a complexified 1d configuration space. A quantum energy level turns out to be a CP¹ structure on the configuration space, and its perturbations are deformations of that geometric structure. This offers a geometric perspective on one of the algebraic perturbation theory results that underpin the exact WKB method.

An expanded version of the Bristol talk, with less background on hyperbolic surfaces and more detail about abelianization.

Hyperbolic surfaces and flat surfaces look very different, but they're linked by a remarkable correspondence. I show two versions of it: a geometric “collapsing” process that flattens hyperbolic surfaces, and a representation-theoretic “abelianization” process that diagonalizes SL₂(R) local systems. Section 10 of “A dynamical perspective on shear-bend coordinates,” which covers much of the same material, might be a serviceable guide to what’s going on in the slides.

It’s long been known that a hyperbolic surface with a maximal measured geodesic lamination is the same thing, loosely speaking, as a half-translation surface: a singular flat surface with a geodesic foliation. I say “loosely” to mean that corresponding hyperbolic and half-translation surfaces are only identified up to isotopy. I present a tighter version of this correspondence, due to Gupta, which maps each hyperbolic surface to its corresponding half-translation surface in a geometrically rigid way. Section 10 of “A dynamical perspective on shear-bend coordinates,” which covers much of the same material, might be a serviceable guide to what’s going on in the slides.

This talk is an older and more accessible version of “Hyperbolic surfaces as singular flat surfaces,” presented in UT Austin’s Jr. Topology seminar. I use the correspondence between hyperbolic and half-translation surfaces to illustrate the Gauss-Bonnet formula and introduce the correspondence between measured geodesic laminations and measured foliations.

As you deform a holomorphic quadratic differential on a punctured Riemann surface, its combinatorial features jump whenever a vertical saddle connection appears. The associated shear coordinates on the Teichmüller space of the surface jump too, giving the algebra of shear coordinates its celebrated cluster structure. When you deform a holomorphic quadratic differential on a compact Riemann surface, you get an even bumpier ride, with jumps at a dense set of times. I'll set out some ideas for keeping track of these jumps, and show hints of a generalized cluster structure on the algebra of shear coordinates.

Spectral networks are tools for cutting up local systems on surfaces and reassembling them into more convenient ones. I show how they work by doing the most hands-on thing I can do with them: cutting up hyperbolic surfaces and reassembling them into flat ones. As an application, I reassemble a hyperbolic torus with a cusp into a flat torus with a cylindrical end, and use the contrast between the hyperbolic and flat structures to investigate the relationship between shear coordinates and Fenchel-Nielsen coordinates.

The space of GL_K(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 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 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.

“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.”

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.

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.

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.

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.

“After a quick review of Frobenius and H*-algebras, I produce explicit constructions of all the two-dimensional algebras of these kinds.”