Bradley Zykoski

This is a document that I occasionally make additions to, so in some sense it will never really be complete. It began as a way for me to study for my preliminary exam (here is my syllabus). The purpose of these notes is for me to explain the classical approach to Teichmüller theory, as best I can, in my own words. By "classical approach," I roughly mean the approach based in complex analysis, rather than the approach based in hyperbolic geometry.

These are slides I made for a lecture on the Ahlfors-Rauch formula I gave to an undergraduate audience learning about Teichmüller theory for an REU project. The slides are accompanied by a collection of exercises.

I wrote these notes for a talk in the UM grad student topology seminar. I first define the spin groups by way of their double covering of SO(n). The proof that the fundamental groups of SO(n) are all the same when n is at least 3 is taken from this Stackexchange answer. I then give general definitions of principal bundles, associated bundles, and reductions of structure group. A good reference is this set of notes by Thomas Walpuski. Finally, I define spin structures and give examples of some spin structures on the torus. A good reference is this set of notes, also by Walpuski.

I wrote these notes for a talk in the UM grad student topology seminar. They first give a quick introduction to Teichmüller and moduli spaces, and then discuss Harer's proof that the moduli space of a punctured surface has a deformation retraction onto a lower-dimensional cell complex. At the end of the notes are an excerpt from reference [H88] illustrating, in the case of once-punctured tori, the cell complex used in the proof.

I wrote these notes for a talk in the UM grad student dynamics seminar. After introducing the relationship between billiards and translation surfaces, the notes give an overview of the dynamical behavior of generic billiards trajectories. At the end, there is a discussion of the Illumination problem, and a proof sketch for cofinite illuminability in rational polygonal rooms, using the Magic Wand theorem and a result of Filip that all affine invariant submanifolds are quasiprojective complex varieties.

I wrote these notes for a talk in the UM grad student topology seminar. The goal of the notes is to show how the Bers embedding is the meeting point two stories: that of Beltrami differentials and that of holomorphic quadratic differentials. The notes then discuss how, along with the Ahlfors-Weill theorem, the Bers embedding gives us a holomorphic system of coordinate charts on Teichmüller spaces.

I wrote these notes for a talk in the UM grad student dynamics seminar. McShane's identity is a striking result on its own, but it is also a jumping-off point for Mirzakhani's recursive computation of the volumes of moduli spaces of bordered Riemann surfaces, in which one of her steps is to come up with a more general form of McShane's original identity. In these notes, I discuss some of the proof of McShane's identity, following McShane's original proof in most places.

I wrote these notes for a graduate course on algebraic geometry taught by David Speyer at the University of Michigan in the Fall semester of 2018. The goal of the notes is to convince the reader why a deformation of complex structure ought to lie in the first Čech cohomology group with coefficients in the sheaf of germs of holomorphic vector fields. I then spend some time at the end discussing how this deformation naturally pops out of the connecting homomorphism of a long exact sequence.

I wrote these notes for a talk in a graduate course on geometric group theory taught by Dick Canary at the University of Michigan in the Winter semester of 2018. The goal of these notes is to explain the "twisted de Rham cohomology" approach to describing tangent spaces to character varieties. The core of these notes is an introduction to principal bundles and connections. The brief "Tying it all together" section probably doesn't do its stated job terribly well, though. I would recommend the cited text by Labourie instead.

I wrote these notes for a talk in a graduate course on Teichmüller theory taught by Dick Canary at the University of Michigan in the Fall semester of 2017. The goal of the notes is to give a sense for what measured geodesic laminations look like, and then to discuss Dehn-Thurston coordinates on the space of such laminations.

I wrote these notes under the direction of Marja Kankaanrinta at the University of Virginia in the Spring semester of 2015 as my Distinguished Major Thesis. The goal of the notes is to introduce the hyperbolic metric as the unique metric, up to scale, preserved by Möbius transformations, and then to talk about Fuchsian groups and their basic properties.

I wrote these notes for a talk in the UM grad student topology seminar. This was my last foray into studying knots and TQFTs before my primary focus shifted to geometric structures. I spent a little time at the beginning talking about TQFTs in the context of physics. The majority of the talk was devoted to motivating the Cobordism Hypothesis by looking at the special case of 2-dimensional TQFTs. I also tried to compile a nice reference list at the end.

I wrote these notes for a talk in the UM grad student representation theory seminar. My goal was to get a bit of a better understanding of the part that Lie algebras play in the story of the Jones polynomial.

I wrote these notes for a talk in the UM grad student topology seminar. I tried to present a narrative in which Khovanov homology can be understood as an "embedded" version of a TQFT.

I wrote these notes for an undergraduate course on knot theory taught by Slava Krushkal at the University of Virginia in the Spring semester of 2016. Two more references I would like to point out are Dror Bar-Natan's beautiful exposition of Khovanov homology (see in particular the cheat-sheet on the last page) and Peter Kronheimer and Tom Mrowka's paper proving that Khovanov homology detects the unknot.

These are short (3-page) notes proving the basic properties of the number e using only the standard facts of first-year calculus.

I wrote these notes for an undergraduate course on permutation groups taught by Thomas Koberda at the University of Virginia in the Spring semester of 2017. The O'Nan-Scott theorem is a beautiful classification theorem, and these notes present part of its proof and a few applications. They only presuppose knowledge of the group theory one might learn in a first course on abstract algebra, and some notions like primitivity which are more specific to permutation groups and are presented in the references [C99] and [DM96].

This is the first set of mathematics notes I ever wrote for a course, so there's a bit of sentimental value here. I wrote these notes for an undergraduate number theory course taught by Andrei Rapinchuk at the University of Virginia in the Fall semester of 2014. In retrospect, I wish I would have included the part of the proof of Dirichlet's unit theorem that depends on Minkowski's theorem; without it, the discussion of Minkowski's theorem feels like all buildup and no payoff. In any case, I hope these notes might be interesting to anyone looking for a nice concrete application of basic algebraic number theory.