Groups, Geometry, and Dynamics
Hierarchically hyperbolic spaces and applications to Teichmüller theory (i.e. the Masur-Minsky machinery)
A tour through the proof of Margulis Superrigidity
The Nielsen-Thurston classification of mapping class elements
These are notes for the talk I gave on Margulis Superrigidity at the topology seminar at the
University of Michigan in Fall 2019. This is more or less based on the sketch of the proof in
Dave Witte Morris' Introduction to Arithmetic Groups
Upgrading ergodicity to mixing
These notes contain somewhat detailed proof of the Nielsen-Thurston classification of the elements of the
mapping class group. Most of the proof is from Bruno Martelli's
An Introduction to Geometric Topology
, although I did need to fill in a few details on my own. However, these notes are still far from self-contained
and should be treated more like a supplement to the section in Martelli's book.
The geodesic flow on symmetric spaces
These are notes I wrote for a talk in the student dynamics seminar at the University of
Michigan on the proof and applications of the Howe-Moore theorem. In a nutshell, the Howe-Moore
theorem says that if the action of a semisimple Lie group is ergodic, it is also mixing, which
provides a quick and easy way to show some related actions are mixing, for instance, the geodesic
flow on a finite volume hyperbolic surface.
Notes on entropy
These are notes I wrote for a talk in the student geometry seminar on the geodesic flow on
symmetric spaces. The main point of this talk was to illustrate the the geometry of symmetric
spaces, and the geodesic flow in particular can be completely understood in terms of the linear
algebra of the associated semisimple Lie groups. As an application, we used just linear algebra to prove
that the geodesic flow on the any finite volume rank 1 locally symmetric space (of non-compact type) is
Constructing complex structures on surfaces via the Beltrami equation
These are notes I took for Ralf Spatzier
course on entropy and its applications in dynamics. These notes are far from complete, and I hope to come
back and complete these at a point in the future when I understand entropy better.
Divisors and the Abel-Jacobi map
These are notes I wrote a talk on the Beltrami equation and how one can use it to construct Riemann surfaces,
and an outline of the proof of when a solution exists.
The moment map for symplectic toric varieties
This is the term paper I wrote on divisors and the construction of the Abel-Jacobi map. The
first goal of this paper was to convince the reader that divisors on Riemann surfaces are
interesting objects, and once that was done, construct the moduli space of all (principal)
divisors, which turns out to be a complex torus (this is where the Abel-Jacobi map comes in).
This is the term paper I wrote outlining the link between toric varieties and convex
polytopes. In particular, the goal of this paper was to work out a lot of examples going from
convex polytopes to toric varieties, and more importantly, from toric varieties to convex
polytopes. The latter as it turns out, can be done by constructing what is called the moment map.
This led me to working out the convex polytopes for a class of toric varieties I did not know
the convex polytopes of, namely the Hirzebruch surfaces.
High Dimensional Probability
Notes on Moduli of Curves
Notes on the measure concentration half of the course on high dimensional probability I took in Winter 2021.
The Laplacian on Riemannian manifolds
These are notes I live-TeXed in the course on Moduli of Curves, taught by Aaron Pixton in Winter 2020 term at the University of Michigan. These notes mostly focus on the topological
and algebro-geometric aspects of the theory, and not the hyperbolic/complex analytic aspects. The TeX files for these notes are here
. I would
appreciate it if people send me corrections to errors in my notes via pull requests.
Notes on homotopy theory
Roth's theorem on 3-term arithmetic progressions
Weyl’s equidistribution theorem for linear and quadratic
This is a condensed version of my senior thesis on the Laplacian on Riemannian manifolds. This article
looks at the Hodge decomposition theorem, and some geometric applications of the theorem. It also outlines
some results relating the eigenvalues of the Laplacian to geometric properties of the manifold like
Ricci and scalar curvature.