Notes
These are some expository notes I’ve written for classes, talks, etc. For the most part they do not contain original material.
pdf abs Equivalent definitions of the tangent space
Abstract: We prove the equivalence of several common definitions of the tangent space to a smooth manifold, and then show that this equivalence is "natural" in the sense that the differentials are related via factorization through the corresponding isomorphism between any two tangent spaces.
pdf abs Lie groups and quotient manifolds
Abstract: We present all of the necessary tools and techniques to state the quotient manifold theorem: given a smooth, proper, free Lie group action on a smooth manifold, the orbit space is a smooth manifold. Then we explore some important applications of the theorem, such as the construction and characterization of homogeneous spaces.
pdf abs Lie algebras and exponential maps
Abstract: We present a proof of the closed subgroup theorem for Lie groups using the machinery of Lie algebras and exponential maps. Along the way, we introduce all of the necessary background information about vector fields and Lie algebras.
pdf abs Shape parameters of ideal tetrahedra
Abstract: We define the notion of ideal tetrahedron and the shape parameters associated with its edges in hyperbolic space and prove some basic facts about them. Final paper for "Algorithmic topology and geometry of 3-manifolds" with Nathan Dunfield.
pdf abs From differential operators to geometric algebras
Abstract: We explore the relationship between the existence of Dirac operators and Clifford module structures on a vector bundle. Final paper for "Differentiable Manifolds II" with Susan Tolman.
pdf abs Topological and operator K-theory
Abstract: Using the $K_{0}$ group of a compact Hausdorff space to motivate the definition of the $K^{0}$ group of a $C^{*}$-algebra, we introduce operator K-theory as a non-commutative analogue of topological K-theory. Final paper for "Hilbert Spaces" with Florin Boca.
