Expository notes I've written for classes, talks, etc. For the most part they do not contain original material.

pdf abs Conservation laws for Yang-Mills Lagrangians

Abstract: Notes on gauge theory for principal bundles, Yang-Mills Lagrangians, and conservation laws. Written during a reading course on gauge theory with Gabriele La Nave at UIUC.

pdf abs Connections and vector bundles

Abstract: Notes on the theory of connections on smooth vector bundles starting from the basics.

pdf abs Complex geometry

Abstract: Notes on the complexification of real vector bundles, complex local coordinates, and almost complex structures.

pdf abs Equivalent definitions of the tangent space

Abstract: We prove the equivalence of several common definitions of the tangent space on a smooth manifold, and then show that this equivalence is "natural" in the sense that the associated differentials are related via factorization through the corresponding isomorphism between any two tangent spaces.

pdf abs Lie groups and quotient manifolds

Abstract: We present the 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. Report for "Algorithmic topology and geometry of 3-manifolds" with Nathan Dunfield.

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.