I am a fourth-year RTG doctoral fellow at UC Santa Barbara. My advisor is Xianzhe Dai.

My interests center around spin geometry and index theory. At present, I would like to understand the application of heat kernel techniques to moduli spaces of flat G-principal bundles over manifolds of dimension greater than two.

**The Kirillov Formula in Equivariant Index Theory**

*Advancement Talk, Aug 6 2015.*

Suppose that a Lie group G acts on a closed, oriented Riemannian manifold M by orientation preserving isometries, and suppose that this action extends to a Clifford module E/M with associated Dirac operator D. The kernel of D forms a finite-dimensional representation of G, and the local equivariant index theorem computes the Z_{2}-graded character of this representation as the integral of certain characteristic classes of E/M. In a neighborhood of the identity of G, this result admits an elegant reformulation in the language of equivariant differential forms. This is the content of the Kirillov formula. In this presentation, I will derive the Kirillov formula from the equivariant index theorem and the localization formula for equivariant differential forms.

**Universals and Limits**

Universal objects abound in mathematics: Free groups, the universal cover, the empty set, the product topology — the field is rife with concrete instances of this categorical abstraction. Roughly speaking, an object in a category C is "universal" when it can be characterized as the "least"/"greatest" object in C with respect to a given condition. Following Mac Lane, we will define universals in terms of initial and terminal objects of a comma category. We will then use this language to develop categorical limits, the instances of which include algebraic/topological products, direct sums, presheaves, and the union/intersection of a family of sets. Familiar examples will be emphasized, and categorical fluency will not be assumed.

**The Cohomology Ring H*(G _{n} ; Z_{2})**

*As part of the Fall 2014 Graduate Geometry Seminar, Nov 1 2014.*

Let γ^{n} be the canonical n-plane bundle over the infinite Grassmann manifold G_{n}(R^{∞}). We will first establish that there are no nontrivial polynomial relations between the Stiefel-Whitney classes w_{1}(γ^{n}),…,w_{n}(γ^{n}). Combining this with our knowledge of the cellular decomposition of G_{n}(R^{∞}), we will show that the cohomology ring H*(G_{n} ; Z_{2}) is the polynomial algebra over Z_{2} freely generated by the canonical classes. We will then prove the uniqueness of w_{1}(ξ),…,w_{n}(ξ) for any n-dimensional vector bundle ξ over a smooth paracompact base space M.