Casey Blacker

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.

Selected Graduate Talks

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 Z2-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

Expository talk for the UCSB Graduate Category Theory Seminar, Nov 19 2014.

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*(Gn ; Z2)

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

Let γn be the canonical n-plane bundle over the infinite Grassmann manifold Gn(R). We will first establish that there are no nontrivial polynomial relations between the Stiefel-Whitney classes w1n),…,wnn). Combining this with our knowledge of the cellular decomposition of Gn(R), we will show that the cohomology ring H*(Gn ; Z2) is the polynomial algebra over Z2 freely generated by the canonical classes. We will then prove the uniqueness of w1(ξ),…,wn(ξ) for any n-dimensional vector bundle ξ over a smooth paracompact base space M.