Casey Blacker




I received my PhD from UC Santa Barbara in 2018, under the guidence of Xianzhe Dai.

My reasearch lies at the intersection of symplectic geometry and gauge theory. I am also interested in spin geometry and index theory. You can find more information in my research statement.


Publications

  1. The Moduli Space of Flat Connections over Higher Dimensional Manifolds, 2018, PhD Dissertation
  2. First eigenvalue of the $p$-Lapacian on Kähler manifolds, (with Shoo Seto, submitted), 2018, arXiv:1804.10876

Selected Graduate Talks


The Moduli Space of Flat Connections over Higher Dimensional Manifolds

JMM San Diego, Jan 10 2018.

While the moduli space MG(Σ) of flat connections on a G-principal bundle over a surface Σ has been extensively studied, the case of a higher dimensional base M remains largely unexplored. Given a Lefschetz symplectic form on M, there is an induced symplectic structure on the moduli space MG(M). We will show that, under the action of the gauge group, MG(M) is a generalized symplectic quotient of the space of all G-connections over M, endowed with a natural vector-valued symplectic form. For special cases of M and G, we also obtain a description of the topology of MG(M), as well as an analytic expression for the symplectic volume.


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.