Casey Blacker




I am a postdoctoral researcher at the East China Normal University. 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. Polysymplectic Reduction and the Moduli Space of Flat Connections (submitted), 2018, arXiv:1810.04924
  2. First eigenvalue of the $p$-Lapacian on Kähler manifolds (with Shoo Seto), Proc. Amer. Math. Soc., 2018, arXiv:1804.10876
  3. The Moduli Space of Flat Connections over Higher Dimensional Manifolds, PhD Dissertation, 2018

Selected Talks


Differential Geometry and Classical Mechanics

NYU-Shanghai, Dec 11 2018.

Symplectic geometry arose in physics as the ideal setting for classical mechanics, and multisymplectic geometry has recently emerged as an analogous candidate in classical field theory. In this talk, I will introduce symplectic geometry according to the perspective that every symplectic manifold is locally the phase space of a classical mechanical system. The related theories of contact and polysymplectic geometry will also be discussed, along with relevant historical background. Working by analogy with the symplectic approach to classical mechanics, I will conclude with a brief introduction to the multisymplectic formalism in classical field theory.


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.