Casey Blacker




I am a China Postdoctoral Science Foundation (CPSF) International Exchange Postdoc at the East China Normal University in Shanghai. I received my PhD from UC Santa Barbara in 2018 under the supervision of Xianzhe Dai.

My interests include symplectic geometry, moduli spaces, and geometric quantization. You can find more information in my Research Statement.


Publications

  1. Reduction of multisymplectic manifolds, (Submitted)
    http://arxiv.org/abs/2002.10062
  2. Quantization of polysymplectic manifolds, J. Geom. Phys., 2019,
    https://doi.org/10.1016/j.geomphys.2019.103480
  3. Polysymplectic reduction and the moduli space of flat connections, J. Phys. A, 2019,
    https://doi.org/10.1088/1751-8121/ab2eed
  4. First eigenvalue of the p-Lapacian on Kähler manifolds (with Shoo Seto), Proc. Amer. Math. Soc., 2019, https://doi.org/10.1090/proc/14395
  5. The Moduli Space of Flat Connections over Higher Dimensional Manifolds (PhD Thesis), 2018, https://escholarship.org/uc/item/0535z0rb

Selected Talks


Polysymplectic Reduction and the Moduli Space of Flat Connections

University of Tokyo, Dec 3, 2019.

In a landmark paper, Atiyah and Bott showed that the moduli space of flat connections on a principal bundle over an oriented closed surface is the symplectic reduction of the space of all connections by the action of the gauge group. By appealing to polysymplectic geometry, a generalization of symplectic geometry in which the symplectic form takes values in a fixed vector space, we may extend this result to the case of higher-dimensional base manifolds. In this setting, the space of connections exhibits a natural polysymplectic structure and the reduction by the action of the gauge group yields the moduli space of flat connections equipped with a 2-form taking values in the cohomology of the base manifold. In this talk, I will first review the polysymplectic formalism and then outline its role in obtaining the moduli space of flat connections.


Quantization of Polysymplectic Manifolds

University of Cologne, July 7, 2019.

Geometric quantization is a method for taking a symplectic manifold and returning a complex Hilbert space. A polysymplectic manifold is a smooth manifold equipped with a symplectic structure taking values in a fixed vector space. Both geometric quantization and polysymplectic geometry have their roots in physics, and have each engendered a rich mathematical literature. In this talk, I will review both formalisms independently and then introduce an extension of geometric quantization to the setting of polysymplectic manifolds. No familiarity with geometric quantization or polysymplectic geometry will be assumed.


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.