Casey Blacker



I am an EIMI International Postdoc at Saint Petersburg State University. Prior to this, I was 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, Lett. Math. Phys., 2021,
    https://doi.org/10.1007/s11005-021-01408-y
  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


Quantization of polysymplectic manifolds

Young Researchers' Virtual Multisymplectic Geometry Conference, 13 July 2021.

Geometric quantization is a method for taking a symplectic manifold and returning a complex Hilbert space. If the symplectic manifold is equipped with the structure of a Hamiltonian G-space, then the Hilbert space inherits a unitary representation of G. A polysymplectic manifold is a smooth manifold equipped with a symplectic structure taking values in a fixed vector space. In this talk, I will briefly review geometric prequantization and introduce an extension to the setting of polysymplectic manifolds.


Reduction of multisymplectic manifolds

Good Morning SFARS, 7 June 2021.

A multisymplectic structure on a smooth manifold is a closed and nondegenerate differential form of arbitrary degree. In this brief presentation, we first review the Marsdeni–Weinstein–Meyer reduction theorem in the original symplectic setting, and then show how this result extends to multisymplectic manifolds.


Polysymplectic reduction and the moduli space of flat connections

University of Tokyo, 3 Dec 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.