"I applied to the Scholars program so that I could focus on my research without having to work an extra job. My goals are to continue to focus on research, and to hopefully publish some new results."
Differential geometry, topology, and relations to physics.
Academic and Other Awards
- University Scholars Program Scholarship (2011-2012)
- Barry M. Goldwater Scholarship (2011)
- W.W. Massey Sr. Presidential Scholarship (2011)
- Kermit Sigmon Scholarship (2010)
- Research Experience for Undergraduates (REU) (2010)
- Pi Mu Epsilon
- Society for Physics Students
- HEP Graduate Student Seminar
- Topology Graduate Student Seminar
- Intramural Sports
Hobbies and Interests
- Basketball, tennis, and golf.
The Seiberg-Witten Equations on 4-Manifolds
The Seiberg-Witten equations are a system of equations arise from a certain gauge theory over a smooth 4-manifold. Gauge theory is the study of vector bundles and connections on them. In physics, one can realize electrodynamics and Maxwell's equations as an abelian gauge theory. Abelian in the sense that its structure group, the circle, is commutative. The generalization of Maxwell's theory to a non-abelian structure group is known as Yang-Mills theory. The study of the solutions to the resulting equations have had great mathematical impacts in the classification of both smooth and topological 4-manifolds. The gauge theory associated to the Seiberg-Witten equations does not come from a straight generalization of electrodynamics, but came up through the study of a N=4 supersymmetric quantum field theory. The gauge theory has been completely formalized mathematically, in terms of spin manifolds, and the analysis of the solutions to the these equations has reproduced and pushed forward the study of 4-manifolds; with the further advantage that the equations are based on an abelian circle structure group. My goal is to study the invariants that these equations define and what these invariants say about the underlying 4-manifold. The study of these invariants have implications in symplectic and complex geometry as well as more applied areas such as quantum field theory of condensed matter systems.