About John V. Pardon's Work
Pardon explores problems in geometric topology and related fields, including symplectic geometry. A longstanding problem known as the Hilbert-Smith conjecture asks whether compact groups of topological symmetries of manifolds (higher dimensional generalizations of points, circles, and surfaces) must be “continuous” (such as the group of rotations of a circle or of a sphere), or whether it is possible to have more exotic “totally disconnected” symmetry groups (these are harder to visualize–indeed, it is expected they do not exist!). He used tools from geometric topology to show that these unexpected exotic symmetries do not exist in dimensions three and lower. Another one of his current projects concerns developing a robust “virtual” (or “algebraic”) intersection theory in certain infinite-dimensional spaces. Such a theory is of fundamental importance in symplectic geometry, where it is used to enumerate pseudo-holomorphic curves in symplectic manifolds.