PhD Public Lecture - Prakash Singh (Math)

Title: The Hofer geometry of Maximal Tori in Hamiltonian Diffeomorphism

Abstract: We can associate to every closed symplectic manifold (M,w), a group of Hamiltonian diffeomorphisms, Ham(M,w).  This group is infinite-dimensional and yet, remarkably, admits a bi-invariant metric, called the Hofer metric. Moreover, for symplectic 4-manifolds, this group contains only finitely many conjugacy classes of maximal tori with respect to the action of the full symplectomorphism group. These observations lead us to believe that (M,w) is, in some sense, much closer to a Lie Group than other wild diffeomorphism groups.

We will be interested in studying some geometric aspects of the Hofer metric on the group Ham(M,w). More specifically, we will investigate how the centralizer C(T) associated to a toric action sits inside Ham(M,w), and how this compares to the case of maximal tori in Lie groups, which are always flat and totally geodesic submanifolds with respect to any bi-invariant metric. 

Another interesting question is the diameter of the full group Ham(M,w) with respect to the Hofer metric. It has been conjectured that the Hofer diameter of Ham(M,w) is infinite for every closed symplectic manifold.This conjecture is mostly open at present and seems out of reach with the currently available methods. We ask whether the centralizer C(T) has infinite extrinsic diameter.

We shall see two approaches to this problem and conclude by discussing the limitations of those methods.