Senior Honors Thesis: "The Complex Geometry and Representation Theory of Statistical Transformation Models"

Abstract: Given a measure space X, we can construct a number of induced structures: eg. its L2 space, the space P(X) of probability distributions on X. If, in addition, X admits a transitive measure-preserving Lie group action, natural actions are induced on those structures. We expect relationships between these induced structures and actions. We study, in particular, the relations between L2(X) and exponential transformation models on X, which are special “submanifolds” of P(X) closed under the induced action, whose tangent bundles are Kähler manifolds (given by Molitor). Geometrically, we show the tangent bundle has, locally, the “same” Kähler metric with the Fubini-Study metric on the projectivization of L2(X). Moreover we show the action on the tangent bundle is equivariant with that on L2(X), which is a unitary representation. Finally, in some cases, when the symplectic action on the tangent bundle is Hamiltonian, we show that any coadjoint orbit in the image of its moment map induces, via Kirillov’s correspondence from orbit method, irreducible unitary representations that are sub-representations of the aforementioned representation in L2(X). We expect further results of this type to illuminate the relations between geometry, representation theory, physics, and more.

Host: Renato Feres