Tuesday, October 15 at 4:00pm to 5:00pm
Kaprielian Hall (KAP), 410
3620 South Vermont Avenue, Los Angeles, CA 90089
Kelley Yang, USC
Abstract: In mathematical models used in sciences, a homogeneous space of a Lie group typically describes a certain system with a group of transformations. Natural structures in such a system are invariant under these transformations; therefore, the study of invariant geometrical objects on Lie groups and homogeneous spaces is indispensable in all sciences.
In this project we work with the case of almost Abelian groups. We first employ general Lie theory methods to explicitly find the left and right Haar measures, as well as the modular function, of a given such group. We proceed by studying an almost Abelian Lie group G as a
manifold with a left and right G-action in order to explicitly describe left and right invariant vector and tensor fields, including Riemannian metrics and symplectic forms.
Since every almost Abelian group with an invariant Riemannian metric gives rise to a natural completely integrable Hamiltonian system, an explicit description of such invariant structures as what we have developed will lead to closed form solutions for this new class of non-compact integrable systems. It also opens doors for the study of invariant partial differential equations on these groups.