Geometry of Grushin Spaces
Michael Albert, UConn
Abstract: Much work has been done recently to try to understand broadly how to do analysis on non-Euclidean geometries. In particular, this focus has yielded many results in the realm of stochastic processes. One interesting class of non-Euclidean spaces are the almost-Riemannian manifolds, which come from a Riemannian structure that degenerates along a "singular set". The canonical example of almost-Riemannian manifold is the Grushin plane, whose geometry arises from requiring that traversal across the y axis occurs with a horizontal tangent vector. We consider a higher dimensional generalization of the Grushin plane and explore the optimality of geodesics. These spaces are modeled on $\mathbb{R}^n$, so no extensive background in differential geometry will be required for this talk.
