Nonlinear classification of Banach spaces

We study the geometric classification of Banach spaces via Lipschitz, uniformly continuous, and coarse mappings. We prove that a Banach space which is uniformly homeomorphic to a linear quotient of lp is itself a linear quotient of lp when p<2. We show that a Banach space which is Lipschitz universal for all separable metric spaces cannot be asymptotically uniformly convex. Next we consider coarse embedding maps as defined by Gromov, and show that lp cannot coarsely embed into a Hilbert space when p> 2. We then build upon the method of this proof to show that a quasi-Banach space coarsely embeds into a Hilbert space if and only if it is isomorphic to a subspace of L0(??) for some probability space (Ω,B,??).