My research is in differential geometry. My dissertation advisor was Professor Guofang Wei at the University of California, Santa Barbara.
In general, I am interested in the relationship between geometry and topology. In particular, I want to know how do local geometric quantities (like curvature!) shape the global topological structure of the manifold? There are many classical theroems addressing this very question. See, for example, the Bonnet-Myers Theorem or Cartan-Hadamard Theorem.
I am currently investigating the fundamental group of smooth metric measure spaces (complete Riemannan manifolds endowed with a weighted measure). The Bakry-Emery Ricci tensor is a natural analogue to Ricci curvature on such spaces. Since Ricci curvature bounded from below gives us information about Riemannian manifolds, we can ask if the Bakry-Emery Ricci tensor bounded from below gives us similar information for smooth metric measure spaces. For more on my current work, please see my paper on the arXiv: Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded Below.
email: maree (dot) jaramillo (at) uconn (dot) edu
office: Hartford Times Building 302D