Research Area: Geometry & Topology
Research Profile: Differential geometry, geometric analysis
My research is about prescribing the curvature of manifolds, which could mean requiring curvature to be positive, zero, or negative in some suitable sense, or requiring some natural differential equation involving curvature to hold on the manifold.
One class of manifolds which I study is motivated from physics. According to Einstein's theory of relativity, gravity exhibits itself mathematically in the bending of space-time. The famous Einstein field equation relates the Ricci curvature, which is the relevant notion of curvature or "bending" in relativity, to the matter fields present. Can Einstein's equation hold on any manifold, or is the topology of space-time restricted, or determined by it ? Is the geometry of relativity also compatible with the elementary particles we observe ? Are there continuous families of solutions of Einstein's equation ? What kind (range) of geometric properties are exhibited by Einstein manifolds? The analogs of these questions in all dimensions and for manifolds with positive definite inner products provide the driving force behind some of my research projects.
Differential geometry, geometric analysis
M2X03 - Fall Term, 2013
M2XX3 - Winter Term, 2014
M766 - Winter Term, 2014