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

2021/2022
Math 2X03
Math 3X03
Math 722

2020/2021
Math 2X03
Math 3X03
Math 761

2019/2020
Math 1XX3
Math 3B03
Math 722

2018/2019
Math 2X03
Math 3B03
Math 4BT3
Math 762

2017/2018
On Research Leave

2016/2017
Math 2X03
Math 3B03
Math 766

2015/2016
Math 2X03
Math 3B03
Math 4B03/6B03

2014/2015
Math 2X03
Math 2XX3
Math 4BT3
Math 762

2013/2014
Math 2X03 
Math 2XX3 
Math 766