**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