I am currently a visiting assistant professor at Purdue University, West Lafayette. This website is intended to give an overview of my research and share anything else that I am interested in through my blog.
Research:
My research is at the intersection of two different areas of math – complex geometry and partial differential equations (PDEs). Although the origins of complex geometry are a bit more obscure, the role of PDEs in math and physics is very clear. Many systems with some type of regularity, symmetry, or self-reinforcing behavior could be described by nonlinear PDEs. In geometry, the basic invariant quantities of surfaces such as curvature are described by tensors that are derivatives of the metric (distance function) on it. This means that it is possible (and a necessity) to study geometry through differential equations.
Differential Forms were introduced into the literature and widely adopted in the early 20th century, mostly because of the efforts of Hodge, Weyl, Chern, Lefschetz among others. In Hodge’s work differential forms became a necessity for integrating functions on a manifold, by providing a consistent orientation. It became clear that these could be studied independently, with an algebraic structure (exterior algebra) and differential operation (exterior derivative). It also became clear that some important topological and geometric information could be encoded through differential forms. Further progress in topology and geometry showed these two seemingly distinct fields originating from different axioms, have deep connections. These were expressed in the language of differential forms.
Monge-Ampere type PDEs of Differential Forms: My current research is exploring a nonlinear PDE theory for differential forms on complex manifolds. The latest two papers on this topic have achieved some success in this direction although much remains unknown.