I work in the broadly defined area of Analysis and its applications. My research follows mainly two interrelated directions. The first direction is more theoretical and it includes the following areas:
Most of my work combines the two general directions mentioned above. The final goal is to obtain concrete, practical applications. This turns out to require not just the classical areas associated to applications, but also more theoretical areas. The reason for this is that the the equations that arise in practice exhibit more structure than the textbook equations. The additional structure presents both challenges and opportunities. It presents challenges in that the classical methods do not work as well as desired. It presents opportunities since one is required to develop new methods, involving more theoretical mathematics.
An example where all the above research areas are useful is given by the problem of finding "quasi-optimal rates of convergence for the Finite Element Method." Another example is provided by the study of the Partial Differential Equations that arise in Option Pricing (stochastic volatility models, for instance). Yet another example is provided by the study of Schrodinger operators, whose very specific singularities in the potential provide a very rich structure that can be exploited using geometric analysis. Several other types of structures studied in, for example, Non-commutative geometry, turn out to appear also in practical applications (families in Uncertainty quantification, group actions in reducing the dimension to simplify numerical methods).