Research Interests

My main current goal is to develop general mathematical techniques and to find concrete, real life applications of mathematics (all kinds of mathematics).

More concretely, my general research interests are centered around

• Analysis and Partial Differential Equations on singular and non-compact spaces,
• Numerical Methods for Partial Differential Equations,
• Operator Algebras and Noncommutative Geometry
• Mathematical Physics,
• Geometric Analysis, and
• Financial Mathematics.

Even more specifically, I am currently working on "quasi-optimal rates of convergence for the Finite Element Method," the "Generalized Finite Element Method," and numerical methods for evolution equations with applications to Option Pricing (stochastic volatility models). I am using these results for "eigenvalue estimates for Schrodinger operators" and other applications. I am also planning some applications to Elastography.

Past projects, I also worked on "Pseudodifferential operators on singular spaces," on "Operator algebras," and "Non-commutative Geometry." I often use ideas from these areas in my current research.

See Preprints and Publications for latest work and publications and reprints for older work.