Overview: My research is concerned with the development of efficient algorithms for scientific computing, primarily methods for solving differential and integral equations, and for analyzing very large matrices and datasets. My interests are broad, but there are several recurring themes:

• Methods based on randomized projections for overcoming the computational challenges associated with problems set in high dimensional spaces. Applications to data analysis, computational statistics, geometry of data-sets in high dimensional spaces, etc.
• PDE solvers that draw on the full arsenal of techniques provided by classical mathematical physics and harmonic analysis.
• The construction of direct (as opposed to iterative) solvers for elliptic PDEs. These solvers directly construct an approximation to the relevant solution operator such as, e.g., a Green's function, an evolution operator, or a Dirichlet-to-Neumann operator.
• Design of computational algorithms that are engineered from the ground up to minimize communication. This is essential for performance in modern multi-core and parallel computing environments.

Resources on randomized methods in linear algebra: If you would be interested to learn more about randomized methods for matrix computations, then let me encourage you to read our 2011 survey. (The survey is very long, but reading the 7 pages in Section 1 would provide a birds eye view of the subject.) There is also a more recent tutorial paper that I prepared for a summer school in 2016. This later monograph has a more informal style, and is focused on practical computing rather than on theory. There is additional material at the webpage for the summer school lectures I prepared.

Resources on Fast Direct Solvers for elliptic PDEs: For a brief introduction, let me point to some slides I prepared for SciCADE in Bath in 2017. A student or junior researcher who wants a more in depth take on this subject may want to look into material I prepared for a summer school in 2014 (a "CBMS conference"), including 10 video taped lectures available on Youtube.