- 2014 -
- Assistant Professor, Polytechnic School of Engineering and Courant Institute, NYU
- 2012 - 2014
- Courant Instructor, Courant Institute, NYU
- 2010 - 2012
- Associate Research Scientist, Courant Institute, NYU
- 2007 - 2010
- Quantitative researcher & assistant trader, Susquehanna International Group, LLP
- 2007
- Ph.D. Applied Mathematics, Yale University
- 2003
- A.B. Mathematics, Cornell University

Most of my research incorporates the development of fast
high-order analysis-based algorithms into problems in
computational physics, statistics, integral equations,
singular quadrature, and in general, computational
science. Almost all problems are rooted in relevant
real-world applications.

**Integral equations, computational physics, and fast
algorithms**

Almost all partial differential
equation occuring in classical mathematical physics can be
reformulated as integral equations with an appropriate
Green's function. Proper integral formulations are usually
very stable, but result in large dense systems which require
fast algorithms to solve. Analysis-based algorithms such as
fast multipole methods, butterfly algorithms, etc. allow
the rapid solution to these systems. I have recently been
working on particular problems in electromagnetics,
acoustics, and magnetohydrodynamics.

**Computational statistics**

Recently it
has been observed that many of the fast analysis-based
algorithms used throughout engineering physics have direct
applications in statistics, machine learning, and data
analysis. In particular, methods for rapidly inverting
similarly structured dense covariance matrices have
immediately found applications in Gaussian processes.

**Numerical analysis**

Complementary to
solving PDEs or integral equations, algorithms which stably
and rapidly compute special functions, invert matrices,
apply operators, etc. must be developed. These schemes fall
broadly under numerical analysis, and constitute
the components that go into necessary software toolboxes for
applied mathematics.

Also visit
the Courant
Mathematics and Computing Laboratory and
the Greengard
research group page for related research.

Siva Ambikasaran (NYU)

Alex Barnett (Dartmouth)

Antoine Cerfon (NYU)

Charlie Epstein (UPenn)

Zydrunas Gimbutas (NIST)

Leslie Greengard (NYU)

David W. Hogg (NYU)

Andreas Klöckner (UIUC)

Jon Wilkening (Berkeley)

- Fast symmetric factorization of hierarchical matrices with applications
- (with S. Ambikasaran), submitted.

[ arXiv:1405.0223 ] - Fast Direct Methods for Gaussian Processes and
the Analysis of NASA
*Kepler*Mission Data - (with S. Ambikasaran, D. Foreman-Mackey, L. Greengard,
and D. W. Hogg), submitted.

[ arXiv:1403.6015 ] - Debye Sources, Beltrami Fields, and a Complex Structure on Maxwell Fields
- (with C. L. Epstein and
L. Greengard), to appear,
*Comm. Pure Appl. Math.*.

[ pdf ] [ arXiv:1308.5425 ] - Exact axisymmetric Taylor states for shaped plasmas
- (with A. Cerfon),
*Phys. Plasmas*21, 064501, 2014.

[ pdf ] [ arXiv:1406.0481 ] - A generalized Debye source approach to electromagnetic scattering in layered media
*J. Math. Phys.*55, 012901, 2014.

[ pdf ] [ arXiv:1310.4241 ]- On the efficient representation of the impedance Green's function for the Helmholtz equation
- (with
L. Greengard and A. Pataki),
*Wave Motion*51(1):1-13, 2014.

[ pdf ] [ arXiv:1109.6708 ] - Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials
- (with A. Klöckner, A. Barnett, and L. Greengard),
*J. Comput. Phys.*252:332-349, 2013.

[ pdf ] [ arXiv:1207.4461 ] - A fast, high-order solver for the Grad-Shafranov equation
- (with A. Pataki, A. J. Cerfon, J. P. Freidberg, and
L. Greengard),
*J. Comput. Phys.*243:28-45, 2013.

[ pdf ] [ arXiv:1210.2113 ] - A consistency condition for the vector potential in multiply-connected domains
- (with C. L. Epstein,
Z. Gimbutas, L. Greengard, and A. Klöckner),
*IEEE Trans. Magn.*49(3):1072-1076, 2013.

[ pdf ] [ arXiv:1203.3993 ] - Debye sources and the numerical solution of the time harmonic Maxwell equations, II
- (with
C. L. Epstein and L. Greengard),
*Comm. Pure Appl. Math.*66(5):753-789, 2013.

[ pdf ] [ arXiv:1105.3217 ] - An algorithm for the rapid evaluation of special function transforms
- (with F. Woolfe and
V. Rokhlin),
*Appl. Comput. Harmon. Anal.*28(2):203-226, 2010.

[ pdf ] - Slow passage through resonance in Mathieu's equation
- (with L. Ng and
R. Rand),
*J. Vib. Control*9(6):685-707, 2003.

[ pdf ]

Below are slides from a few selected talks that I've given over the years.

The largest computational task encountered when modeling
using Gaussian processes is the inversion of a (dense)
covariance matrix. Often, these matrices have a systematic
structure that can be exploited. *george* is a Python
interface for a `C++` implementation of the HODLR
factorization. An optimized `Fortran` version is
currently in development.

george -
HODLR

Two-dimensional and three-dimensional fast multipole codes
developed by Leslie Greengard and Zydrunas Gimbutas for
Laplace, Helmholtz, elastostatic, and Maxwell potentials can
be downloaded on the CMCL webpage.

CMCL

Tentatively I will teach a course on integral equations at NYU-Poly Spring 2015.

Spring 2015

A projects mentoring course for students concentrating on a data science track
within the computational science masters program at the Courant Institute.

Fall 2013

Introductory linear algebra.

Fall 2012 & Spring 2012

This is the crowning project course for students enrolled
in the Data Science masters program at NYU through the
Center for Data Science. Working with industry and/or
faculty mentors, students complete and present a thorough
treatment of a real-world data science problem.

Fall 2014

This course is a junior/senior level introduction to the mathematical
theory of statistics to be taken after a similarly focused course on the
theory of probability has been taken.

Spring 2014 & Spring 2013