Melvin Leok

Associate Professor of Mathematics
University of California, San Diego

Department of Mathematics
University of California, San Diego
9500 Gilman Drive, Dept. 0112
La Jolla, CA 92093-0112, USA.

Office:
Email:
Phone:
Fax:

APM 5763
mleok@math.ucsd.edu
+1(858)534-2126
+1(858)534-5273

Teaching
Research
Honors
Thesis
Publications
Presentations
Service
Education
Conferences

Announcements

Promotion to (Full) Professor, Step I
My promotion to (Full) Professor, Step I, with a half-step bonus, has been approved and will take effect July 1, 2013. The half-step bonus is in recognition of my extraordinary research productivity during the review period.
Kavli Fellow, National Academy of Sciences
I have been invited in participate in the German-American Kavli Frontiers of Science symposium, which is jointly organized by the Alexander von Humboldt Foundation and the National Academy of Sciences. The symposium bring together outstanding young scientists to discuss exciting advances and opportunities in a broad range of disciplines. A committee of NAS members selects the participants from among young researchers who have already made recognized contributions to science, including recipients of major fellowships and awards. [ Poster ]
Joining the Editorial Board of the SIAM Journal on Control and Optimization
I have accepted an invitation to serve on the editorial board of the SIAM Journal on Control and Optimization, for a 3 year term, from January 2012 to December 2014.
Joining the Editorial Board of the LMS Journal of Computation and Mathematics
I have accepted an invitation to serve on the editorial board of the London Mathematical Society's Journal of Computation and Mathematics, and will be replacing Prof. Sebastian Reich in covering the area of mathematical and computational modelling starting December 2011. This is an electronic, open access journal, established in 1998.
FRG Mathematical and Numerical Relativity Seminar
Michael Holst and I, together with Don Estep (Colorado State) and Lee Lindblom (Caltech), recently received a NSF FRG grant to study error quantification and control for gravitational wave simulations. To facilitate interaction between participants, we will be holding a Mathematical and Numerical General Relativity Seminar this quarter.
New Journal: Journal of Computational Dynamics
I am on the editorial board of a new AIMS journal, the Journal of Computational Dynamics, which is focused on the intersection of computation with deterministic and stochastic dynamics. In particular, this entails new computational methods for analyzing dynamical systems, or novel applications of dynamical systems techniques to improve computation.
Accelerated Merit Advancement to Associate Professor, Step III
I have received a two-year accelerated merit advancement from Associate Professor, Step I, to Associate Professor, Step III, which will take effect July 1, 2011. Accelerations are typically awarded on the basis of unusual achievement and exceptional promise of continued growth.
In Memory of Jerrold E. Marsden (1942-2010)
On September 21, 2010, my dear advisor, mentor, role model, collaborator, colleague, and friend, Jerry Marsden, lost his battle with cancer. He will be missed and remembered always. [ ICIAM Memorial Events | SIAM News Obituary ]
Computational Geometric Mechanics at UCSD
Melvin Leok is an associate professor of mathematics at the University of California, San Diego, and directs the computational geometric mechanics group, which is affiliated with the Center for Computational Mathematics, the Program in Computational Science, Mathematics, and Engineering, and the Cymer Center for Control Systems and Dynamics.
Research Vignette on Streaming Video
Prior to joining UCSD, I was an assistant professor at Purdue University, and the department of mathematics there commissioned a short video vignette where I describe my research in a broadly accessible fashion. This is available as a streaming video.
NSF CAREER Award DMS-0747659
I received a National Science Foundation Faculty Early Career Development (CAREER) Award in support of my work on Computational Geometric Mechanics: Foundations, Computation, and Applications. This is funded by the Applied Mathematics Program of the Division of Mathematical Sciences. [ Purdue University News Service Release | Inside Purdue, Pages 8-9 | Notices of the AMS, Pages 269-270 ]
SciCADE New Talent Prize
I was awarded the SciCADE New Talent Prize, in recognition of my work on Lie group variational integrators and their applications to optimal control, and gave a plenary lecture at the SciCADE 2007 International Conference on Scientific Computation and Differential Equations, in Saint-Malo, France. This prize recognizes the best paper submitted by a candidate under the age of 35, within 4 years of the Ph.D.

Research Opportunities

Graduate Positions Available in the Computational Geometric Mechanics group
Research positions for graduate students are available in the broad area of geometric numerical methods in geometric mechanics and control. A description of my current research may be found in my research statement, bibliography, collected publications, or the abstracts below. These positions are funded by the following research grants from the National Science Foundation in applied and computational mathematics:
- DMS-1065972, FRG: Error Quantification and Control for Gravitational Waveform Simulation, 2011-2014.
- CMMI-1029445, Computational Geometric Uncertainty Propagation for Hamiltonian Systems on a Lie Group, 2010-2013.
- DMS-1010687, CAREER: Computational Geometric Mechanics: Foundations, Computation, and Applications, 2008-2013 (transferred from DMS-0747659).
- DMS-1001521, LTB: Generalized Variational Integrators for Large-Scale Scientific Computation, 2007-2011 (transferred from DMS-0714223).
- DMS-0726263, Computational Geometric Mechanics and its Applications to Geometric Control Theory, 2005-2009 (transferred from DMS-0504747).
Please contact me by email if you are interested in any of these positions. These are full-time positions, and students who are interested in internship opportunities should read my policy on internships.
Academic and Industrial, Science and Engineering Collaborations Welcomed
I am constantly on the lookout for interesting and challenging collaborations with scientists and engineers from academia or industry. If you have an interesting application for which geometric numerical methods might be relevant, please feel free to contact me.
Much of my theoretical work is motivated by specific applications. To paraphrase William Morris, "Have nothing in your [mathematics] that you do not know to be useful, or believe to be beautiful."

Teaching

Spring 2013, MATH 170C, Introduction to Numerical Analysis: Ordinary Differential Equations. [ Course Website ]
Spring 2013, MATH 270C, Numerical Ordinary Differential Equations. [ Course Website ]
Previous courses

Research Interests

Computational Geometric Mechanics: Applying discrete differential geometry and discrete Lagrangian and Hamiltonian mechanics to the construction of geometric structure-preserving numerical algorithms.
Computational Geometric Control Theory: Construction of real-time digital feedback control of mechanical systems using techniques from geometric control theory and computational geometric mechanics.
Numerical Analysis: Derivation of accurate and efficient numerical schemes with good long-time geometric stability properties by combining geometric integration with adaptive, spectral, and multiscale techniques.

Selected Honors

National Science Foundation Faculty Early Career Development (CAREER) Award, DMS-0747659, 2008-2013.
SciCADE New Talent Award, International Conference on Scientific Computation and Differential Equations, 2007.
National Science Foundation Computational Mathematics LTB Research Grant, DMS-0714223, 2007-2010.
National Science Foundation Applied Mathematics Research Grant, DMS-0504747 (transferred to DMS-0726263), 2005-2008.
Margaret and Herman Sokol Spring/Summer Research Grant, University of Michigan, Ann Arbor, MI, 2006.
International Congress of Mathematicians Travel Grant, NSF DMS-0514413, American Mathematical Society, 2006.
Horace H. Rackham Faculty Fellowship and Grant, University of Michigan, Ann Arbor, MI, 2004-2005.
SIAM Student Paper Prize, Society for Industrial and Applied Mathematics, 2003.
Leslie Fox Prize in Numerical Analysis (second prize), Institute of Mathematics and its Applications, UK, 2003.
Josephine de Kármán Fellowship (established by Theodore von Kármán), 2003-2004.
International Fellowship, Agency for Science, Technology, and Research, Singapore, 2002-2004.
Poincaré Fellowship (Betty and Gordon Moore Fellowship), Caltech, 2000-2004.
Tau Beta Pi Fellowship, Tau Beta Pi, National Engineering Honor Society, 2000-2001.
Tan Kah Kee Foundation Postgraduate Scholarship, Singapore, 2000.

Editorial Boards

Journal of Nonlinear Science (Springer).
Journal of Geometric Mechanics (American Institute of Mathematical Sciences).
Journal of Computational Dynamics (American Institute of Mathematical Sciences).
Journal of Computation and Mathematics (London Mathematical Society).
Journal on Control and Optimization (SIAM).

Thesis

Foundations of Computational Geometric Mechanics, Control and Dynamical Systems, California Institute of Technology, 2004.
A preliminary version of this thesis received the SIAM Student Paper Prize, and the Leslie Fox Prize in Numerical Analysis (second prize).
[ Abstract | PDF | Order a bound copy ]

Publications and Preprints (search the ArXiv, AMS MathSciNet) [ BibTeX ]

Discrete Poincaré Lemma (with M. Desbrun, and J.E. Marsden), Appl. Numer. Math. 53 (2-4), 231-248, 2005.
[ PDF | Journal Link | BibTeX ]
A Lie Group Variational Integrator for the Attitude Dynamics of a Rigid Body with Applications to the 3D Pendulum (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Control Applications, 962-967, 2005.
[ PDF | IEEE Xplore | BibTeX | Animation: Chaotic Motion ]
Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Proc. IEEE Conf. on Decision and Control, 6579-6584, 2005.
[ PDF | IEEE Xplore | BibTeX ]
Discrete Routh Reduction (with S.M. Jalnapurkar, J.E. Marsden, and M. West), J. Phys. A: Math. Gen. 39, 5521-5544 (Geometric Integration special issue, invited paper), 2006.
[ PDF | arXiv:math.NA/0508330 | Journal Link | BibTeX ]
Attitude Maneuvers of a Rigid Spacecraft in a Circular Orbit (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 1742-1747, 2006.
[ PDF | arXiv:math.OC/0509299 | IEEE Xplore | BibTeX | Animation: Fig2(a), Fig2(b), Fig2(c), Fig3(a) ]
Polyhedral Potential and Lie Group Variational Integrator Computations for the Full Two Body Problem (with E. Fahnestock, T. Lee, N.H. McClamroch, D.J. Scheeres), Proc. AIAA/AAS Astrodynamics Specialist Conf., AIAA-2006-6289, 2006.
[ PDF | arXiv:math.NA/0608695 | Journal Link | BibTeX ]
Optimal Control of a Rigid Body using Geometrically Exact Computations on SE(3) (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 2710-2715, 2006.
[ PDF | arXiv:math.OC/0602588 | IEEE Xplore | BibTeX | Animation: Fig1(a), Fig2(a), Fig3(a), Fig4(a) ]
Deterministic Global Attitude Estimation (with T. Lee, A.K. Sanyal, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 3174-3179, 2006.
[ PDF | arXiv:math.OC/0602589 | IEEE Xplore | BibTeX ]
Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Proc. IEEE Conf. on Decision and Control, 3333-3338, 2006.
[ PDF | arXiv:math.OC/0602590 | IEEE Xplore | BibTeX ]
A Discrete Variational Integrator for Optimal Control Problems in SO(3) (with A.M. Bloch, I.I. Hussein, and A.K. Sanyal), Proc. IEEE Conf. on Decision and Control, 6636-6641, 2006.
[ PDF | arXiv:math.OC/0509536 | IEEE Xplore | BibTeX ]
Global Attitude Estimation using Single Direction Measurements (with T. Lee, N.H. McClamroch, and A.K. Sanyal), Proc. American Control Conf., 3659-3664, 2007.
[ PDF | arXiv:math.OC/0609481 | IEEE Xplore | BibTeX ]
Optimal Attitude Control for a Rigid Body with Symmetry (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 1073-1078, 2007.
[ PDF | arXiv:math.OC/0609482 | IEEE Xplore | BibTeX | Animation: Fig2(a), Fig3(a), Fig4(a), Fig5(a) ]
Lie Group Variational Integrators for the Full Body Problem, (with T. Lee, and N.H. McClamroch), Comput. Methods Appl. Mech. Engrg. 196(29-30), 2907-2924, 2007.
[ PDF | arXiv:math.NA/0508365 | Journal Link | BibTeX | Animation: Fig4 ]
Lie Group Variational Integrators for the Full Body Problem in Orbital Mechanics (with T. Lee, and N.H. McClamroch), Celestial Mechanics and Dynamical Astronomy 98(2), 121-144, 2007.
[ PDF | Journal Link | BibTeX | Animation: Fig2 ]
Propagation of Uncertainty Sets for Rigid Body Attitude Flows (with N.A. Chaturvedi, T. Lee, N.H. McClamroch, and A.K. Sanyal), Proc. IEEE Conf. on Decision and Control, 2689-2694, 2007.
[ PDF | arXiv:math.DS/0702737 | IEEE Xplore | BibTeX ]
A Combinatorial Optimal Control Problem for Spacecraft Formation Reconfiguration (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 5370-5375, 2007.
[ PDF | arXiv:math.OC/0702738 | IEEE Xplore | BibTeX | Animation: Fig2 ]
Global Optimal Attitude Estimation using Uncertainty Ellipsoids (with A.K. Sanyal, T. Lee, and N.H. McClamroch), Systems and Control Letters, 57(3), 236-245, 2008.
[ PDF | arXiv:math.OC/0606083 | Journal Link | BibTeX ]
Time Optimal Attitude Control for a Rigid Body (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 5210-5215, 2008.
[ PDF | arXiv:0709.2514 | IEEE Xplore | BibTeX | Animation: Fig1(a), Fig2(a) ]
Matching and stabilization of discrete mechanical systems (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Proc. Appl. Math. Mech., 7, 1030603-1030604, 2007.
[ PDF | Journal Link | BibTeX ]
Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on SO(3) (with T. Lee, and N.H. McClamroch), Journal of Dynamical and Control Systems, 14(4), 465-487, 2008.
[ PDF | arXiv:math.OC/0601424 | Journal Link | BibTeX | Animation: Fig2(a), Fig3(a), Fig4(a), Fig5(a) ]
Global Symplectic Uncertainty Propagation on SO(3) (with T. Lee, and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 61-66, 2008.
[ PDF | arXiv:0803.1515 | IEEE Xplore | BibTeX ]
Discrete Control Systems (with T. Lee, and N.H. McClamroch), invited article for Springer Encyclopedia of Complexity and System Science, 2002-2019, 2009.
[ PDF | arXiv:0705.3868 | Journal Link | BibTeX ]
Geometric Structure-Preserving Optimal Control of the Rigid Body (with A.M. Bloch, I.I. Hussein, and A.K. Sanyal), Journal of Dynamical and Control Systems, 15(3), 307-330, 2009.
[ PDF | arXiv:0712.4400 | Journal Link | BibTeX ]
Lagrangian Mechanics and Variational Integrators on Two-Spheres (with T. Lee, and N.H. McClamroch), International Journal for Numerical Methods in Engineering, 79(9), 1147-1174, 2009.
[ PDF | arXiv:0707.0022 | Journal Link | BibTeX | Animation: Fig3.1, Fig3.2, Fig3.3, Fig3.4, Fig3.5, Fig3.6, Fig3.7 ]
Computational Geometric Optimal Control of Rigid Bodies (with T. Lee, and N.H. McClamroch), Brockett Legacy Special Issue, Communications in Information and Systems, 8(4), 445-472, 2008.
[ PDF | arXiv:0805.0639 | Journal Link | BibTeX ]
Dynamics of Connected Rigid Bodies in a Perfect Fluid (with T. Lee, and N.H. McClamroch), Proc. American Control Conf., 408-413, 2009.
[ PDF | arXiv:0809.1488 | IEEE Xplore | BibTeX | Animation: Fig1 ]
Controlled Lagrangians and Stabilization of Discrete Mechanical Systems (with A.M. Bloch, J.E. Marsden, and D.V. Zenkov), Discrete and Continuous Dynamical Systems - Series S, 3(1), 19-36, 2010.
[ PDF | arXiv:0704.3875 | Journal Link | BibTeX ]
Nonlinear Dynamics of the 3D Pendulum (with N.A. Chaturvedi, T. Lee, and N.H. McClamroch), Journal of Nonlinear Science, 21(1), 3-32, 2011.
[ PDF | arXiv:0707.1196 | Journal Link | BibTeX ]
Variational and Geometric Structures of Discrete Dirac Mechanics (with T. Ohsawa), Foundations of Computational Mathematics, 11(5), 529-562, 2011.
[ PDF | arXiv:0810.0740 | Journal Link | BibTeX ]
Dynamics of a 3D Elastic String Pendulum (with T. Lee and N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 3347-3352, 2009.
[ PDF | arXiv:0903.0332 | IEEE Xplore | BibTeX | Animation: Fig2 ]
Computational Dynamics of a 3D Elastic String Pendulum Attached to a Rigid Body and an Inertially Fixed Reel Mechanism (with T. Lee, N.H. McClamroch), Nonlinear Dynamics, 64(1-2), 97-115, 2011.
[ PDF | arXiv:0909.2083 | Journal Link | BibTeX | Animation: Fig2(a), Fig3(a), Fig4(a) ]
Computational Geometric Optimal Control of Connected Rigid Bodies in a Perfect Fluid (with T. Lee, N.H. McClamroch), Proc. American Control Conf., 5985-5990, 2010.
[ PDF | arXiv:0909.3871 | BibTeX | Animation: Fig2(a), Fig3(a) ]
Discrete Hamilton-Jacobi Theory (with A.M. Bloch, T. Ohsawa), SIAM Journal on Control and Optimization, 49(4), 1829-1856, 2011.
[ PDF | arXiv:0911.2258 | Journal Link | BibTeX ]
Discrete Hamiltonian Variational Integrators (with J. Zhang), IMA Journal of Numerical Analysis, 31(4), 1497-1532, 2011.
[ PDF | arXiv:1001.1408 | Journal Link | BibTeX ]
Geometric Tracking Control of a Quadrotor UAV on SE(3) (with T. Lee, N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 5420-5425, 2010.
[ PDF | arXiv:1003.2005 | IEEE Xplore | BibTeX ]
Discrete Hamilton-Jacobi Theory and Discrete Optimal Control (with T. Ohsawa, A.M. Bloch), Proc. IEEE Conf. on Decision and Control, 5438-5443, 2010.
[ PDF | IEEE Xplore | BibTeX ]
Stokes-Dirac Structures through Reduction of Infinite-Dimensional Dirac Structures (with J. Vankerschaver, H. Yoshimura, J.E. Marsden), Proc. IEEE Conf. on Decision and Control, 6265-6270, 2010.
[ PDF | arXiv:1010.2547 | IEEE Xplore | BibTeX ]
Discrete Dirac Structures and Implicit Discrete Lagrangian and Hamiltonian Systems (with T. Ohsawa), XVIII International Fall Workshop on Geometry and Physics, 91-102, AIP Conference Proceedings 1260, 2010.
[ PDF | Journal Link | BibTeX ]
Geometric Numerical Integration of Complex Dynamics of Tethered Spacecraft (with T. Lee, N. H. McClamroch), Proc. American Control Conf., 1885-1891, 2011.
[ PDF | arXiv:1010.1724 | BibTeX ]
Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3) (with T. Lee, N. H. McClamroch), Asian Journal of Control, 15(3), 1-18, 2013,
[ PDF | Journal Link | BibTeX ]
Prolongation-Collocation Variational Integrators (with T. Shingel), IMA Journal of Numerical Analysis, 32(3), 1194-1216, 2012.
[ PDF | arXiv:1101.1995 | Journal Link | BibTeX ]
On the Geometry of Multi-Dirac Structures and Gerstenhaber Algebras (with J. Vankerschaver, H. Yoshimura), Journal of Geometry and Physics, 61(8), 1415-1425, 2011. 2011.
[ PDF | arXiv:1102.2835 | Journal Link | BibTeX ]
General Techniques for Constructing Variational Integrators (with T. Shingel), Frontiers of Mathematics in China (Special issue on computational mathematics, invited paper), 7(2), 273-303, 2012.
[ PDF | arXiv:1102.2685 | Journal Link | BibTeX ]
Stable Manifolds of Saddle Points for Pendulum Dynamics on S^2 and SO(3) (with T. Lee, N.H. McClamroch), Proc. IEEE Conf. on Decision and Control, 3915-3921, 2011.
[ PDF | arXiv:1103.2822 | BibTeX ]
Variational Integrators, invited article for Springer Encyclopedia of Applied and Computational Mathematics, in press, 2012.
[ PDF | BibTeX ]
The Hamilton-Pontryagin Principle and Multi-Dirac Structures for Classical Field Theories (with J. Vankerschaver, H. Yoshimura), Journal of Mathematical Physics, 53(7), 072903 (25 pages), 2012.
[ PDF | arXiv:1207.2814 | Journal Link | BibTeX ]
Nonlinear Robust Tracking Control of a Quadrotor UAV on SE(3) (with T. Lee, N.H. McClamroch), Proc. American Control Conf., 4649-4654, 2012.
[ PDF | arXiv:1109.4457 | BibTeX ]
Hamilton--Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints (with T. Ohsawa, D. Sosa), Journal of Mathematical Physics, 53(7), 072905 (29 pages), 2012.
[ PDF | arXiv:1109.6056 | Journal Link | BibTeX ]
Generating Functionals and Lagrangian PDEs (with C. Liao, J. Vankerschaver), Journal of Mathematical Physics, submitted, 2013.
[ PDF | arXiv:1111.0280 | BibTeX ]
Dynamics and Control of a Chain Pendulum on a Cart (with T. Lee and N. H. McClamroch), Proc. IEEE Conf. on Decision and Control, 2502-2508, 2012.
[ PDF | arXiv:1211.4604 | BibTeX ]
Hamel's Formalism and Variational Integrators on a Sphere (with A.M. Bloch, and D.V. Zenkov), Proc. IEEE Conf. on Decision and Control, 7504-7510, 2012.
[ PDF | arXiv:1211.4607 | BibTeX ]
High-Fidelity Numerical Simulation of Complex Dynamics of Tethered Spacecraft (with T. Lee, and N.H. McClamroch), Acta Astronautica, submitted, 2013.
[ PDF | arXiv | BibTeX ]
Dirac Structures and Hamilton-Jacobi Theory for Lagrangian Mechanics on Lie Algebroids (with D. Sosa), Journal of Geometric Mechanics, 4(4), 421-442, 2012.
[ PDF | arXiv:1211.4561 | BibTeX ]
A novel formulation of point vortex dynamics on the sphere: geometrical and numerical aspects (with J. Vankerschaver), Journal of Nonlinear Science, submitted, 2012.
[ PDF | arXiv:1211.4560 | BibTeX ]
Spectral Variational Integrators (with J. Hall), Numerische Mathematik, submitted, 2012.
[ PDF | arXiv:1211.4534 | BibTeX ]
Symplectic Semiclassical Wave Packet Dynamics (with T. Ohsawa), Journal of Physica A, submitted, 2013.
[ PDF | arXiv | BibTeX ]
The Discrete Hamel's Formalism and Energy-Momentum Integrators for the n-dimensional Spherical Pendulum (with A.M. Bloch, D.V. Zenkov), Discrete and Continuous Dynamical Systems A (Special issue in honor of Arieh Iserles' 65th birthday), submitted, 2013.
[ PDF | arXiv | BibTeX ]
Generalized Galerkin Variational Integrators, preprint, 2004.
[ PDF | arXiv:math.NA/0508360 | BibTeX ]
A Discrete Theory of Connections on Principal Bundles (with J.E. Marsden, and A.D. Weinstein), preprint, 2004.
[ PDF | arXiv:math.DG/0508338 | BibTeX ]
Discrete Exterior Calculus (with M. Desbrun, A.N. Hirani, and J.E. Marsden), preprint, 2003.
[ PDF | arXiv:math.DG/0508341 | BibTeX ]
Estimating the Attractor Dimension of the Equatorial Weather System, Acta Phys. Pol. A 85, S27-S35, 1994.
[ PDF | Journal Link | BibTeX ]

Papers in Preparation

Dirac Structures in Lagrangian Mechanics on Lie Groupoids (with D. Sosa).
Discrete Augmented Variational Principles in Mechanics and Control (with T. Ohsawa).
On the Decomposition of Forces (with D.E. Chang, J.E. Marsden, T.S. Ratiu).

Upcoming and Recent Invited Presentations

Variational Techniques in Structure-Preserving Methods for PDEs, SciCADE, Valladolid, Spain, Sep 2013.
Geometric Mechanics and Control Theory, Indian Institute of Science, Bangalore, India, Aug 2013.
Smooth and Discrete Geometrical Approaches to Control, SIAM Controls, San Diego, CA, Jul 2013.
Variational Principles, SIAM Dynamical Systems, Snowbird, UT, May 2013.
Geometric Approximation, Approximation Theory 14, San Antonio, TX, Apr 2013.
Special seminar, Rutgers University, Piscataway, NJ, Feb 2013.
IRSES Meeting on DIfferential Geometry and Mechanics, Ghent, Belgium, Jan 2013.
Geometric methods in mechanical and dynamical systems special session, AMS Fall Western Section Meeting, Tucson, AZ, Oct 2012.
Center for Applied Mathematics Colloquium, Cornell University, Ithaca, NY, Feb 2012.
Mathematics Colloquium (Oliver Club), Cornell University, Ithaca, NY, Feb 2012.
Mathematics Colloquium, Penn State University, State College, PA, Sep, 2011.
Marsden Memorial minisymposium, ICIAM, Vancouver, Canada, Jul 2011.
Symmetry in Variational Problems minisymposium, SIAM Applications of Dynamical Systems, Snowbird, UT, May 2011.
Mathematics Colloquium, University of Illinois, Chicago, Apr 2011.
Geometric Control on Nonlinear Manifolds special session, IEEE CDC, Atlanta, GA, Dec 2010.
PIMS Seminar, University of British Columbia, Vancouver, Canada, Aug 2010.
New Geometric and Numeric Tools for the Analysis of Differential Equations Workshop, BIRS, Canada, Aug 2010.
Nonlinear Dynamics and Control minisymposium, CAIMS Annual Meeting, St. John's, Newfoundland, Canada, Jul 2010.
Computational Methods for Dynamical Systems Analysis minisymposium, USNCTAM 2010, State College, PA. Jun 2010.
Applied Mathematics Seminar, CSU Northridge, Northridge, CA, Apr 2010.
Sixth Annual Structured Integrators Workshop, University of California, San Diego, CA. Apr 2010.
Center for Computational Mathematics Seminar, University of California, San Diego, CA. Apr 2010.
Cymer Center for Control Systems and Dynamics Seminar, University of California, San Diego, CA. Apr 2010.
Colloquium, School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ, Oct 2009.
Computational and Applied Mathematics Colloquium, Penn State University, State College, PA, Oct 2009.
XVIII Fall Workshop on Geometry and Physics, Benasque, Spain, Sep 2009.
Mechanics and Control Seminar, Beijing Institute of Technology, May 2009.
Applied Mathematics Seminar, Stanford University, May 2009.
Applied Mathematics and Statistics Seminar, Johns Hopkins University, Baltimore, MD, Feb 2009.
Physical Mathematics Seminar, Massachusetts Institute of Technology, Cambridge, MA, Feb 2009.
Mathematics Special Colloquium, University of Minnesota, MN, Dec 2008.
Applied Mathematics Seminar, Texas A&M University, College Station, TX, Dec 2008.
Mathematics Special Colloquium, University of California, San Diego, CA, Nov 2008.
CDS Seminar, Caltech, Pasadena, CA, Nov 2008.
Center for Applied Mathematics Colloquium, University of Notre Dame, Notre Dame, IN, Nov 2008.
Applied and Interdisciplinary Mathematics Seminar, University of Michigan, Ann Arbor, MI, Nov 2008.
Geometric Mechanics, Control, and Integrability minisymposium, AMS Southeastern Sectional Meeting, Huntsville, AL, Oct 2008.
Dynamics Seminar, Department of Mathematics, Cornell University, NY, Oct 2008.
Advances in Time-Integration minisymposium, World Congress on Computational Mechanics, Venice, Italy, Jul 2008.
Geometric Integration and Computational Mechanics Workshop, Foundations of Computational Mathematics, Hong Kong, China, Jun 2008.
Scientic Computing Seminar, Department of Mathematics, Tsinghua University, Beijing, China, Jun 2008.
Institute of Computational Mathematics Seminar, Chinese Academy of Sciences, Beijing, China, Jun 2008.
Fourth Annual Structured Integrators Workshop, Stanford, CA, Apr 2008.
Geometric Numerical Integration Special Session, Joint Meeting of the AMS-NZMS, Welington, New Zealand, Dec 2007.
Discrete Differential Geometry, Berlin, Germany, Jul 2007.
Geometric and Symplectic Integration minisymposium, SciCADE, Saint-Malo, France, Jul 2007.
Effective Computational Methods for Highly Oscillatory Problems, Newton Institute, University of Cambridge, UK, Jul 2007.
Minicourse, First International Summer School on Geometry, Mechanics, and Control, Cantabria, Spain, Jun 2007.
Geometric Methods in Dynamical Systems minisymposium, SIAM DS07, Snowbird, UT, May 2007.
Applied Mathematics Seminar, Imperial College, London, UK, May 2007.
Highly Oscillatory Problems Seminar, Newton Institute, University of Cambridge, UK, May 2007.
Center for Applied Mathematics Colloquium, University of Notre Dame, Notre Dame, IN, Apr 2007.
Mathematics Colloquium, University of Iowa, Iowa City, IA, Mar 2007.
CDS/CIMMS Lunchtime Seminar, Caltech, Pasadena, CA, Dec 2006.
Applied Mathematics Colloquium, University of Maryland, Baltimore County, Nov 2006.
Mathematical Sciences Colloquium, Rensselaer Polytechnic Institute, Troy, NY, Oct 2006.
Applications of the Geometric Phase in Classical Mechanics minisymposium, SIAM Annual Meeting, Boston, MA, Jul 2006.
Geometry and Dynamical Systems with Applications Seminar, Arizona State University, Tempe, AZ, Apr 2006.
Dynamics Seminar, University of Colorado, Boulder, CO, Feb 2006.
Mathematics Colloquium, Colorado School of Mines, CO, Feb 2006.
Applied and Interdisciplinary Mathematics Seminar, University of Michigan, Ann Arbor, MI, Feb 2006.
Mathematics Colloquium, University of South Carolina, Columbia, SC, Feb 2006.
Mathematics Colloquium, Texas A&M University, College Station, TX, Feb 2006.
Mathematics Colloquium, Colorado State University, Fort Collins, CO, Feb 2006.
Mathematics Special Colloquium, Purdue University, West Lafayette, IN, Feb 2006.
Joint Numerical Analysis and Differential Equations Seminar, North Carolina State University, NC, Jan 2006.
Contemporary Dynamical Systems special session, AMS National Meeting, San Antonio, TX, Jan 2006.
Applied Mathematics Seminar, Mathematics, University of Waterloo, Canada, Dec 2005.
Control Seminar, College of Engineering, University of Michigan, Ann Arbor, MI, Oct 2005.
Differential Geometry and Analysis Seminar, University of Toledo, Toledo, OH. Sep 2005.
Computer and Computational Sciences, CCS-2, LANL, Los Alamos, NM. Aug 2005.
Geometric Dynamics and its Applications minisymposium, SIAM DS05, Snowbird, UT, May 2005.
Special Seminar, Mathematics, University of California, Berkeley, CA, Apr 2005.
Mathematical Physics Seminar, University of Minnesota, Twin Cities, MN, Mar 2005.
Joint Numerical Analysis and Differential Equations Seminar, North Carolina State University, NC, Jan 2005.
Mathematics Colloquium, University of California, San Diego, CA, Dec 2004.
Flight Dynamics and Control Seminar, University of Michigan, Ann Arbor, MI, Oct 2004.
Geometry Seminar (two sessions), University of Michigan, Ann Arbor, MI, Oct 2004.
Applied Mathematics Seminar, University of California, San Diego, CA, Aug 2004.
Temasek Laboratories, National University of Singapore, Jul 2004.
Mathematics Seminar (four sessions), National University of Singapore, Jul 2004.
Geometric Dynamics and Applications minisymposium, AIMS Conference on Dynamical Systems and Differential Equations, Pomona, CA, June 2004.
Applied Mathematics and Numerical Analysis Seminar, University of Minnesota, Twin Cities, MN, Jan 2004.

Professional Service

Co-organizer, computational methods for geometric PDEs minisymposium, ICIAM, Vancouver, Canada, July, 2011.
Organizing committee member, geometric methods for optimal control semester, Madrid, Spain, Jul-Dec 2010.
Co-organizer, Sixth Annual Structured Integrators Workshop, University of California, San Diego, CA, Apr 2010.
Co-organizer, 26th Pacific Coast Gravity Meeting, Mar 2010.
Organizer, computational mechanics minisymposium, SciCADE 2009, Beijing, China, May 2009.
Co-organizer, geometric mechanics and its applications minisymposium (MS 121), SIAM DS09, May 2009.
Co-organizer, Fifth Annual Structured Integrators Workshop, Caltech, Pasadena, CA, May 2009.
Co-organizer, nonlinear dynamics and control of mechanical systems invited session, IEEE Conference on Decision and Control, Dec 2006.
Co-organizer, contemporary dynamical systems special session, AMS National Meeting, Jan 2006.
Co-organizer, geometric dynamics and its applications minisymposium (MS 59, 70), SIAM DS05, May 2005.
Co-organizer, CIMMS Workshop on Discrete Geometry for Mechanics, Caltech, Pasadena, CA, Oct 2003.
Co-organizer, discrete geometry and geometric integration minisymposium (MS 38, 62), SIAM DS03, May 2003.
Co-organizer, CIMMS Workshop on Networks, Optimization and Duality, Caltech, Pasadena, CA, Jul 2002.
Co-organizer, geometric integration minisymposium (MS 51, 69), SIAM DS01, May 2001.
Referee, Advances in Difference Equations, Aerospace Science and Technology, Applied Numerical Mathematics, Celestial Mechanics and Dynamical Astronomy, Communications in Contemporary Mathematics, Communications in Numerical Methods in Engineering, Computational Science & Discovery, Discrete and Continuous Dynamical Systems, ESAIM: Control, Optimisation and Calculus of Variations, Foundations of Computational Mathematics, IEEE Transactions on Automatic Control, IET Control Theory & Applications, IMA Journal of Numerical Analysis, Journal of Computational Physics, Journal of Mathematical Physics, Journal of Nonlinear Science, Journal of Physics A, Journal of Symplectic Geometry, Mathematical and Computer Modelling, Nonlinearity, Numerical Algorithms, Numerische Mathematik, Physica D, Physics Letters A, Proceedings of the Royal Society A, SIAM Applied Dynamical Systems, SIAM Multiscale Modeling and Simulation, SIAM Journal on Numerical Analysis, SIAM Journal on Scientific Computing, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications), Soft Computing and Automation Journal, Transport in Porous Media.
Reviewer, IEEE Conference on Decision and Control 2005, 2006, 2007, 2008, IEEE Multi-conference on Systems and Control 2007, American Control Conference 2008, Mathematical Reviews, Springer Books, Swiss National Science Foundation.
Panel member, Applied Mathematics, National Science Foundation, Mar 2009.
Panel member, Computational Mathematics, National Science Foundation, Mar 2008.

Education

Ph.D., Control and Dynamical Systems (minor in Applied and Computational Mathematics), California Institute of Technology, 2004.
M.S., Mathematics, California Institute of Technology, 2000.
B.S. with honor, Mathematics, California Institute of Technology, 2000.

Upcoming Conferences (Confirmed)

German-American Frontiers of Science Symposium, Potsdam, Germany, May 10-13, 2012.
Symmetries of Differential Equations: Frames, Invariants and Applications, Minneapolis, MN, May 17-19, 2012.
Geometry, Symmetry, Dynamics, and Control: The Legacy of Jerry Marsden, Fields Institute, Toronto, July 23-27, 2012.
Structured Matrix Computations in Non-Euclidean Geometries: Algorithms and Applications, CIRM, Luminy, France, October 8-12, 2012.
AMS Western Section Meeting, Tucson, AZ, October 27-28, 2012.
IEEE Conference on Decision and Control, Maui, Hawaii, December 10-13, 2012.
IRSES Meeting on Differential Geometry and Mechanics, Ghent, Belgium, January 14-16, 2013.
14th International Conference on Approximation Theory, San Antonio, Texas, April 7-10, 2013.
SIAM Conference on Applications of Dynamical Systems, Snowbird, Utah, May 19-23, 2013.
International Linear Algebra Society 2013 Meeting, Providence, RI, June 3-7, 2013.
Workshop on Reduced-Order Modeling in General Relativity, Caltech, Pasadena, CA, June 6-7, 2013.
Topics in Numerical Analysis for Differential Equations, ICMAT, Madrid, Spain, July 8-12, 2013.
SciCADE 2013, Volladolid, Spain, September 16-20, 2013.
Modern Applications of Homology and Cohomology, IMA, Minneapolis, MN, October 28-November 1, 2013.
Geometric Mechanics and Control Theory, Indian Institute of Science, Bangalore, India, January 2-10, 2014.

Interesting Upcoming Conferences

Manifolds and Geometric Integration Colloquium (MAGIC) 2013, Geilo, Norway, February 26-March 1, 2013.
Weinstein Symposium, Institute Henri Poincare, Paris, France, July 18-20, 2013.
Conference in Honor of Alan Weinstein, EPFL, Lausanne, Switzerland, July 22-26, 2013.

Recent Conferences Attended

IMA Tutorial/Workshop: New Paradigms in Computation, Minneapolis, MN, March 28-30, 2005.
IPAM Grand Challenge Problems in Computational Astrophysics, N-Body Problems in Astrophysics, Los Angeles, CA, April 18-22, 2005.
IPAM Grand Challenge Problems in Computational Astrophysics, Relativistic Astrophysics, Los Angeles, CA, May 2-6, 2005.
SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, May 22-26, 2005.
Summer School on Control in Quantum Systems, California Institute of Technology, Pasadena, CA, Aug 8-14, 2005.
IMA New Directions Short Course: Quantum Computation, Minneapolis, MN, Aug 15-26, 2005.
Workshop in Celebration of Bjorn Engquist 60th Birthday: International Forum on Multiscale Methods and Partial Differential Equations, Los Angeles, CA, August 26-27, 2005.
Frontiers of Applied Analysis, Pittsburgh, PA, September 8-10, 2005.
IPAM Bridging Time and Length Scales in Materials Science and Bio-Physics, Multiscale Analysis and Computation, Los Angeles, CA, November 14-16, 2005.
Multiscale Modeling and Computation - Basic Theory and the Geosciences, Pasadena, CA, November 17-18, 2005.
AMS National Meeting, San Antonio, TX, January 12-15, 2006.
Geometric Numerical Integration Workshop, Oberwolfach, Germany, March 19-25, 2006.
Advanced Computational Electromagnetics Workshop, Boston, MA, May 4-6, 2006.
SIAM Annual Meeting, Boston, Massachusetts, July 10-14, 2006.
International Congress of Mathematicians, Madrid, Spain, August 22-30, 2006.
IMA Tutorial: Algebraic Geometric Methods in Engineering, Minneapolis, Minnesota, September 15-16, 2006.
MSRI Workshop on Application of topology in science and engineering, Berkeley, California, September 18-22, 2006.
IEEE Conference on Decision and Control, San Diego, California, December 13-15, 2006.
Geometry of Mechanism Science, Notre Dame, Indiana, March 1-3, 2007.
Highly Oscillatory Problems: Computation, Theory and Application, Newton Institute, Cambridge, United Kingdom, May 7-20, 2007.
SIAM Conference on Applications of Dynamical Systems, Snowbird, Utah, May 28-June 1, 2007.
First International Summer School on Geometry, Mechanics and Control, Castro Urdiales, Spain, June 25-29, 2007.
Effective Computational Methods for Highly Oscillatory Problems, Newton Institute, Cambridge, United Kingdom, July 2-6, 2007.
SciCADE, Saint-Malo, France, July 9-13, 2007.
Discrete Differential Geometry, Berlin, Germany, July 16-20, 2007.
BIRS Geometric Mechanics, Banff, Canada, August 12-17, 2007.
Joint AMS-NZMS Meeting, Wellington, New Zealand, December 12-15, 2007.
Gene Golub Symposium, Urbana, IL, February 29 - March 1, 2008.
Fourth Annual Structured Integrators Workshop, Stanford, CA, April 24-25, 2008.
Foundations of Computational Mathematics, Hong Kong, China, June 16-26, 2008.
World Congress on Computational Mechanics, Venice, Italy, June 30 - July 4, 2008.
AMS Southeastern Sectional Meeting, Huntsville, AL, October 24-26, 2008.
Groupoidfest 2008, Riverside, CA, November 22-23, 2008.
Fifth Annual Structured Integrators Workshop, Pasadena, CA, May 7-8, 2009.
SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, May 17-21, 2009.
SciCADE 2009, Beijing, China, May 25-29, 2009.
XVIII International Fall Workshop on Geometry and Physics, Benasque, Spain, September 6-11, 2009.
IEEE Conference on Decision and Control, Shanghai, China, Dec 16-18, 2009.
Pacific Coast Gravity Meeting, La Jolla, CA, March 26-27, 2010.
Sixth Annual Structured Integrators Workshop, La Jolla, CA, April 26-27, 2010.
HarrisFest 2010, Ann Arbor, MI, May 22, 2010.
16th US National Congress of Theoretical and Applied Mechanics, State College, PA, June 27-July 2, 2010.
Geometry, Mechanics, and Dynamics: A Workshop celebrating the 60th Birthday of Tudor Ratiu, CIRM, France, July 12-16, 2010.
CAIMS Annual Meeting, St. John's, Newfoundland and Labrador, Canada, July 17-20, 2010.
BIRS Workshop: New Geometric and Numeric Tools for the Analysis of Differential Equations, Banff, Canada, August 13-15, 2010.
Symmetry, Separation, Super-integrability and Special Functions, University of Minnesota, September 17-19, 2010.
Geometric and Topological Methods in Control and Robotics, Madrid, Spain, October 4-6, 2010.
NSF CMMI Research and Innovation Conference, Atlanta, GA, January 4-7, 2011.
Geometric Numerical Integration Workshop, Oberwolfach, Germany, March 20-26, 2011.
SIAM Conference on Applications of Dynamical Systems, Snowbird, UT, May 22-26, 2011.
BIRS Workshop: KAM theory and Geometric Integration, Banff, Canada, June 5-10, 2011.
Computational Methods in Dynamics, ICTP, Trieste, Italy, July 4-8, 2011.
International Congress on Industrial and Applied Mathematics, Vancouver, BC, Canada, July 18-22, 2011.
Applied Dynamics and Geometric Mechanics Workshop, Oberwolfach, Germany, August 14-20, 2011.

Other Interesting Conferences

BIRS Numerical Relativity, Banff, Canada, April 16 - 21, 2005.
SciCADE 2005, Nagoya, Japan, May 23-27, 2005.
Dundee Numerical Analysis Conference, Dundee, Scotland, June 28 - July 1, 2005.
Geometric Integration and Computational Mechanics Minisymposium, Foundations of Computational Mathematics, Santander, Spain, June 30 - July 9, 2005.
Numerical Relativity, Global Problems in Mathematical Relativity, Newton Institute, Cambridge, UK, August 15-21, 2005.
IEEE Conference on Control Applications, Toronto, Canada, August 28-31, 2005.
Compatible Discretizations for Partial Differential Equations, Oslo, Norway, September 26-28, 2005.
IEEE Conference on Decision and Control, Seville, Spain, December 12-15, 2005.
Workshop on Structured Integration, Palo Alto, California, March 10-11,2006.
American Control Conference, Minneapolis, Minnesota, June 14-16, 2006.
IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control, Nagoya, Japan, July 19-21, 2006.
International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan, July 24-28, 2006.
AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Keystone, Colorado, August 21-24, 2006.
IPAM Workshop on Mathematical Challenges and Opportunities in Sensor Networking, Los Angeles, California, January 8-12, 2007.
Analysis on Graphs and its Applications, Newton Institute, Cambridge, United Kingdom, January 8 - June 29, 2007.
International Congress on Industrial and Applied Mathematics, Zurich, Switzerland, July 16-20, 2007.
IMA Summer Program: Classical and Quantum Approaches in Molecular Modeling, Minneapolis, Minnesota, July 23-August 3, 2007.
Trimester on Combinatorics and Control, Madrid and Benasque, Spain, April-June, 2010.
Discrete Differential Geometry and Applications, University of Tennessee, Knoxville, TN, May 17-21, 2010.
Highly Oscillatory Problems: From Theory to Applications, Newton Institute, Cambridge, United Kingdom, September 12-17, 2010.
Foundations of Computational Mathematics, Budapest, Hungary, July 4-14, 2011.
Stochastic Systems, Simulation and Control, ICMAT, Madrid, Spain, November 5-9, 2012.

Erdös Number

Erdös, P.; Graham, R. L.; Montgomery, P.; Rothschild, B. L.; Spencer, J.; Straus, E. G.; Euclidean Ramsey theorems. I., J. Combinatorial Theory Ser. A, 14 (1973), 341-363.
Golubitsky, M. A.; Rothschild, B. L.; Primitive subalgebras of exceptional Lie algebras. Bull. Amer. Math. Soc. 77 1971 983-986.
Golubitsky, M. A.; Marsden, J. E.; The Morse lemma in infinite dimensions via singularity theory. SIAM J. Math. Anal. 14 (1983), no. 6, 1037-1044.
Desbrun, M.; Leok, M.; Marsden, J. E.; Discrete Poincaré Lemma, Appl. Numer. Math. 53(2-4), 231-248, 2005.

Interestingly, three of my collaborators have Erdös number 3. They are, Anthony M. Bloch, Jerrold E. Marsden, and Alan D. Weinstein.
Useful sites in this regard are the Erdös Number Project and the AMS MathSciNet Collaboration Distance Page.

Collaborators

Anthony Bloch, Mathematics, University of Michigan, Ann Arbor.
Mathieu Desbrun, Computer Science, California Institute of Technology.
Anil Hirani, Computer Science, University of Illinois, Urbana-Champaign.
Islam Hussein, Mechanical Engineering, Worcester Polytechnic Institute.
Sameer Jalnapurkar, Mathematics, Indian Institute of Science, Bangalore.
Jerrold Marsden, Control and Dynamical Systems, California Institute of Technology.
N. Harris McClamroch, Aerospace Engineering, University of Michigan, Ann Arbor.
Amit Sanyal, Mechanical Engineering, University of Hawaii at Manoa.
Alan Weinstein, Mathematics, University of California, Berkeley.
Matthew West, Aeronautics and Astronautics, Stanford University.
Dmitry Zenkov, Mathematics, North Carolina State University.

Students

Taeyoung Lee, Ph.D. in Aerospace Engineering (Distinguished Achievement Award, Ivor K. Mclvor Award, Rackham Predoctoral Fellow, and Rackham International Student Fellow), University of Michigan, Ann Arbor. (co-advised with N. Harris McClamroch)
Masako Kishida, M.S. in Applied and Interdisciplinary Mathematics, University of Michigan, Ann Arbor.

Biographical Sketch

Melvin Leok is a tenured associate professor of mathematics at the University of California, San Diego, where his research is supported in part by grants from the National Science Foundation in applied and computational mathematics, including a Faculty Early Career Development (CAREER) award. He serves on the editorial boards of the Journal of Nonlinear Science, the SIAM Journal on Control and Optimization, the LMS Journal of Computation and Mathematics, the Journal of Geometric Mechanics, and the Journal of Computational Dynamics.

Prior to joining UCSD, he was a tenure-track assistant professor of mathematics at Purdue University, a visiting assistant professor of control and dynamical systems at the California Institute of Technology, and a T.H. Hildebrandt research assistant professor of mathematics at the University of Michigan, Ann Arbor. At Purdue, he was a nominee for the Packard Fellowship for Science and Engineering, and at Michigan, he received a Horace H. Rackham Faculty Fellowship and Grant, and a Margaret and Herman Sokol Spring/Summer Research Grant.

He received his B.S. with honors and M.S. in Mathematics in 2000, and his Ph.D. in Control and Dynamical Systems with a minor in Applied and Computational Mathematics under the direction of Jerrold Marsden in 2004, all from the California Institute of Technology.

His primary research interests are in computational geometric mechanics, computational geometric control theory, discrete geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry and multiscale systems.

He was the recipient of the SciCADE New Talent Prize in 2007 for his work on Lie Group and Homogeneous Variational Integrators, and the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis, both in 2003, for his work on Foundations of Computational Geometric Mechanics. While a doctoral student at Caltech, he held a Poincaré Fellowship (2000-2004), a Josephine de Kármán Fellowship (2003-2004), an International Fellowship from the Agency for Science, Technology, and Research (2002-2004), a Tau Beta Pi Fellowship (2000-2001), and a Tan Kah Kee Foundation Postgraduate Scholarship (2000).

As a Caltech undergraduate, he received the Loke Cheng-Kim Foundation Scholarship (1996-2000), the Carnation Scholarship (1998-2000), the Herbert J. Ryser Scholarship (1999), the E.T. Bell Undergraduate Mathematics Research Prize (1999), and the Jack E. Froehlich Memorial Award (1999).