Publications, Preprints and E-prints

News

Our paper "Optimal Homologous Cycles, Total Unimodularity, and Linear Programming" (described in the January 5 item below) has been accepted for publication in STOC10 (the 42nd ACM Symposium on Theory of Computing).

Problem statement : given a simplicial complex with weights on its simplices, and a chain, find the chain with minimal weight which is homologous to the given one. With binary coefficients (1 if a simplex is included, 0 if not) the problem has been known to be NP-hard. We show that if integer coefficients are used instead, the problem can be solved in polynomial time. Our main result is a theorem characterizing precisely when the boundary matrix of a simplicial complex is totally unimodular. Read all about it in our E-print on arXiv at http://arxiv.org/abs/1001.0338.

This seemingly simple problem had been open for many years : triangulate a cube into tetrahedra such that each triangle has acute angles and all angles between triangles are acute. It was not even known if such a triangulation exists ! Our paper gives the solution of this problem with the help of algorithms and software we developed for well-centered triangulation (WCT). WCT is described in many papers available on my publications page. This is work of my student Evan VanderZee (co-advised with Vadim Zharnitsky) in collaboration with Damrong Guoy. Edgar Ramos is another collaborator in the WCT project. The vertex data and program for computing the angles is available. Paper appears in journal Computational Geometry: Theory and Applications. Also available as a preprint.

Awards

• 2007 : NSF CAREER Award (Algebraic Topology and Exterior Calculus in Numerical Analysis)
• 2003 : SIAM Student Paper Competition, Honorable Mention (Discrete Exterior Calculus: Theory and Applications)

Research Interests

• Computational algebraic topology: computation of topological invariants of simplicial complexes with applications e.g. in numerical analysis, mesh topology, machine learning, and sensor networks coverage analysis.


• Optimized triangulations: creation of optimized triangulations for numerical PDEs and computational topology, e.g. well-centered triangulations (in which each simplex contains its circumcenter) and acute triangulations.


• Discretization of exterior calculus (DEC): vector calculus generalizes to nonlinear manifolds as exterior calculus, and in DEC we are attempting to build a complete mathematical and computational framework in which theorems from the smooth world have discrete analogs on simplicial complexes.


• Numerical solution of partial differential equations (PDEs): formulation using differential forms and solution using DEC, for PDE vector problems for which traditional nodal vector finite element methods are unstable, e.g. Poisson's equation in first order form or flow in porous media.


• Computational astrodynamics: adaptive representation and computation of gravitational field of small irregular bodies like asteroids and comets for fast trajectory propagation; applications of dynamical systems to low thrust space mission trajectory design.

Teaching

• Spring 2010: CS 555 / MATH 552 / CSE 510 Numerical Methods for PDEs
• Fall 2009: CS 558 / CSE 513 Laplacians on Networks and Meshes (Topics in Numerical Analysis)
• Spring 2009: CS 450 / CSE 401 / MATH 450 / ECE 491 Introduction to Numerical Analysis
• Fall 2008: CS 558 / CSE 513 Calculus on Meshes (Topics in Numerical Analysis)
• Spring 2008: CS 450 / CSE 401 / MATH 450 / ECE 491 Introduction to Numerical Analysis
• Fall 2007: CS 257 / MATH 257 Numerical Methods
• Spring 2007: CS 257 / MATH 257 Numerical Methods
• Fall 2006: CS 598ANH Symplectic Integrators and Discrete Mechanics
• Spring 2006: CS 450 / CSE 401 / MATH 450 / ECE 491 Introduction to Numerical Analysis
• Fall 2005: CS 598ANH Calculus on Meshes

Group Alumni

• Evan VanderZee, Completed Ph.D in Mathematics, December 2009. Heading to Argonne National Labs as a postdoc.
  Coadvised with Vadim Zharnitsky of Mathematics department.
• Andrew Colombi, Graduated with Ph.D in CS, August 2008. Now at Palantir.
• John Jossey (Postdoc), September 2006 - August 2007

Theses

• Ph.D Thesis, Discrete Exterior Calculus, Caltech, 2003 (advisor Jerrold E. Marsden). (BibTeX entry)
• M.S. Thesis, Linearization Methods for Variational Integrators and Euler-Lagrange Equations, Caltech, 2000.

CV

• 2005-current, Assistant Professor, University of Illinois at Urbana-Champaign, Department of Computer Science
• 2004-05, Senior Engineer, Jet Propulsion Laboratory Guidance, Navigation and Control Section
• 2003-04, CIMMS Postdoctoral Scholar, California Institute of Technology (Caltech), Control and Dynamical Systems
• 2003, Ph.D, California Institute of Technology (Caltech), (Advisor: Jerrold E. Marsden),
  Ph.D in Computer Science with minors in Mathematics and Control and Dynamical Systems
  Thesis : Discrete Exterior Calculus
• 2000, M.S, Computer Science, Caltech
• Engineer, Sony Corporation
• Software Engineer, Sun Microsystems
• M.S, Computer Science (Theoretical track), Stanford University
• Undergraduate degree, Computer Science Birla Institute of Technology and Science (BITS), Pilani, India
  

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