Applied Topology and Matrix Methods

SpeakerDr. Anil Hirani
Organization Dept. of Computer Science - University of Illinois at Urbana-Champaign
LocationEngineering Building II Room 1230
DateSeptember 7, 2012 12:50 PM

Abstract:

Applied topology is a relatively new field. This talk is a survey of matrix based methods which extend its reach in new directions. (1) Given a sparse set of pairwise comparisons (between objects, sports teams, movies, candidates, etc.) a global ranking can be computed as a least squares problem. Digging deeper, newer applications of such data become possible with Hodge decomposition. I will show how to do this efficiently on many types of graphs using various matrix methods. One application will be to rank NCAA Men's Basketball teams. One surprise is the failure of a technique which is otherwise quite popular in traditional scientific computing. (2) Harmonic forms are one piece of Hodge decomposition. These have been widely used in computer graphics for texture mapping and vector field design.  I will show their use in locating holes in idealized sensor networks by solving certain linear systems. Then I'll show how to use eigenvector and least squares methods to find harmonic forms on meshes. These can then be used to solve vector elliptic partial differential equations in topologically interesting domains.  (3) The above problems are both related to L2 (2-norm) minimization. In contrast L1 (1-norm) minimization has sometimes been hailed as the technique to supplant least squares for this century's problems. I will show how the use of 1-norm transforms two NP-hard topological problems into polynomial time solvable problems.

Bio:

Anil Hirani is an applied mathematician on the faculty in the Computer Science department at University of Illinois at Urbana-Champaign. He works on formulating problems using topology and geometry and solves them by using numerical methods. Before Illinois he spent a year at NASA's Jet Propulsion Lab in Pasadena working on a mission to the icy moons of Jupiter. He has several years of experience working as a graphics software engineer at Sun Microsystems in California and as a researcher in image processing at Sony Corporation in Japan. He received his Master’s degrees in Computer Science from Stanford and Caltech.  Additionally, Dr. Hirani received his Ph.D. from Caltech in Computer Science with minors in Mathematics and Control and Dynamical Systems.

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