This talk presents a strategy for computational wave propagation that consists in decomposing the solution wavefield onto a largely incomplete set of eigenfunctions of the weighted Laplacian, with eigenvalues chosen randomly. The recovery method is the ell-1 minimization of compressed sensing. For the mathematician, we establish three possibly new estimates for the wave equation that guarantee accuracy of the numerical method in one spatial dimension. For the engineer, the compressive strategy offers a unique combination of parallelism and memory savings that should be of particular relevance to applications in reflection seismology. Joint work with Gabriel Peyre.

We explain how to construct knot and tangle invariants (such as the Jones polynomial or Khovanov homology) by studying holomorphic vector bundles on certain compact, complex manifolds. Topologically these complex manifolds are just products of the same projective space P1. Conjecturially, if one used Grassmannians Gr(k,n) instead of projective spaces this would give a series of new knot invariants.

Each of the items in the title is a big area with a number of known
results and open questions. I will explain that these areas are iso-
morphic. The talk is based on the following joint works with Richard Montgomery: Geometric approach to Goursat flags; Points and curves in the Monster tower; Resolving singularities with Cartans prolongation.

Monte Carlo simulations of the radiative transport equation provide a gold standard of computational 
accuracy for many problems in biomedical optics, but their slow convergence (as dictated by the central 
limit theorem) prevents their routine use. In the past decade, there has been a concerted effort to 
develop adaptively modified Monte Carlo algorithms that converge geometrically to solutions of 
radiative transport equations. Our group has concentrated on algorithms that extend to integral 
equations methods first proposed for matrix equations by Halton in 1962. This was accomplished by 
expanding the solution in suitable basis functions and estimating a finite number of expansion 
coefficients by random variables, based on either correlated sampling or importance sampling, and 
designing  strategies to lower the variance recursively. Geometric convergence has been rigorously 
established for these first generation adaptive algorithms, but their practical utility is degraded by  the 
expansion technique itself. More recently we have developed new adaptive algorithms that overcome 
most of the computations shortcomings of the earlier algorithms, and we have demonstrated the 
geometric convergence of these second generation algorithms.  We will outline the major ideas 
involved and illustrate their advantages over conventional Monte Carlo methods. These algorithms will 
play a significant role in providing real\time computational support for biophotonics applications at the 
Beckman Laser Institute and Medical Clinic.