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Laurent Demanet

Assistant Professor of Applied Mathematics

Mathematics
MIT
Math
Applied Math
 
 
 

What Was New?
  • Rejecta Mathematica is an experimental journal started recently by M. Wakin et al. where you can submit any math paper provided it has been rejected elsewhere before. You need to provide an open letter disclosing in full honesty the reasons for rejection. Link. November 2007.
  • A new preprint by D. Needell and R. Vershynin presents an elegant algorithmic solution to compressed sensing. In short, ``regularized orthogonal matching pursuit'' (ROMP) has the speed of matching pursuit and the convergence guarantees of ell-1 minimization. See their paper. August 2007. Update October 2007: J. Romberg has a clever primal-dual version of ell-1 minimzation for the Dantzig selector that offers similar complexity and convergence estimates.
  • The problem of adapting fast multipole ideas to the Helmholtz equation (high-frequency electromagnetic scattering) has received some attention lately. V. Rokhlin and coworkers have developed a low-complexity method, and more recently L. Ying and B. Engquist presented an extension based on multiscale directional ideas. See their paper. August 2007.
  • The Abel prize goes to S. Varadhan of the Courant Institute, for his contributions to the theory of large deviations. It is the second time for probabilists to be in the spotlight over the past year, after W. Werner's Fields medal. March 2007.
  • CMV: the unitary analogue of Jacobi matrices. R. Killip and I. Nenciu present a remarkable Householder algorithm for unitary matrices, where the reduction is not to a tridiagonal matrix as in the symmetric case, but to a CMV matrix (pentadiagonal with every other element equal to zero on the +2 and -2 diagonals). The algorithm has antecedents in the linear algebra literature (A. Bunse-Gerstner and L. Esner), but here the algorithm is only an offshoot of a much deeper analysis. Link added December 2006. The paper in CPAM.
  • Compressed sensing, also known as compressive sampling, has become fertile ground for research in signal-oriented applied math over the past two years. The main idea is to formulate inverse problems using an ell-1 regularization, and measurements in an "incoherent" basis, even for tasks that do not look like they require this kind of formulation. Recent applications include A/D conversion, one-pixel cameras, and sensor networks. Link added December 2006. Three resource webpages are L1-magic at Caltech, Sparselab at Stanford, and the CS page at Rice.
  • Given a finite set of scattered point values, the Whitney extension problem is to find lower bounds on the derivatives of all smooth functions passing through those points. Charles Fefferman recently made progress on this tough math problem. While not all engineers may care, high-level mathematics flirting with the limits of sampling theory — or how simple a continuous model can explain data — is rare enough to be noticed. Link added September 2006. The paper in Annals of Mathematics.
  • The Fields medal to Tao, Okounkov, Perelman and Werner. The Nevanlinna prize to Kleinberg. The Gauss prize to Ito (of the Ito calculus). August 2006.
  • Perhaps the first conclusive numerical simulation of a binary black hole merger, by J.G. Baker and collaborators, February 2006. The paper on ArXiv.
  • Quantum chemistry is scientific computing in high dimensions. Some insights from applied mathematics to break the complexity of the full Schroedinger equation, without ``modeling'' or choice of basis functions:
    • Separated Representations, by G. Beylkin, M. Mohlenkamp, R. Harrison and collaborators. Link added April 2005. The project webpage.
    • Sparse Grids, by H. Yserentant. Link added July 2005. One paper.
  • Analyzing the US political landscape with mathematics looks like a good recipe to get media attention:
    • Statistical ``proof'' that the range of political opinions in the US is truly one-dimensional—at least in the Supreme Court. A study of the (dis)agreements among justices, by L. Hubert and D. Steinley, September 2005.
    • Network visualization of the ties between the congressional committees, by M. Porter et al., May 2005. The ScienceNOW news release.
    • How to skew the US electoral map to make the area proportional to population, by M. Gastner, C. Shalizi, and M. Newman, November 2004.The project webpage.
  • In the news: Peter Lax to Receive Abel Prize, March 2005.
  • Robust uncertainty principles: how to recover a sparse sequence from its very incomplete spectrum, by E. Candes, J. Romberg and T. Tao, May 2004. Many extensions have been considered since then by E. Candes, D. Donoho and their collaborators, in particular on sparse solutions of large underdetermined systems.
  • Statistically optimal and computationally fast detection of filamentary and sheet-like structures in 3-D point clouds, by D. Donoho, April 2004. (Professor Donoho, please come back soon and write a beautiful theory.)
  • Computational geometric mechanics: how to consistently discretize and integrate PDE's with strong geometric structure, by M. Leok, 2003. PDF.
  • A Fast Multipole Method based only on point evaluations of the kernel, by L. Ying, G. Biros, and D. Zorin, October 2003. PDF.
  • Primes is in P! A deterministic polynomial-time algorithm to determine whether a number is prime or composite. By Agrawal, Kayal and Saxena, August 2002.
  • Near-optimal computation of the Fourier transform in O(polylog(N)), without even browsing all the data. By A. Gilbert et al., 2002. Homepage.
  • One-bit sigma-delta quantization of bandlimited signals is exponentially accurate. By S. Gunturk, 2002. Homepage.

Disclaimer: this column will be forever incomplete, and has no ambition of being representative of the field. My views are my own.