Brown Analysis Seminar
Department of Mathematics Colloquium
Abstract: The famous Plateau problem consists in finding the surface of least area spanning a given contour in a given Riemannian manifold. In a cel- ebrated work of the sixties, Federer and Fleming introduced the theory of currents, the most successfull framework for proving the existence of a solution using the direct methods of the calculus of variations. Due to the work of De Giorgi, Almgren, Simons, Federer and Si- mon (among others), there is a quite satisfactory regularity theory for area-minimizing currents of codimension 1, which are in fact regular up to a singular set of codimension 7. In codimension higher than 1 the situation is dramatically different, because branching (and hence singularities of codimension 2) can occur. A deep regularity theory has been developed 30 years ago by Almgren and it is contained in a monograph of 900 pages. In two recent works with Emanuele Spadaro we have found much shorter proofs of a good portion of Almgren's results and we have im- proved upon some of his theorems. These results come as a combina- tion of his ideas with new techniques, give some new points of view on Almgren's theory and provide links to other problems.
Center for Computational Molecular Biology Seminar
Abstract: Several next-generation sequencing techniques produce very large numbers of short sequences (reads), which needs to be mapped to a genome. Most existing methods use fast indexing and match the reads with up to N mismatches, where N=2 is typical. However, often a sequence read comes with a quality score for each nucleotide, which can be translated to a probability of error. In this talk, I describe how such probabilities can be used directly in the search through position weight matrices, and a data structure that makes it feasible to use the weight matrices in the search. Finally, I show results on simulated data.
Scientific Computing Seminar
Abstract: Standard Galerkin finite element discretizations applied to advection- dominated, elliptic PDEs can lead to highly oscillatory solutions, unless the grid is sufficiently fine. Over the years a number of stabilized methods, such as streamline upwind/Petrov Galerkin (SUPG) methods or Galerkin/Least-Squares (GALS) methods, were developed. These methods are frequently applied to advection-dominated elliptic PDEs. Local and global error estimates for these methods are well known. In this talk we are interested in local and global error estimates of the SUPG stabilized numerical solution of optimal control problems. We show that the discretization error for this method in the optimal control context behaves differently as a function of the mesh size than it does for scalar advection-dominated elliptic problems. This is true even for local error estimates in the regions of smoothness, away from interior or boundary layers. We will provide error estimates for the computed solution of the optimal control problem and we will present numerical results to illustrate our findings. This talk is based on joint work with M. Heinkenschloss, Rice University.
PDE Seminar
Abstract: In the early ninetees Scheffer produced a complicated example of a nontrivial weak solution to the incompressible Euler equations, hav- ing compactly supported in space and time. Subsequent papers by Shnirelman produced other examples of quite irregular solutions by different, yet complicated, methods. In a recent joint work with L´aszl´o Sz´ekelyhidi we have used a suit- able "h-principle" to produce solutions with the same behavior in a relatively simple way. Our approach answers to further questions left open by the works of Scheffer and Shnirelman and might be relevant in understanding a long?standing conjecture of Onsager. The same kind of analysis has supripising applications also to the theory of hyperbolic systems of conservation laws and shares some striking similarities with the theory of fully developped turbulence.