Applied Math Colloquium

Bilayer vesicles mimic the behavior and chemistry of real biomembranes, for example the plasma membranes surrounding cells and their constituent organelles. Their equilibrium conformations have been well studied, but our understanding of their dyanmics has remained comparatively primitive due to the lack of appropriate experimental techniques. Since real biological processes are usually far from equilibrium, examples of well-controlled dynamical phenomena in bilayers are especially interesting.

Recently Bar-Ziv and Moses discovered a dramatic, dynamical shape transformation induced in cylindrical lipid bilayer vesicles by the action of laser tweezers. We develop a hydrodynamic theory of fluid bilayers in interaction with the surrounding water to explain their video micrographs. Applying the marginal stability criterion to this situation gives us predictions for the selected initial wavelength and the propagation velocity, both in agreement with the experimental values. In particular we show that the instability initially propagates as a front at constant velocity, as observed. Finally we introduce an approximate hydrodynamic model applicable to the fully nonlinear regime. This model exhibits propagating fronts as well as fully-developed "pearled" vesicles similar to those seen in the experiments.

A model for the interaction of surface or internal waves and bottom sediment is proposed and analyzed. Numerical simulations are compared with field observations.

A variety of pattern recognition methods including polynomial discriminants and neural networks have a feature of universal statistical consistency for arbitrary joint distributions of inputs and outputs. Indeed, the probability of error of the estimated discriminant converges to the Bayes optimal probability of error in the limit of large sample size when the size of the model is chosen adaptively. An index of resolvability quantifies the rate of convergence in terms of a complexity versus approximation trade-off. Though the convergence is not uniform over all distributions, this index of resolvability quantifies interesting nonparametric classes for which the convergence is uniform at a polynomial rate. We discuss the relationship of these conclusions to computational learning theory.

A computer is generally considered to be a universal computational device; that is, it is believed able to simulate any physical computational device with a cost in computation time of at most a polynomial factor. It is not clear whether this is still true when quantum mechanics is taken into consideration. A quantum computer is a hypothetical machine which uses quantum mechanics to perform computations. We will give algorithms for prime factorization on a quantum computer that take a number of steps which is polynomial in the number of digits of the integer to be factored. This problem are generally considered hard on a classical computer and has been used as the basis of several proposed cryptosystems. So far, quantum computers are purely thought experiments; it is not clear whether it will be possible to build one. This result does show that simulating arbitrary quantum mecanical systems is at least as hard as factoring.

Experimental observations of an Isotropic -> Smectic/A (liquid crystal) phase transition show the rapid growth of slender, buckling filaments. This creates beautiful, space-filling patterns that apparently arise from the combination of filament elasticity and growth, with hydrodynamic self-interaction.

Accordingly, I have been studying the Stokes hydrodynamics of stretching buckling filaments. Our modelling combines slender-body theory with filament elasticity and tension, and gives an integro-differential system for the filament dynamics. Solving numerically such a system is very delicate, and brings forward issues of the interaction of asymptotics with constructing numerically tractable models. This is joint work with Ted Ueda (Courant).

Semidefinite programming refers to the optimization of a linear function over an affine subspace of the cone of positive semidefinite matrices. It generalizes linear programming, is a special case of convex optimization, and is closely related to eigenvalue problems. It has a powerful duality theory. Recently, semidefinite programming has been the focus of much attention, partly because of its applications to control theory and combinatorial optimization.

After presenting the scenery surrounding semidefinite programming, we will explore several areas where semidefinite programming has been particularly useful. We will start by visiting Lyapunov theory, John's minimum volume ellipsoids, and other areas, and then head towards more combinatorial applications, including the approximability of maximum cuts and Lovasz's seminal work on perfect graphs. During our journey, we will encounter several interesting combinatorial or geometric open questions.

Partially based on joint work with David Williamson, Uri Feige, and Jon Kleinberg.

Conjugate gradient methods and their relatives (such the minimum residual method) are widely used to solve large sparse linear systems and to regularize ill-posed problems. The usual approach to the analysis of these methods is to recognize that the successive approximate solutions are polynomials in the matrix of the system operating on the residual vector. Since the approximations satisfy an optimality condition, one can obtain upper bounds on the accuracy of the approximations by selecting polynomials that are nearly optimal. Unfortunately, the results are averaged over the spectrum of the matrix, so that it is difficult to see how the method converges along the individual eigenvectors.

In this talk we describe a new approach. Here the approximate solutions (in the coordinates of the eigensystem of the matrix) are expressed in terms of orthogonal projections involving certain modified Krylov subspaces. Convergence in a component is signaled by the approach of the corresponding row of the projection to a row of the identity matrix, and bounds can be given for the rate of convergence. This componentwise approach may be more suitable than the usual techniques for analyzing the behavior of the methods for ill-posed problems.

Recent experiments on single molecules of DNA and interpretive theory, will be used to illustrate how biomaterials permit one to test novel aspects of polymer statistical mechanics. Our current understanding of the physical structure of chromosomes will illustrate the challenges involved in doing any physical theory on a meaningful biological problem.

Boltzman's ergodic hypothesis underlies statistical mechanics and much of physical thinking. Yet in 1954, Kolmogorov announced that there are no ergodic Hamiltonian systems in a neighborhood of completely integrable ones. By contrast, in 1967, Anosov found the first open sets of ergodic systems i.e. stably ergodic diffeomorphisms and flows. These systems have the property that time averages converge to the same constant for almost every initial point, not only for the system itself but also for any sufficiently close perturbation. Anosov systems are uniformly hyperbolic while completely integrable systems have no hyperbolic behavior at all.

In this talk , we study the mixed situation which is only partially hyperbolic. Our main themes are first, that a little hyperbolicity goes a long way in guaranteeing stable ergodic behavior (which is more prevalent than one might have imagined) and secondly, that in fact it may be necessary for it. In both cases we use the accessibility property from control theory applied to the hyperbolic part of the derivative. We are led in this direction to formulate some new genericity conjectures.

In a sense, our main theorem may be interpreted to say that even for systems which are not uniformly hyperbolic, the same phenomenon which produces chaotic behavior i.e. some hyperbolicity may also guarantee robust statistics in the form of stable ergodicity.

Various parts of this talk represent joint work with Matt Grayson, Charles Pugh, Jonathan Brezin and Amie Wilkinson.