Past

It became apparent during the last decade that in several questions in classical complex analysis extremal configurations are fractal, making them very difficult to attack: it is not even clear how to construct or describe extremal objects. We will argue that the most promising approach is to consider conformally self-similar random configurations, which should be extremal almost surely.

Fragmentation processes model the evolution of a particle that split as time goes on. When small particles split fast enough, the fragmentation is intensive and the initial mass is reduced to dust in finite time. We encode such fragmentation into a continuum random tree (CRT) in the sense of Aldous. When the splitting times are dense near 0, the fragmentation CRT is in turn encoded into a continuous (height) function. Under some mild hypotheses, we calculate the Hausdorff dimension of the CRT, as well as the maximal H

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Given a countable set of sites S an a transition matrix p(x,y) on that set, we consider a process of particles evolving on S according to the following rule: each particle waits an exponential time and then jumps following a Markov chain governed by p(x,y); the particle keeps jumping until it reaches an empty site where it remains for another exponential time. Unlike most interacting particle systems, this process fails to

have the Feller property. This causes several technical difficulties to study it. We present a method to prove that certain measures are invariant for the process and exploit the Kolmogorov zero or one law to study some of its unusual path properties.

According to the Stokes-Einstein law, microscopic particles subject to intense bombardment by (much smaller) gas molecules perform Brownian motion with a diffusivity inversely proportion to their radius. Smoluchowski, shortly after Einstein's account of Brownian motion, used this model to explain the behaviour of a cloud of such particles when, in addition their diffusive motion, they coagulate on collision. He wrote down a system of evolution equations for the densities of particles of each size, in particular identifying the collision rate as a function of particle size.

We give a rigorous derivation of (a spatially inhomogeneous generalization of) Smoluchowski's equations, as the limit of a sequence of Brownian particle systems with coagulation on collision. The equations are shown to have a unique, mass-preserving solution. A detailed limiting picture emerges describing the ancestral spatial tree of particles making up each particle in the current population. The limit is established at the level of these trees.

Spectral projections enjoy high order convergence for globally smooth functions. However, a single discontinuity introduces O(1) spurious oscillations near the discontinuity and reduces the high order convergence rate to first order, Gibbs' Phenomena. Although a direct expansion of the function in terms of its global moments yields this low order approximation, high resolution information is retained in the global moments. Two techniques for the resolution of the Gibbs' phenomenon are discussed, filtering and reprojection methods. An adaptive filter with optimal joint time-frequency localization is presented, which recovers a function from its N term Fourier projection within the error bound \exp(-Nd(x)), where d(x) is the distance from the point being recovered to the nearest discontinuity. Symmetric filtering, however, must sacrifice accuracy when approaching a discontinuity. To overcome this limitation, Gegenbauer postprocessing was introduced by Gottlieb, Shu, et al, which recovers a function from its N term Fourier projection within the error bound \exp(-N). An extension of Gegenbauer postprocessing with improved convergence and robustness properties is presented, the robust Gibbs complements. Filtering and reprojection methods will be put in a unifying framework, and their properties such as robustness and computational cost contrasted. This research was conducted jointly with Eitan Tadmor and Anne Gelb.

The use of weighted matchings is becoming increasingly standard in the

solution of sparse linear systems. While non-symmetric permutations based on these

matchings have been the state-of-the-art for

several years (especially for direct solvers), approaches for symmetric

matrices have only recently gained attention.

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In this talk we discuss results of our work on using weighted matchings in

the preconditioning of symmetric indefinite linear systems, following ideas

introduced by Duff and Gilbert. In order to maintain symmetry,

the weighted matching is symmetrized and the cycle structure of the

resulting matching is used to build reorderings that form small diagonal

blocks from the matched entries.

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For the preconditioning we investigated two approaches. One is an

incomplete $LDL^{T}$ preconditioning, that chooses 1x1 or 2x2 diagonal pivots

based on a simple tridiagonal pivoting criterion. The second approach

targets distributed computing, and is based on factorized sparse approximate

inverses, whose existence, in turn, is based on the existence of an $LDL^{T}$

factorization. Results for a number of comprehensive test sets are given,

including comparisons with sparse direct solvers and other preconditioning

approaches.

Coherent structures, or defects, are interfaces between wave trains with

possibly different wavenumbers: they are time-periodic in an appropriate

coordinate frame and connect two, possibly different, spatially-periodic

travelling waves. We propose a classification of defects into four

different classes which have all been observed experimentally. The

characteristic distinguishing these classes is the sign of the group

velocities of the wave trains to either side of the defect, measured

relative to the speed of the defect. Using a spatial-dynamics description

in which defects correspond to homoclinic and heteroclinic orbits, we then

relate robustness properties of defects to their spectral stability

properties. If time permits, we will also discuss how defects interact with

each other.

We describe a nice example of duality between coagulation and fragmentation associated with certain Dirichlet distributions. The fragmentation and coalescence chains we derive arise naturally in the context of the genealogy of Yule processes.

We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which don't, at present, incorporate all the competitive strategies that a population might adopt. The second is a simplification of the first in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching-annihilating random walk. For each model, using a comparison with N-dependent oriented percolation, we show that for certain parameter values both populations will coexist for all time with positive probability.

As a corollary we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates.

We also present conjectures relating to the role of space in the survival probabilities for the two populations.