Wednesdays at 4 PM in Eckhart 202.
The management of dispersion in optical fibers has emerged as a key technique in stabilizing pulse propagation. A mathematical approach is developed here that exposes the mechanism underlying this striking effect. Is shown to be effective in surprising cases such as critical nonlinearities, random dispersion in the media, managed higher order dispersion and negative residual dispersion.
In 1952 Turing's paper on the chemical basis of morphogenesis initiated an important approach to the mathematical analysis of spontaneous pattern formation. In 1973 Wilson and Cowan introduced a similar formulation in nets of interacting neurons and in 1979 Ermentrout and Cowan developed the mathematical analysis of such nets using local bifurcation theory and symmetry groups. Recently Bressloff, Cowan, Golubitsky, Thomas and Wiener have further developed this approach to characterize and analyze some of the circuitry of the primate visual cortex. So far the symmetry group used is E2) X Z2 under a novel rotation action. Such an action is related to the fact that the visual cortex is a network of oriented edge detectors. However it is clear that much more than the orientation of a local edge is detected in the visual cortex: movement, texture and surface information, color and depth, for example. In this talk I will describe our approach and how to extend it to more complex symmetry groups that incorporate some of these features, and apply the analysis to two topics: (a) how geometric visual hallucinations are formed, and (b) how the cortex responds to external stimuli.
Coupled networks of systems of differential equations are models for a variety of physical and biological systems (such as animal gaits and the beating pattern of the leech heart). One important feature of such networks is that they support solutions in which some components vary synchronously and some with well defined time lags (spatio-temporal symmetries). In this lecture we build on the example of animal gaits and discuss the mathematics of spatio-temporal symmetries and how these solutions arise in coupled cell networks. We describe a theorem that gives necessary and sufficient conditions for a given network to support solutions with given spatio-temporal symmetries. We end with a discussion of the beating of the leech heart that when interpreted naively seems to contradict the theory --- but in fact leads to interesting mathematical questions (and answers).
Dynamical systems on infinite dimensional spaces satisfying conditions compatible with those of dissipative parabolic PDEs such as the 2-D Navier-Stokes system are considered. These systems are subjected to random forcing. Results on invariant measures and their mixing properties will be discussed. Elementary proofs will be presented.
In 1883, while studying and modeling the atmomic structure J.J.\ Thomson investigated the {\it linear} stability of corotating point vortices in the plane (see J.J. Thomson , ``A Treatise of the Motion of Vortex Rings'', Macmillian (1883), pag 94-108). In particular, his interest was in configurations of identical vortices equally spaced along the circumference of a circle, i.e. located at the vertices of a regular polygon. He proved that for six of fewer vortices the polygonal configurations are stable, while for seven vortices -- the Thomson heptagon -- he erroneously concluded that the configuration is slightly unstable (Morikawa $\&$ Swenson (1971)). It took over slightly more than a century to make some progresses on this problem! In his PhD thesis (Princeton, 1985), D.G.\ Dritschel proved that the Thomson heptagon is neutrally stable and that for eight or more vortices the corresponding polygonal configurations are linearly unstable. Recently (1999) H.E.\ Cabral and D.S.\ Schmidt proved that for seven or fewer vortices the polygonal configurations are {\it non-linearly} stable in the plane.\\ For the spherical case the results are much more recent! In 1993 D.G. Dritschel and L.M. Polvani determined the ranges of linear stability -- in terms of the latitude-- of polygonal configurations. By a similar method to the one used by Dritschel in the planar case, Dritschel and Polvani showed that at the pole, for $N<7$ the configuration is stable, for $N=7$ it is neutrally stable and for $N>7$ it is unstable. In 1998 J.E.\ Marsden and S.\ Pekarsky proved that for $N=3$ the range of non linear stability is the whole sphere (they also had stability results for vortices with different vorticities $k_1$, $k_2$ and $k_3$). H.E.\ Cabral and myself (2001) determined the ranges of non-linear stability for all $N$.