# CAMP/Nonlinear PDE's Seminar

Wednesdays at 4 PM in Eckhart 202.

## Fall 2001 Schedule

October 3,
Christopher Jones, Brown University
Optical Pulse Propagation and Averaged Variational Principles

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.

October 10,
Jack Cowan, University of Chicago
Towards a mathematically tractable model of the action of the visual cortex

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.

October 16, Tuesday 3pm in Ryerson 251
Andrea Bertozzi, Duke University
Navier-Stokes, fluid dynamics, and image and video inpainting

October 22, Monday 3:45pm in Eckhart 308
Edward Spiegel, Columbia University
On First Looking into Chapman and Cowling

October 31,
Martin Golubitsky, University of Houston
Coupled Oscillators and Symmetry

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).

November 7,
Lai-Sang Young, New York University
Ergodic theory of randomly forced dynamical systems with application to certain PDEs

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.

November 14,
Stefanella Boatto, Institute of Celestial Mechanics, Paris,
Dynamics of point vortices in the plan and on the sphere: the non-linear stability of relative equilibria

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 $N$.

November 21,
TBA, TBA University,
TBA

November 28,
Vadim Kaloshin, MIT and AIM
Dynamics of an oil spill

We consider an oil spill $\Omega$ on the surface of ocean $R^2$. Ocean evolves in time and the ultimate goal is to remove the spill $\Omega_t$ after time $t>0$ from the surface. The problem gives rise to various questions about size and shape of the spill $\Omega_t$. We model motion of an oil spot by a stochastic differential equation driven by a finite-dimensional Brownian motion. Let $\Omega \subset R^2$ be an open set and let $\Omega_t$ evolves according to that equation. Then $\cdot$ Central Limit Theorem (CLT) for distribution of the spill holds true. This, in particular, says that for any $t>0$ and some large $C>0$ with probability 99\% the ball of radius $C \sqrt{t}$ contains 99\% of the spill $\Omega_t$. $\cdot$ Eventhough most of the points escape to infinity with speed $~ \sqrt{t}$ diameter of the spill $d(\Omega_t)$ grows linearly in time almost surely (Cranston-Scheutzow-Steinsaltz). $\cdot$ (Shape Theorem) Define the set of poison points by time $t$ by $W_t=\{x\in R^2: d(x,\Omega_s)<1\ for\ 0 0$ almost surely we have $$(1-\epsilon) t B \subset W_t \subset (1+\epsilon) t B.$$ This is a joint work with D. Dolgopyat and L. Koralov.

December 5,
Roger Temam, Indiana University
Mathematical Problems in Meteorology and Oceanography

In this lecture we will present some recent and some less recent mathematical results concerning the equations of the atmosphere, the ocean and the coupled atmosphere ocean (the so-called Primitive Equations first considered by Richardson)

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