Geometric methods play a central role in dynamical systems and its applications. To showcase recent theoretical breakthroughs and to outline challenges and opportunities for applications, in particular in areas in which dynamical systems have not been utilized much, a conference on geometric methods in infinite-dimensional dynamical systems will be held November 4-6, 2011, at Brown University. The conference will be centered around four mathematical themes: data assimilation, geometric singular perturbation theory, nonlinear optics, and traveling waves. There will be around 15 long and 10 short talks. Several of the longer talks and at least half of the short talks will be given by younger participants. A poster session will give other younger participants the opportunity to present their results, and a future-directions panel discussion will feature a broad-scale discussion of the challenging theoretical questions and applied problems involving infinite-dimensional dynamical systems.
Geometric methods are essential for the development of mathematical tools and for applications in areas such as oceanography, data assimilation, neuroscience, nonlinear optics, and pattern formation. At present, new geometric methods are being developed to expand our knowledge of multi-dimensional waves, pulses propagating in structured media, the dynamics of defects and other localized patterns, and in the dynamics on neuron populations. Another emerging challenge to dynamical systems theory is to develop mathematical techniques for data assimilation, which is used in weather forecasts and climate research. By advertising these opportunities to younger and senior participants alike, this conference could stimulate additional research in this field and its applications.
The conference `Geometric Methods for Infinite-Dimensional Dynamical Systems' was held November 4-6, 2011 at Brown University. It was attended by 81 participants, most of whom came from the US. The scientific program consisted of 14 longer plenary talks and 8 shorter topical talks that were held in the lecture hall of the new Institute for Computational and Experimental Research in Mathematics (ICERM) at Brown University. In addition, a poster session and a panel discussion about open problems and important mathematical techniques were held. Overview of conference themes and topics: The plenary and topical talks formed the heart of the conference. They featured the four mathematical themes of the conference: data assimilation, geometric singular perturbation theory, nonlinear optics, and traveling waves. The goal was to showcase recent theoretical breakthroughs and to outline challenges and opportunities for applications, in particular in areas in which dynamical systems have not been utilized much. By advertising these opportunities to younger and senior participants alike, this conference should stimulate additional research in this exciting field. Data assimilation: Climate predictions rely crucially on mathematical models, both for small-scale weather forecasts and for long-term ocean-atmosphere predictions. Typically, these models are based on mathematical equations for the relevant physical variables, such as pressure, temperature, precipitation, and fluid velocity fluids. Data assimilation is the key process by which past and current observations are incorporated into mathematical models, typically by adapting previously obtained initial data. Most often, observed data are sparsely distributed in space and time. Hence, these data need to be extrapolated to improve on or to generate initial conditions that are defined everywhere on the numerical mesh. Given the uncertainty in both the observations and the model, the focus of recent studies has been on the initialization and forward-propagation of probability distributions. The invited speakers discussed successful, recently-developed approaches to data assimilation using Lagrangian- and Bayesian-based techniques. Geometric singular perturbation theory: Dynamical systems with multiple time scales often result in singularly perturbed differential equations. Geometric methods have proved extremely useful in analyzing such systems. The theory to describe the dynamics near and between normally hyperbolic slow manifolds has basically been completed. Over the course of the past decade, the focus in this field has shifted to applications and to systems in which the invariant manifolds fail to be normally hyperbolic. The invited speakers addressed applications to problems in mathematical biology, focusing on action potential propagation in neuroscience and the dynamics of neuronal networks, and discussed extensions to non-hyperbolic invariant manifolds. Nonlinear Optics: Wave propagation in waveguide arrays, photonic crystals, dispersion-managed fibers, and mode-locked fiber lasers continue to be of significant interest in the nonlinear optics community, as does the generation of stable ultra-short waves. These problems are often tackled successfully using dynamical-systems methods. The speakers discussed recent results in this area, and specifically on optical pulse propagation in waveguide gratings, the effects of noise on light localization in nonlinear fiber arrays, collision of optical pulses along fibers, collision-induced timing jitter, and ultra-short waves. Traveling Waves: The existence, stability, and dynamics of traveling waves continues to be a highly-active research area. Over the past years, the focus has shifted from nonlinear waves of partial differential equations (PDEs) posed in one spatial dimension to discrete and nonlocal models, as well as to equations that are posed on multi-dimensional domains. The speakers discussed recent progress on multi-dimensional or nonlocal systems as well as applications to biological and physical problems.
