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The theory of infinite dimensional dynamical systems is a vibrant field of mathematical development and has become central to the study of complex physical, biological, and societal processes. The most immediate examples of a theoretical nature are found in the interplay between invariant structures and the qualitative behavior of solutions to evolutionary partial differential equations (PDEs) of parabolic or hyperbolic types. Insight has also been gained from the theory of infinite dimensional dynamics into the solution structure for nonlinear elliptic equations, including those arising in geometry. Other important and general topics, besides PDEs and dynamics in abstract spaces, addressed by the theory of infinite dimensional dynamical systems, include delay differential equations, lattice dynamics, and evolutionary systems with spatially nonlocal interaction.

(1)  Dynamical Systems in Studies of Partial Differential Equations

The theory of partial differential equations (PDEs) is a broad research field, rapidly growing in close connections with other mathematical disciplines and applied sciences. In this workshop, connections between the theories of dynamical systems and PDEs will be explored from several points of view. Infinitedimensional dynamical systems generated by evolutionary PDEs provide the most immediate examples of interplay between the two theories. Extensions of well-established results and techniques from finitedimensional dynamical systems (invariant manifolds, bifurcations, KAM theory) have proved very useful in qualitative studies of PDEs. On the other hand, specific questions for PDEs brought about stimulating problems in the theory of dynamical systems, such as the existence of finitedimensional attractors and their behavior under (regular or singular) perturbations. Entire (or eternal) solutions, which emerged as key objects in these problems, have long served as organizing centers for qualitative investigations of dissipative evolutionary PDEs and they continue to play an important role in other modern approaches to PDEs (for example, entire solutions of spatially extended systems have been examined in connection with traveling waves, and classification of entire solutions for specific classes of PDEs have been shown to have important implications on the structure of singularities of solutions). In quite a different way, dynamical systems have been used for the investigation of solutions of PDEs, which are not originally set up as models of evolution phenomena. The spatial dynamics of elliptic equations on unbounded cylinders is an example of such an approach. The key underlying idea that interesting solutions can be found by studying ODEs on manifolds in the state space has been successfully applied in such problems and in many other PDEs. As it often happens in studies of evolutionary PDEs, in particular those on unbounded spatial domains, applications of standard results from dynamical systems may be hindered by obstacles, such as the presence of the essential spectrum of the linearized problem. Yet, even when standard results do not apply, the conclusions they would lead to can often be proved by other methods, like the renormalization (rescaling) techniques. Examples of such conclusions can be found in studies of blow up of solutions of parabolic equations. In these results, the role of dynamical systems is in the guideline they provide for the investigation. Rigorous computational approaches to PDEs will also form an important part of the workshop. We believe that a workshop focusing on these and related applications of dynamical systems in the theory of PDEs will provide a platform for an exciting exchange of ideas between specialists with different backgrounds in PDEs, dynamical systems, and their applications. It is our intention to have different backgrounds and approaches represented in the lectures and informal discussions, and we expect the workshop to stimulate new collaborations.

(2) Random Dynamical Systems

Random events occur in the physical world and throughout our everyday experiences. Taking stochastic effects into account is of central importance for the development of mathematical models of complex phenomena under uncertainty arising in applications. Macroscopic models in the form of differential equations for these systems contain randomness in many ways, such as stochastic forcing, uncertain parameters, random sources or inputs, and random initial and boundary conditions. The theory of random dynamical systems and stochastic differential equations provides fundamental ideas and tools for the modeling, analysis, and prediction of complex phenomena. Solutions to the important dynamical problems increasingly require techniques from several areas of mathematics. The research in these areas is becoming increasingly collaborative, especially crossing disciplines. In fact, rapid progress requires an organized collaborative effort like this workshop. It will serve as a venue for developing communication and establishing collaborative research among research groups. Theoretical development in nonlinear analysis, dynamics, and stochastic differential equations has been at the forefront of science progress in a vast number of areas. Environmental fluid dynamics, geophysical flows, climate dynamics, electronic and telecommunication systems, networks of neurons, genetics, physiology, and finance are just a few of the important fields being influenced. On the other hand, these areas of science and engineering are sources of interesting mathematical problems. This workshop will focus on nonlinear and stochastic dynamics with applications to models in fluid flow, nonlinear waves, nonlinear optics, telecommunication, and related fields. Much of the analysis employed is based on the dynamical systems approach to stochastic differential equations, which is currently a very fruitful area of research and this will be a significant aspect of the purely mathematical part of the program.

(3) Lattice and Nonlocal Dynamical Systems and Applications

Over the past two decades, lattice differential equations and nonlocal evolution equations have been widely studied, both for their interesting mathematical properties and because of the plethora of applications. For instance, one finds long-range interaction in polymeric science, quantum mechanics, neuroscience, genetic regulation, ecology, and image processing. Longrange interactions were included in models formulated by Van der Waals and others, but finding it more convenient to view the range of interaction as infinitesimal, one is lead to PDEs. On the other hand, many systems exhibit genuine longrange spatial and/or temporal interactions and ignoring them results in models poorly matching experimental evidence. Lattice differential equations naturally arise when one discretizes continuum models but they also arise in modeling physically discrete systems, such as interactions on a single strand of DNA, pulses along myelinated neuronal axons, waves in lattice gases, or dispersal or evolution in patchy media or environments, for instance. Since lattice differential equations and nonlocal evolution equations can be cast as infinitedimensional dynamical systems, this workshop will bring together mathematicians having expertise in different theoretical aspects of this field and scientists with a deeper physical understanding of the applications.