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"Fredholm property and global bifurcation of quasilinear elliptic systems"

A powerful tool to understand the global structure of the solution sets of nonlinear elliptic systems is Rabinowitz's global bifurcation theorem. As people with experience in this know, the first thing to do is to convert the PDEs into a nonlinear functional equation whose principal part is a "compact perturbation of the identity" in a function space. When the elliptic system has boundary conditions that are either nonlinear or involve the bifurcation parameter, this conversion part is often cumbersome.

For nonlinear Fredholm mappings with zero index, Fitzpatrick, Pejsachowicz and Rabier have recently established an abstract theory, in particular, a global bifurcation result that allows us to tackle the elliptic system directly. However, to apply this theory, we need to check if the Fredholm index of the linearized elliptic system and the boundary condition is zero. We prove this for elliptic systems that satisfy Agmon's condition. We also prove an abstract "unilateral global bifurcation" result in the new framework that is needed when we study positive solutions of reaction-diffusion systems. We have thus built a bridge between the abstract theory and applications. I will end this talk with two examples: a chemotactic diffusion system with a nonlinear boundary condition, and a cross diffusion model.

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"Quasi-periodic breathers in Hamiltonian networks"

Hamiltonian networks form an  important class of infinite dimensional Hamiltonian  systems arising in solid state physics, cell biology,  and many other areas of science and technology. They also arise naturally in the discretization of Hamiltonian PDEs but the physical interest in Hamiltonian networks mainly lies in dynamics which are far away from those of Hamiltonian PDEs. Among interesting dynamics of a Hamiltonian network, of physical importance is a robust coherent structure known as breathers or quasi-periodic breathers which are self-localized, time periodic or quasi-periodic solutions.  In this lecture, several models of Hamiltonian networks of long-range, weakly coupled anharmonic oscillators will be considered. It will be shown that  corresponding to any fixed number of sites in such a Hamiltonian network, there is a positive Lebesgue measure set of linear stable, quasi-periodic breathers with the number of oscillating frequencies equal to the number of excited sites.

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"Paradigm Shifts in Science Based Simulations"

Today's scientific world is experiencing a paradigm shift where the sophistication of mathematical models, the accuracy and efficiency of numerical algorithms, the robustness of computer software, and the power of computation have become so great that numerical simulations are now considered a third pillar, along with theory and experiment, in the triad of tools used for scientific discovery.  The rate of advances in these fields, and our ability to simulate complex physical systems, will increasingly be the limiting factors in our ability to solve many of our most pressing scientific challenges.  I will describe recent advances in mathematical models, numerical algorithms, software, and hardware that have allowed computer simulations of complex multidisciplinary problems to have unprecedented impact in guiding scientific discoveries and policy decisions with global consequences. 

Gibson Hall 310

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