By Roberto Camassa, University of North Carolina at Chapel Hill

The interplay between structures in relative motion with respect to an ambient fluid and density stratifications, including such extreme cases as free surfaces, are a constant source of interesting physical an mathematical phenomena. This talk will outline examples in this area of research involving a combination of analytical, numerical and experimental techniques, stemming from interdisciplinary projects centered around the facilities of the UNC Joint Fluids Lab in collaboration with science departments at UNC.

By Richard Mclaughlin, University of North Carolina at Chapel Hill

The behavior fluids is greatly complicated by the presence of stratification:  Bodies and jets may move ambient fluids into regions creating buoyancy forces which either are mitigated by mixing or create strong flows.  We overview some of our theoretical, computational, and experimental studies of the motion of bodies and buoyant fluids moving through a stratified background density field, focusing on the vertical transport.  Interesting critical phenomena are observed in which bodies and buoyant fluids may either escape or be trapped as parameters (such as the propagation distance) are varied.  Two models are discussed:  First, we discuss the Morton-Taylor-Turner model for which we (Camassa, McLaughlin, Tzou) have rigorously established, within this hierarchy, the optimal mixer within a broad class of ambient stratifications.  Second, we examine wall induced mixing in the evolution of passive scalar skewness in channels and ducts, joint work with Aminian, Bernardi, and Camassa.

By Nick Ercolani, University of Arizona

Methods of nonlinear steepest descent have proved effective in developing normal form for the propogation of oscillations in a nonlinear environment. This talk will focus on the birth of such oscillations within the framework of random matrix theory and its applications to generating functions for random combinatorial structures relating to older perspectives on dispersive limits of integrable PDE such as the nonlinear Schrodinger equation. 

By Jared Bronski, University of Illinois at Urbana-Champaign

There are many situations where it is necessary to count the number of negative eigenvalues of some operator, for instance if one is trying to compute the dimension of an unstable manifold to some solution. Often these kind of counting problems lend themselves to a geometric or topological approach. We discuss some problems of this type that have come up in the authors work in recent years.

By Kenneth McLaughlin, University of Arizona

The goal is to understand the asymptotic behavior of integrable nonlinear PDEs in 2 spatial dimensions, and the sticking point is semi-classical analysis of Complex Geometrical Optics solutions of d-bar problems.

By Peter Miller, University of Michigan

In a 20-year period beginning in the early 1970's, Dave McLaughlin's career trajectory intersected with a vibrant team effort at the University of Arizona to understand nonlinear wave propagation in the setting of integrable equations and their perturbations.  Many papers generated from this group have continued to inspire new work on some old problems.  This talk will highlight some of the contributions from this team and some related recent work joint with Robert Buckingham on the semiclassical limit for the sine-Gordon equation.

By Roy Goodman, New Jersey Institute of Technology

We will talk about two physical systems where the dynamics, though infinite-dimensional, is well approximated by a small Hamiltonian system of ODE. The first is the nonlinear Schrödinger/Gross-Pitaevskii equation with a multiple-well localized potential. Here we show how how Hamiltonian Hopf bifurcations lead to novel quasiperiodic orbits and chaotic dynamics. The second is vortex interactions in a two-dimensional Bose-Einstein condensate, where we find homoclinic chaos leading to stochastic reversals in the direction of rotation of a pair of rotating vortices.

By Constance Schober, University of Central Florida

Breather type (homoclinic) solutions of the Nonlinear Schrodinger equation have been widely used as models for rogue waves. In this talk we examine the generation of rogue waves for random sea states characterized by JONSWAP spectra using an approach based on the NLS equation and its inverse spectral theory. We introduce a spectral quantity, the ``splitting distance'' between consecutive simple points in the associated discrete Floquet spectrum. Using the splitting distance we correlate the development of rogue waves for JONSWAP data with the proximity in spectral space to instabilities and homoclinic data of the NLS equation.

By Annalisa Calini, College of Charleston

The Vortex Filament Equation or VFE, a model for the self-induced motion of a vortex filament in an ideal fluid, is a simple but important example of integrable geometric evolution equation for space curves. Its connection with the cubing focusing Nonlinear Schrödinger equation through the well-known Hasimoto transformation allows the use of many of the tools of soliton theory to construct and investigate finite-gap and soliton solutions.  I will discuss linear and nonlinear stability properties of some of these solutions, including filaments in the shape of torus knots and solitons on vortex filaments. This is joint work with with Tom Ivey and Stephane Lafortune.

By Alan Newell, University of Arizona

Man's curiosity and fascination with the wonderful architectures seen near the shoot apical meristems (SAM's) of plants goes back almost two thousand years. Renaissance scientists such as Kepler and daVinci were intrigued and Kepler was one of the first to note that spiral plant patterns had connections with Fibonacci sequences. It is remarkable, despite continued interest over the centuries, that only recently have quantitative explanations emerged which enjoy broad acceptance. In this talk, I will review the progress to date, and discuss both teleological explanations based upon the observations of Hofmeister and encoded in the works of Douady and Couder, and mechanistic explanations which seek to model the relevant biochemistry and mechanics at work near the SAM. The former approach argues that new phylla (flowers, seeds, bracts, etc.) are placed according to some optimization principle. The latter approach leads to instability driven pattern forming systems in which either the plant growth hormone auxin field or the local stress field have quasiperiodic structures at whose maxima new phylla are likely to be initiated. Whereas the latter model is richer than the former in that one obtains field rather than configuration information and because it addresses the connection between phyllotactic configurations and surface morphologies, one of the stunning and surprising outcomes of our work is that both approaches lead to absolutely consistent outcomes. It may very well be that nature employs pattern forming systems to achieve optimal outcomes not just in plants but in many organisms.

By Greg Forest, University of North Carolina at Chapel Hill