
Asymptotic and Multiple scale analysis
The major defining challenges in almost all areas of modern science 
whether we consider the physical, the biological or even the social sciences 
require reliable methods for analyzing how macroscopic structures can emerge
from microscopic dynamics.
There are fundamental mathematical questions, in the disciplines of multiscale
analysis and asymptotic analysis, that remain unanswered; the ultimate goal being to understand the emergence of large scale order out of what appears to be incoherent microscopic dynamics or randomness. Such questions motivate many of the core areas of research in Applied Analysis at Arizona.
Topics that we are investigating include envelope descriptions of nonlinear wave phenomena in fluids and optics, as well as what happens at the ultrashort or highly oscillatory extremes, when such descriptions start to break down; nanoscale growth and fabrication processes that can produce materials with remarkable and useful properties; the universality of pattern formation and defect nucleation in physical processes, both near and far from critical thresholds; the striking collective behaviors of organisms ranging from microscopic creatures such as bacteria to flocks of birds or schools of fish.
At the same time, by studying the asymptotic behavior on the large scale of more purely mathematical assemblies such as random tilings and networks, constrained equilibria of charges, equilibria of thin elastic object, or interface growth models, we discover new analytical tools that bring into play rich ideas from geometry, representation theory, statistics, number theory and numerical analysis. These form the basis of the new applied mathematics needed for the scientific challenges of the 21^{st} Century.

Ultraintense shortpulse lasers
A major focus of the research in Nonlinear waves at the University of Arizona is on ultraintense, short laser pulse propagation phenomena in condensed media and in the atmosphere. Recently, we won a major instrumentation award to buy our own high intensity femtosecond laser  this lab will be located in the College of Optical Sciences and will provide the first experimental facility where experiments with ultraintense short laser pulses will be guided by theory and simulation.
These phenomena occur in an extremely nonlinear regime, and we have to develop new models that go beyond the standard Nonlinear Schrödinger equation to describe them. The figure shows a billboard in Como, Italy depicting the experimental results of our collaborators on the Nonlinear Xwave. Beside it we have the results of our modeling and simulations of this phenomenon.
This and other fascinating behaviors in (short and intense) light beams may be understood with the tools of the modern applied mathematician, namely analysis, modeling, simulations and experiments.

Patterns in nature and in the laboratory
You can see patterns all over the place. Look at the epidermal ridges on your fingertips on which your fingerprints are encoded. These features, each unique in its spatial distribution of ridge ends and beginnings, called defects, were laid down while you were in your first ten to seventeen weeks as a fetus! Have you ever looked closely at how the individual buds are arranged on a sunflower or a cactus? On a cactus they are far less flower like. They are sticker spines. But these plants share in common the fact that these flowers and stickers, called phylla, almost always lie on families of spirals and the numbers in each family all belong to the famous Fibonacci sequence.
Have you observed how the surfaces of plants are deformed into hexagonal (many cacti) or diamond shaped (the bracts on a pinecone) tilings? Or when you were on a long sandy beach washed by the ebb and flow of tide and waves, have you noticed the sandripples and wondered why it is that the pattern of sandripples is not at all unlike the patterns of epidermal ridges? Or have you looked skyward on a fine day or earthward from the airplane window and seen streets of cirrus clouds? At the zoo you have seen the coats of leopards whose spots form hexagonal patterns and those lucky enough to explore the undersurface of the sea have swum with fish whose skins are decorated with stripes and square and hexagonal patterns.
The reason we see such similarity in patterns arising in so many different contexts is that the equations describing all these different situations have all their most important properties in common. We usually give that kind of property the name of symmetry. It is a rather wonderful feeling to gain understanding of things you see around you by expressing them in mathematical clothing which reveals to you their inner workings.
A wide range of patterns and their defects can be described in terms of simple nonlinear partial differential equations (PDEs). For those of you familiar with dynamical systems, you can think of these equations as the PDE equivalent of normal forms. Their simplicity often makes them tractable from an analytical point of view. In particular, the nature and stability not only of patterns but also of a variety of coherent structures (fronts, pulses, defects) is often understood by studying these equations. This is an area where the use of analytical tools and methods of applied mathematics combines with numerical simulations of PDEs. Computer experiments lead to an intuitive understanding of the dynamics of nonlinear patternforming systems and guide the analytical investigation of these phenomena.
You might think that all that is meant to be discovered about such patterns has already been discovered. That is far from true. We do not fully understand what makes the phylla arrangements on plants. We do not fully understand fingerprints. We do not understand much at all the wonderful patterns we are seeing in materials at nanoscales, the patterns in light (yes, there are fascinating patterns in light beams!). Much work also remains to be done on understanding how simple patterns can destabilize into structures that are disorganized both in space and in time, and on the dynamics of such "turbulent" states. What is grand altogether about the subject is that it usually involves situations you can readily imagine, even do experiments on, to check whether your ideas are right and whether you are really gaining understanding. Come join us in the grand adventure of teasing out the mysteries of patterns and of their applications to physics, mechanics, chemistry and biology.
Nanotechnology
"There's plenty of room at the bottom"  In a famous talk at the American Physical Society meeting in 1959, Richard Feynman speculated about the potential applications of miniaturization. Even he did not foresee what recent advances in nanofabrication have achieved.
It is now possible to artificially engineer structures with electromagnetic properties that don't exist in nature, for instance, materials with a negative index of refraction!
These materials open up fundamentally new regimes of lightmatter interaction, and unique opportunities for applications. Among potential applications are subwavelength imaging, cloaking (invisibility) and complex light manipulations.
In general, light scattering from objects can be represented as a sum of near and far fields. The near field carries information about fine structure of the object which can be smaller than the light's wavelength. The far field carries information on the object geometry at scales greater than that wavelength. The near field decays exponentially with the distance from the object, therefore information about the object which is at scales below the wavelength cannot be resolved using conventional optic devices. Negative refractive index materials can transmit near fields over large distances without degradation. The properties of these artificial materials are very counterintuitive: Snell´s law and Doppler´s shift have the "wrong" sign, phase velocity and transport of energy are in opposite directions. This gives new opportunities for manipulation of light. Even more control of refractive index may allow one to achieve invisibility for objects covered with such artificial materials.
In the Department of Mathematics at the University of Arizona we are developing mathematical models of light interaction with such materials as well as studying various potential applications. Another direction of research interests is mathematical modeling of the properties of the smallest artificial materials: carbon nanotubes. These carbon nanotubes have unique properties, they are much stronger than conventional materials, they have better conductivity than silver and gold and they are very light. The modeling of their electric and mechanical properties as well as properties of materials assembled from nanotubes is a great challenge, requiring that we take into account both the classical and the quantum nature of these objects.
Estimation of eigenvalues
A classical approach to nonlinear problems involving equilibria is to first
linearize and analyze the linearization. The linearization often results in
an eigenvalue problem, as is the case in vibrations of mechanical systems,
deformation of structures, and various "exchange of stabilities" or
"stability/instability" problems in fluid mechanics. A change of sign (from negative to positive)
in the real part of an eigenvalue signals an instability, and it is such transitions that we want to identify.
The most common means of approximating these eigenvalues is the RayleighRitz
method in which one picks "trial vectors" to replace a differential problem by
a generalized matrix eigenvalue problem. This technique is commonly implemented
via a spectral or finite element method, or in the high dimensional problems of
atomic physics by the HartreeFock method of trial vector selection. Accurate
and reliable techniques for the solution of the resulting spectral problems
have been widely studied.
But use of the approximate eigenvalueeigenvector pair to estimate other quantities
of physical interest, such as local construction of a bifurcation curve, does not
immediately follow. The problem is that if the approximation of the eigenvector only converges
weakly, then the relevant nonlinear functionals are no longer continuous with respect to this
weak convergence.
Our work on remedying this is well underway in the case of self adjoint linearizations,
and this is also useful for obtaining error estimates for RayleighRitz calculations by
leading to complementary bounds. Future work will
involve extensions to the important case of linearizations that are not self adjoint.

Geometric Asymptotics
Our research group represents a wide range of interests related to spectral theory and its applications.
Rather recently there has been a flurry of research activity centered around the study of random geometry or random transformations.
Examples of the kinds of questions that come up in these studies are,
 How many (topologically distinct) graphs can be constructed on a surface of genus g with n vertices each having a specified valence?
 What is asymptotic "shape" of a generic partition of N as N becomes arbitrarily large?
 How can one predict the appearance of order versus disorder in large random tilings of planar domains?
A classical result of Hermann Weyl says that eigenvalues of the Laplacian in a domain,
on a surface, or on a more general manifold behave asymptotically according to the power law.
A natural question is how they deviate from the power law. Are they distributed more or less
uniformly or do they come in clusters? Differences between eigenvalues are of special importance
because they are related to radiation frequencies in quantum systems. It turns out
that the answers depend on dynamic properties of the geodesics flow (billiards in the
case of domains with boundary.) There are many open questions in this area of research.
One of the approaches that has been pursued recently is to study operators on graphs
("quantum graphs".)