Departmental Colloquium

I will discuss three interesting and related problems in the dynamics of complex fluids at low Reynolds number. The first concerns how visco-elasticity affects the peristaltic pumping of fluids -- which arises in many biological settings -- especially as related to loss of flow reversibility associated with the Stokes equations. The second seeks to understand, in a relatively simple setting, recent observations of complex flow structures and efficient mixing in low Reynolds number visco-elastic flows. And the third studies the dynamics of an elastic fiber moving in a simple cellular flow. Here the fiber can act as a spatially extended test particle whose internal dynamics, particularly buckling at hyperbolic points, leading to complex transport properties across space.

Einstein metrics are expected to provide the optimal connection between the local geometry and the global properties of a manifold. After an introduction, the talk describes the construction of new Einstein metrics on odd dimensional spheres using complex algebraic orbifolds.

The asymptotic cone of a metric space X is the Gromov-Hausdorff limit of rescaled copies X/n of X, n=1,2,.... They were introduced by Gromov in his celebrated paper about groups of polynomial growth. Every finitely generated group becomes a metric space if we equip it with the word metric. Asymptotic cones of a group capture both algorithmic and geometric (coarse) properties of the group. Asymptotic cones of a group also provide information about the subgroups of the group, solving equations over the group, etc. I am going to survey recent results about asymptotic cones obtained jointly with Cornelia Drutu, Shahar Mozes, Alexander Olshanskii and Denis Osin.

Everyone knows how to integrate However it is impossible to integrate using only elementary functions. In this talk we will show that the reason for this is due to the geometry of the underlying algebraic curve, most especially the meromorphic forms on the curve. We will then show how the behaviour of the holomorphic forms on any algebraic curve controls the geometry, differential geometry, topology, and arithmetic of the curve. Finally we will explain how recent progress in higher dimensional geometry shows that the relatively simple picture for curves carries over to birtional geometry in higher dimensions.

Everyone knows how to integrate...

What does the local differential geometry of a subvariety of projective space tell us about its global geometry? I will survey results on the rigidity of homogeneous varieties and the Hwang-Mok program for Fano manifolds in the projective setting.

Unlike classical telecommunication, whose fundamental principles can be described via commutative harmonic analysis, the analysis and design of advanced wireless communication systems requires methods from noncommmutative harmonic analysis. We will see that the modeling of the mobile radio channel leads to interesting problems in pseudodifferential operator theory. Their treatments lies outside the standard theory of PDEs, and requires tools from time-frequency analysis and Banach algebra theory. For instance, I will show that Gabor systems provide an optimally sparse representation of these non-standard pseudodifferential operators. These results can even be generalized to the abstract setting of locally compact abelian groups. Finally, I will show how the theoretical framework presented in this talk leads to powerful numerical algorithms for designing multicarrier communication systems, which has already been utilized in practice. -->

Metamaterials with strong dispersion that can be tailored, magnetic properties from engineered structures, negative permittivity and/or permeability and negative refractive index opened up an exciting research field less than a decade ago when Veselago's long forgotten paper on negative property electromagnetic materials was rediscovered. Applications such as super lenses that can overcome the diffraction barrier, near field focusing, flat lenses and the ability to 'cloak' or hide objects give rise to visions of a 21st century emperor's new clothes. This seminar is on experimental studies of metamaterials fabricated with printed circuit board and LTCC techniques and explores some of the intriguing applications that have been proposed for metamaterials. A brief introduction will be given on a free space, focused microwave beam measurement system that mimics free field conditions on a table top and permits the use of small samples. Some well known and less well known experimental results on the microwave properties (5-110 GHz) of such materials will be presented. The current debate regarding the allowed sign of the imaginary part of complex electromagnetic properties in a no gain material will be discussed in the context of causality and energy conservation. Kramer-Kronig reconstruction of the imaginary part using just experimental data for the real part and vice versa will be shown. The measured anomalous dispersion of the complex permittivity and permeability is fitted with resonance models and it is shown that the properties are analytic functions in the upper half frequency plane and that the singularities are in the lower half plane even if the imaginary part is negative. It is also shown that the material is a net loss system independent of the sign of the imaginary parts for metamaterials and other biisotropic and bianisotropic materials described by both permittivity and permeability, independent of the constitutive model. Some recent experimental demonstrations of cloaking by a metamaterial as proposed by Milton and Nicorovici (2006) and earlier Nicorovici, McPhedran & Milton (1994) will help us conclude if indeed metamaterials will be used to stitch the emperor's new cloak. -->

In this talk we will first discuss positive-weight quadrature formulas for the sphere that the author and co-workers have recently developed. We will also present a new class of frames on the sphere; these frames use these quadrature formulas. Frames resemble wavelets in many ways, and in fact are used in connection with them. Because there is no uniform grid on the sphere, it is not possible to construct direct analogues of, say, the Daubechies wavelets. Various types of wavelets for the sphere have been introduced, but all of them suffer from defects, such as poor localization properties and difficult implementation. Concerning the frames we discuss here, there are several novel features. They are tight, which makes reconstruction simple, and they have excellent localization properties.

We give a mathematician's view of evolutionary biology literature concerning stochastic models for phylogenetic trees. We spotlight a model for the tree on n extant species that would be observed if macroevolution were purely random. The model can be extended in two ways -- to time series of observed taxa in a fossil record, and to different levels of the taxonomic hierarchy -- and provides a logically consistent (``coherent'') framework for simultaneous study thereof. We illustrate with a variety of theoretical calculations and simulations, and propose a variety of real-data projects suggested by our analysis. (Joint work with Lea Popovic and Maxim Krikun)