Colloquium index

Vigre seminars

Undergrad seminar

Clifford lectures

Sept 9

Prof. Jens Lorenz, Dept. of Math. & Stat., University of New Mexico

Topic: "The Role of the Pressure for Navier-Stokes Blow-Up"

Abstract:

In this talk I consider the blow-up question for the solutions of the Navier-Stokes equations, one of the millennium problems. After reviewing blow-up conditions for some simpler models, I will focus on the particular difficulties posed by the Navier-Stokes equations.

To analyze the equations, it is common to first eliminate the pressure by applying the Helmholtz projector. However, since pressure differences drive the flow, the pressure term is essential for understanding blow-up conditions. In this lecture, I will describe properties of the pressure, assuming that blow-up does occur.

Prof. Laszlo Lempert, Purdue University

Topic: "Complex approximation"

Abstract:
Originally, complex approximation was about approximating a holomorphic function, defined on an open subset of the complex plane, by polynomials. Subsequently one was led to consider more general problems: the function to be approximated had several, perhaps infinitely many variables, and the approximating functions could come from classes of functions other than polynomials.

In my talk I will describe one line of research, starting with Carl Runge's discovery from 1885 and leading up to recent results. I will also explain why approximations are in a sense inevitable in complex analysis/geometry.

Prof. Steven Cox , Rice University

Topic: "The Art and Science of Achieving Harmonics on Stringed Instruments"

Abstract:
One may elicit the q^th tone of a string by applying the 'correct touch' at one of its q-1 nodes during a simultaneous pluck or bow. This effect was first scored for violin by Mondonville in 1735. Though it captured the attention of the 19th century masters, Chladni, Tyndall, Helmholtz and Rayleigh, it remained for Bamberger, Rauch and Taylor in 1982 to develop and analyze the first mathematical model of harmonics Their 'touch' is a damper of magnitude b concentrated at the node p/q. The `correct touch' is that b for which the modes, that do not vanish at p/q, are maximally damped. We here examine the associated spectral problem. We find the spectrum to be periodic and determined by a polynomial of degree q-1. We establish lower and upper bounds on the spectral abscissa and show that the set of associated root vectors constitutes a Riesz basis in the natural energy space and so identify 'correct touch' with the b that minimizes the spectral abscissa.

Prof. Constantin Georgescu, Tulane University

Topic: "Using Auxiliary Information in Survey Sampling A Quantile Regression Approach"

Abstract
Improved estimates of a survey population parameter can be obtained by using information about co-related auxiliary variable. There is a huge literature on such methods for estimating the mean. In this talk we explore some median-versions of these mean-based methods.

Following a brief overview on quantile regression, the new quantile based estimator is introduced and some of its properties are examined. A proof for consistency of the marginal quantile estimator and for Bahadur-Expansion validity of the conditional quantile estimator is included. Then, a look by means of the conditional double exponential likelihood model reveals the quantile base estimator connections to some of the already classical estimators.

Some simulation results exploring the performance of the new estimator concludes the presentation.

Chris Jones , University of North Carolina

Topic: "Can viscosity and forcing in a fluid flow result in chaotic advection?"

Abstract
The Lagrangian (fluid particle) trajectories of a steady Euler flow in 2D are determined by an integrable system and hence exhibit no chaotic motion. The question then naturally arises as to whether a perturbed flow, which incorporates the physical effects of viscosity and forcing, can be chaotic. This problem is non-trivial since the perturbation is added at the level of the full partial differential equation but the potential chaos is at the level of the particle trajectories. It brings up many issues including long-time existence for 2D Navier-Stokes and finite-time Melnikov theory.

Dan Burns, University of Michigan

Topic: "Renormalized Chern Forms"

Abstract
Renormalized Chern forms are invariants of a complex manifold with strongly pseudoconvex boundary M. They give rise to numerical invariants because though X may have infinite (invariant) volume, the characteristic numbers are given by convergent integrals. Prompted by numerous examples in two complex dimensions, JS Ryu and the speaker have shown that if the boundary M is locally spherical (CR equivalent to the sphere), then these forms give rational cohomology classes. The methods of proof are analytic continuation and showing that monodromy eigenvalues are roots of unity, and a residue calculaton. Speculations on the real analogue, and analytic applications will also be discussed, as well as further applications of the analytic continuation technique.

Speaker, University

TBA

Speaker, University

TBA

John Mayer, University of Alabama at Birmingham

Topic: “Do shadows leave impressions?”

Abstract:
A simply-connected open set U in the plane R² can have a "nice" boundary (a circle, or homeomorphic image thereof) or a "nasty" one --- with lots of interesting possibilities (to a topologist) for "nasty". There is a theory, useful in plane topology and in dynamics of the complex plane, that views even nasty boundaries from the point of view of the nicest one, a circle. Imagine that you have a map of U with polar coordinates provided by the Riemann Mapping Theorem. That is, there is a complex analytic homeomorphism h from the open unit disk D (with nice radial rays and concentric circles as its coordinates) onto U (with images of those rays and circles to provide coordinates). You stand at the image h(0) in U of the center 0 of D and walk along an image of a radial ray toward the boundary of U. Assume the sun is at h(0) and its illuminating rays follow the radial coordinates that h imposes on U. You are thicker than a ray, so you cast a shadow on the boundary. Suppose this shadow gets narrower as you get further from the sun and closer to the boundary. If the boundary is "nice" (like a circle, say), then your limiting shadow is a point on the boundary --- surely a trivial impression! What is your limiting shadow if the boundary is "nasty" in the direction you are walking? Further progress on understanding the connected Julia sets of certain polynomials requires understanding shadows that leave nasty impressions.

Graeme Milton, Distinguished Professor and Chairman Department of Mathematics, University of Utah

Topic: "Composite materials: An old field of study full of new surprises"

Abstract
Composite materials have been studied for centuries, and have attracted the interest of reknown scientists such as Poisson, Faraday, Maxwell, Rayleigh, and Einstein. Their properties are usually not just a linear average of the properties of the constituent materials and can sometimes be strikingly different. The beautiful red glass one sees in old church windows is a suspension of small gold particles in glass. Sound waves travel slower in bubbly water than in either water or air. In the last few decades composites have been found to have some surprising properties. Most materials, such as rubber, get thinner when they are stretched, but it is possible to design composites which get fatter as they are stretched. Electromagnetic signals can travel faster in a composite than in the constituent phases. It is possible to combine materials which expand when heated to obtain a material which contracts when heated. It is still an open question as to what properties can be achieved when one mixes two or more materials with known properties. This lecture will survey some of the progress which has been made.

Prof. Lex Oversteegen, University of Alabama at Birmingham

Topic: "Complex Dynamics and Geometry"

Abstract:

THANKSGIVING HOLIDAY

Prof. Randy LeVeque , University of Washington,Seattle

Topic: "High-Resolution Finite Volume Methods for Modeling Volcanos and Tsunamis"

Abstract
Hyperbolic systems of partial differential equations often arise when modeling phenomena involving wave propagation or advective flow. Finite volume methods are a natural approach for conservation laws of this form since they are based directly on integral formulations and are applicable to problems involving shock waves and other discontinuities. High-resolution shock-capturing methods developed originally for compressible gas dynamics can also be applied to many other hyperbolic systems. A general formulation of these methods has been developed in the CLAWPACK software that allows application of these methods, with adaptive mesh refinement, to a variety of problems in fluid and solid dynamics.

I will describe these methods in the context of some recent work on modeling geophysical flow problems, particularly in the study of volcanos and tsunamis. Volcanos generate many challenging flow problems, and accurate simulation is required both to further scientific understanding and to aid in hazard assessment and mitigation. The initial blast wave can cause devastation in a large region around the volcano, the continuing eruption leads to lava flows or pyroclastic flows on the flanks of the volcano and ash plumes that are a danger to aircraft far away. Melting glaciers on snow-capped volcanos can lead to debris flows endangering nearby cities. Tsunamis generated by earthquakes or underwater landslides can cause damage and loss of life far away from the source, and accurate prediction of their propagation through the ocean and interaction with coastal topography is essential in issuing early warnings.