Applied Mathematics Colloquia Series
Archive

Spring 2008

January

01/18
Steven Strogatz
Center for Applied Mathematics and Department of Theoretical and Applied Mechanics
Cornell University

Arthur Winfree Memorial Lecture: Synchronization in Nature

Art Winfree's first scientific paper, published in 1967 and based on research he did as a senior in college, was about synchronization of biological oscillators. In his honor, this talk will survey what we know (and don't know) about synchronization, 40 years later.

The tendency to synchronize is one of the most mysterious and pervasive drives in all of nature. Every night along the tidal rivers of Malaysia, thousands of fireflies flash in silent, hypnotic unison; the moon spins in perfect resonance with its orbit around the Earth; the intense coherence of a laser comes from trillions of atoms pulsing together. All these astonishing feats of synchrony occur spontaneously --- almost as if the universe had an overwhelming desire for order.

On the surface, these phenomena might seem unrelated. After all, the forces that synchronize fireflies have nothing to do with those in a laser. But at a deeper level, they are all connected by the same mathematical theme: self-organization, the spontaneous emergence of order out of chaos. Video footage of synchronous fireflies and the notorious crowd synchrony that triggered the wobbling of London's Millennium Bridge will be shown.

Refreshments served at 3:30 in 401N.
01/25
Vakhtang Putkaradze
Department of Mathematics
Colorado State University

Models of Dissipation and Self-Organization in Physical Systems: From Kinetic Equations to Self-Organization of Magnetic Particles to Protein

The engineering community has been actively pursuing the development of self-organization of particles in order to design smaller, faster and more efficient electronic elements. In order to achieve theoretical understanding of these processes, we suggest a dissipative analogue of kinetic equations that describe the motion of probability distribution in the momentum-coordinate space. This work is based on the double bracket dissipation ideas that were originally suggested for astrophysical applications.

We then show how to extend the double bracket method to include particles with interaction dependent on orientation, for example, magnetized particles in colloidal solution. We derive evolution equations for density and magnetization that reduce to the celebrated Landau-Lifshitz-Gilbert equations for non-moving magnets and to Debye-Huckel equations for particles without orientation. We also show how our equations naturally give the motion for of an elastic self-interacting curve, and discuss the application of our technique to folding of biologically-relevant strands.

Collaborators: Darryl D. Holm, Cesare Tronci (Mathematics, Imperial College, London

Refreshments served in Math 401N at 3:30

February

02/08
Becca Thomases
Department of Mathematics
University of California, Davis

Singularities and Transport in Viscoelastic Fluids

In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow. For example, experiments have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time-dependent dynamics and show efficient mixing. The corresponding experiments using Newtonian fluids do not show such nontrivial dynamics. To better understand these phenomena we study numerically the 2D Oldroyd-B Viscoelastic model at low Reynolds number. A background force is used to create a periodic cell with four-roll mill vertical structure around a hyperbolic fixed point. We consider both steady and time-periodic forcing. For low Weissenberg number (Wi) the elastic stresses are bounded to the forcing, with mixing confined to small sets near the hyperbolic point. At larger Wi an analog to the coil-stretch transition occurs, yielding large stresses and stress gradients concentrated on sets of small measure. The flow then becomes very sensitive to perturbations in the forcing and there is a transition to global mixing in the fluid.

Refreshments served in Math 401N at 3:30
02/15
Isaac Klapper
Department of Mathematicsl Sciences
Montana State University

Microbial Biofilms

Compared to the plant and animal kingdoms, diversity of microbial life is considerably less explored and less understood (even the notion of microbial species is a current topic of debate). Prokaryotes (bacteria and archaea) are estimated to make up approximately half of extant biomass; for example, each human harbors about 100 trillion microbes (bacteria and archaea), ten times more microbial than human cells. The familiar view of microbes in their free (planktonic) state is however not the norm; rather it is believed that much of the microbial biomass, perhaps 95-99%, is located in close-knit communities, designated biofilms and microbial mats, consisting of large numbers of organisms living within self-secreted matrices constructed of polymers and other molecules. (Microbes in collective behave very differently from their planktonic state; even genetic expression patterns change.) These matrices serve the purposes of anchoring and protecting their communities in favorable locations while providing a framework in which structured populations can differentiate and self-organize.

One can and will find biofilms in almost any damp or wet environment, and they are often key players in problems such as human and animal infections, fouling of industrial equipment and water systems, and waste remediation, just to name a few. Medical relevance is quite dramatic. Quoting from the National Institutes of Health: "Biofilms are clinically important, accounting for over 80 percent of microbial infections in the body. Examples include: infections of the oral soft tissues, teeth and dental implants; middle ear; gastro-intestinal tract; urogenital tract; airway/lung tissue; eye; urinary tract prostheses; peritoneal membrane and peritoneal dialysis catheters, in-dwelling catheters for hemodialysis and for chronic administration of chemotherapeutic agents (Hickman catheters); cardiac implants such as pacemakers, prosthetic heart valves, ventricular assist devices, and synthetic vascular grafts and stents; prostheses, internal fixation devices, percutaneous sutures; and tracheal and ventilator tubing."

Viewed as materials, biofilms are quite interesting: they are living, growing viscoelastic fluids with surprising ability to respond to and defend against their environments. This talk will present a general overview of efforts to characterize and model biofilms on a continuum macroscale, addressing some of the issues mentioned above.

Refreshments served in Math 401N at 3:30
02/22
Michael Kuecken
Technische Universitaet Dresden

Modeling Cnidarians: Oscillations in Hydra and Growth Rules in Corals

Cnidaria is an ancient phylum that includes solitary organisms like Hydra and sea anemones and colonial organisms like corals. Because of their apparent lack of bilaterality and their simple body plan, Cnidarians were considered very unsophisticated organisms that should be easy to understand. This view, however, has changed recently due to the discovery of numerous Cnidarian genes and signaling molecules that are also present in "higher" organisms.

For a long time Hydra has been a model system for developmental biology and a favorite pet for theorists. It is remarkable for its extraordinary regeneration capabilities that enable the survival of the organism from only 1% of the body tissue. In the course of regeneration Hydra forms a hollow sphere that undergoes cycles of oscillations. The purpose of the oscillations has not yet been completely understood but is likely due to osmoregulation, as I will argue in my talk.

Corals give rise to one of the world's most diverse ecosystems and fascinate because of their bright colors and fantastic shapes. How these different shapes are created is still very much unclear. On the one hand, genetic dispositions must be important; on the other hand, environmental factors such as light and water flow modulate the growth significantly. In my talk I will summarize experimental work geared towards deciphering the rules of growth in the coral Stylophora pistillata and present a model setup for studying these rules.

Refreshments served in Math 401N at 3:30

March

03/07
Timothy J. Healey
Theoretical and Applied Mechanics
Cornell University

CANCELED: Some Problems in Second-Gradient Nonlinear Elasticity

CANCELED

Due to a threatened snowstorm in upstate New York, Friday's Colloquium has been cancelled.
03/28
Julia Arciero
Program in Applied Mathematics
The University of Arizona

Al Scott Prize & Lecture: Theoretical Model of Metabolic Blood-flow Regulation

Al Scott Lecture. The ability of the circulatory system to adequately match blood supply to tissue demand implies the existence of regulatory mechanisms that communicate tissue status to blood vessels. For example, red blood cells have been shown to respond to low tissue oxygen levels by releasing ATP. The ATP triggers a conducted response signal to travel upstream and cause arterioles to dilate so that more blood is delivered to the region of demand. A theoretical model focusing on the role of this mechanism in blood flow regulation is presented here. In the model, arterioles control blood flow by dilating or constricting in response to changes in metabolism as well as to changes in pressure and wall shear stress. The model predicts that responses to these three stimuli can account for the increase in blood flow that occurs with increased oxygen demand.

Refreshments served in Math 401N at 3:30

April

04/04
Tom Chou
Departments of Biomathematics and Mathematics
UCLA

Stochastic Models in Biophysics

I will develop and analyze stochastic models for two important processes in cellular biophysics. The first problem concerns mRNA translation and protein production, and is modeled as an interacting particle system in 1D. The effects of "slow codons," or defects in the mRNA, on protein production rates are addressed by asymptotic matching of mean-field solutions of the problem. In the second problem, a stochastic model for viral entry into cells is developed. The entry of viruses turns out to be a competition between membrane fusion and endocytosis. The probabilities for entry via each of these pathways are calculated within one- and two-surface receptor models. Conditions for endocytosis are mapped. Time permitting, I will also briefly introduce a stochastic inverse problem where transition rates of a Markov process can or cannot be reconstructed from first passage time distributions.

Refreshments served in Math 401N at 3:30
04/11
Dmitry Pelinovsky
Department of Mathematics
McMaster University

CANCELED: Advection-Diffusion Equations with Forward-Backward Diffusion

(This week's colloquium has been canceled.)

We study the spectrum of a linear advection-diffusion equation in a periodic domain, where the diffusion coefficient changes its sign. We prove that the spectrum of an associated linear operator consists of an infinite set of simple eigenvalues on the imaginary axis and the set of corresponding eigenfunctions is complete. However, we also show, assisted with numerical approximations, that the complete set of linearly independent eigenfunctions does not form a basis in a space of square integrable functions and that the Cauchy problem for the advection-diffusion equation is ill-posed.

04/18
Maria-Carme T. Calderer
School of Mathematics
University of Minnesota

[CANCELED] Elastic and Ferroelectric Properties of Liquid Crystals: Modeling and Analysis

(This colloquium has been canceled.)

Since the development and commercialization of the first nematic liquid crystal display devices in the middle of the last century, mathematical modeling and analysis of liquid crystals has experienced significant progress. Liquid crystals are phases intermediate between solid and liquid; they occur in synthetic as well as in organic compounds. The Kevlar fiber is an example of a highly employed liquid crystal polymer; many virus and bacteria colonies as well as biological tissues present liquid crystal ordering.

Liquid crystals of small molecular weight consist of rigid, rod-like molecules that tend to follow preferential directions of alignment. Their interaction with electric and magnetic fields is at the core of application to display devices. Recently developed liquid crystals exhibit more complicated molecular shapes able to sustain permanent dipoles that result in ferroelectric coupling with applied electromagnetic fields. The speed of switching of such devices is about 10^3 to 10^4 times that of the nematic cell. Equilibrium states of ferroelectric liquid crystals result from minimizing the total energy subject to packing and electrostatic constraints. I will present an application of such a theory to predicting shape of material filaments.

Liquid crystal elastomers are nonlinear elastic solids that may also present liquid crystal phases. One remarkable feature is their capability to undergo unusually large deformations along preferential directions. Upon analyzing mathematical issues of such models, I will address their gel states and discuss the potential matrix role in modeling cell motility in the brain.

Refreshments served in Math 401N at 3:30
04/25
Yalchin Efendiev
Department of Mathematics
Texas A&M University

Multiscale Numerical Methods for Flow and Transport in Heterogeneous Porous Media and Their Applications

Typical porous media processes are affected by heterogeneities at different length scales. In this talk, I will describe multiscale finite element methods for flow and transport in heterogeneous porous media. The main focus of the talk is on subgrid capturing using various local and global methods.

I will discuss the use of local boundary conditions and the use of global information in capturing subgrid effects. The upscaling of the transport equation and its coupling to the flow equation will be presented. The mathematical analysis of these methods will be discussed.

Refreshments served in Math 401N at 3:30

May

05/02
Michael Holst
Department of Mathematics
University of California, San Diego

Some New Existence Results for a Geometric PDE Arising from General Relativity and an Approximation Theory Framework

There is currently tremendous interest in geometric PDEs, due in part to the geometric flow program used recently to solve the Poincare conjecture. Geometric PDEs also play an expanding role in many other applications, such as understanding the gravitational wave models of Einstein. The need to validate these models has led to the construction of gravitational wave detectors in the last several years, such as the NSF-funded LIGO project. In this lecture, we consider the coupled nonlinear elliptic constraints in the Einstein equations, a geometric flow which describes the propagation of gravitational waves generated by collisions of massive objects such as black holes. The constraint equations must be solved numerically to produce initial data for gravitational wave simulations and to enforce the constraints during dynamical simulations. In the first part of the lecture, we consider a thirty-year-old open question involving existence of solutions to the constraint equations on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic curvature, and we give a partial answer using a priori estimates and a new type of topological fixed-point argument.

In the second part of this lecture, we develop some adaptive numerical methods for which we can prove a number of useful results on convergence, optimality, and scalability. Based on the a priori estimates developed in the first part of the talk, we first establish some critical discrete estimates. We then derive error estimates for Galerkin approximations and describe a class of nonlinear approximation algorithms based on adaptive finite element methods (AFEM). We establish some new AFEM convergence and optimality results for geometric PDE problems with non-monotone nonlinearities such as the Einstein constraints.

We finish by illustrating the algorithms with some examples using the Finite Element ToolKit (FETK).

Refreshments served in Math 401N at 3:30