Albert Cohen, Laboratoire J.-L. Lions at Université Pierre et Marie Curie

Geometric Approximation of Images and Surfaces

Approximation theory plays a pivotal role
in the mathematical analysis of image processing applications
such as compression and denoising. This is particularly
well illustrated by non-linear approximation using wavelet bases
which is a central ingredient in the state of the art compression
standard JPEG 2000. In this talk, we shall first review some
results which connect image application with nonlinear
approximation, bounded variation functions and wavelet bases.
We shall then focus on recent developments which aim
to incorporate the geometry of edges within the image
model and the approximation-compression-denoising strategies.
This talk will aim to be accessible to the non-specialists.

February 4

Xiantao Li, Program in Applied and
Computational Mathematics (PACM), Princeton University

Multiscale modeling and computation for solids

Material properties can be
described by continuum equations or atomistic
models with different precision. I will present
a new multiscale method that makes use of both
continuum model and local atomistic simulations.
The new method resolves constitutive deficiency
in the continuum models, and limit the atomistic
simulations to very localized regions within
very short time, which makes the overall scheme
computationally efficient. In the second part of
my talk, I will talk about a defect tracking
method to bypass usual ad hoc kinetic laws.
Examples include dislocation dynamics and phase
transformations. This is joint work with Weinan
E.

February 11

Gilad Lerman, Courant Institute of Mathematical Sciences,
New York University

Identifying Differentially Expressed Genes via Multiscale
Geometric Analysis

We confront some problems of data analysis (in
particular outlier detection with applications
to bioinformatics) and use ideas developed in
harmonic analysis (specifically stopping-time
constructions and multiscale geometric
analysis).

The first problem is the identification of
differentially expressed genes or, more
generally, the nonparametric detection of
outliers in heteroscedastic data. We begin by
assuming the data is normalized so that it is
concentrated around a line. We suggest a
multiscale construction for a "strip" with
varying width around the line. The strip is
intended to separate the "deviating" points or
genes from the rest. We may generalize to the
case where the data is not normalized a priori,
so that we construct a strip of varying width
around a curve. We also discuss briefly more
general constructions, possible applications,
difficulties, and relevant geometric information
theory developed by Peter Jones and the speaker.

This is a joint work with Joe McQuown and Bud
Mishra as well as continuing theoretical work
with Peter Jones.

February 19

(Note special
date and time)

**THURSDAY**
at 3:30PM

Shouhong Wang, Department of Mathematics,
Indiana University

Bifurcation and Stability of Rayleigh-Benard Convection
JOINT
CSCAMM/Applied Mathematics seminar

In this talk, I shall present my recent work on bifurcation and stability of the
solutions of the Boussinesq equations, and the onset of the Rayleigh-B\'enard convection.
This problem goes back to the Benard's experiments in 1900, and the pioneering work of
Rayleigh in 1916. The best nonlinear theory for this problem was done by Rabinowitz and
Yudovich in the 60's. However, most, if not all, known results on the bifurcation and
stability analysis for this problem are restricted to certain subspaces of the entire
phase space obtained by imposing certain symmetry when the first engenvalue of the linear
problem is simple.

I shall present a nonlinear theory for this problem, which is established using a new notion
of bifurcation and its corresponding theorem. This theory includes 1) the existence of
bifurcation from the trivial solution when the Rayleigh number $R$ crosses the first
critical Rayleigh number $R_c$ for all physically sound boundary conditions, regardless
of the multiplicity of the eigenvalue $R_c$ for the linear problem, 2) aymptotical
stability of the bifurcated solutions, and 3) the roll structure and its stability in the
physical space. This is joint work with Tian Ma.

February 25

Qing Nie, Department of Mathematics, University of California, Irvine

Computational Analysis of Morphogen Gradients

Many patterns of cell and tissue organization
are specified during development by gradients of
morphogens, substances that assign different
cell fates at different concentrations. The
central questions in developmental biology
include what mechanisms are responsible for
morphogen transport, and what are the sources of
robustness in morphogen gradients. We attempt to
answer those questions by mathematical modeling,
analysis and computations in addition to examing
recent experimental data in ways that
appropriately capture the complexity of the
systems. In particular, we will study the BMP
activity gradients in the Drosophila embryo.
Applications of the models and computations to
various other systems will be discussed as well.

March 3

Alexander Katsevich, Department of Mathematics at
The University of Central Florida

Image Reconstruction in Cone Beam Computer Tomography

Computer tomography (CT) is a very common medical imaging
modality. Most of CT scans today are done in the spiral mode
with two-dimensional detector arrays. It turns out that image
reconstruction from the data provided by such scans is a very
complicated problem. On one hand, very good image quality is
required. On the other hand, the algorithm has to be highly
efficient. In this talk we will present a reconstruction
algorithm that would combine these two features. Mathematically,
image reconstruction in spiral CT can be described as the
problem of recovering an unknown function f knowing its
integrals along lines intersecting a spiral. We will describe
also a more general inversion formula, which recovers f knowing
its integrals along lines intersecting a fairly arbitrary curve
in R^3. This formula is of interest in other versions of CT, most
notably in C-arm scanning.

March 10

Gerhard Hummer, Laboratory of Chemical Physics at NIDDK, National Institutes of Health

Water in Confinement: from Nanotubes to Proteins

Water exclusion from the hydrophobic core is a paradigm of protein stability. Protein function, in contrast,
often requires water penetration into the nonpolar interior. Biomolecular proton conduction occurs through
transiently solvated hydrophobic channels, as in the proton pumps cytochrome c oxidase and bacteriorhodopsin.
Water itself is selectively transported across biological membranes through predominantly hydrophobic, not
hydrophilic channels, as in aquaporin-1. Why do water molecules occupy such narrow hydrophobic channels where
they can form only few hydrogen bonds, and thus lose many kT in energy compared to bulk solution? How do
water molecules get into, through, and out of such hydrophobic channels? An analysis of the water, proton,
and solute transport through the simplest molecular hydrophobic channel, a carbon nanotube, addresses these
questions and sheds new light on the functional role of hydrophobic channels in proteins for water and proton
transport. Molecular dynamics simulations show that nanotubes in contact with a water reservoir can fluctuate
in sharp transitions between water-filled and empty states. In the filled state, water molecules move rapidly
and in a highly concerted fashion through the sub-nanometer pores. An osmotic setup will be used to study the
nearly frictionless nanoscale flow through nanotubes assembled into membranes, and to characterize the properties
of two-dimensionally confined water monolayers that form between the nanotube membranes. Transport of small
hydrophobic molecules through nanotubes illustrates how selective binding leads to selective transport of
low-concentration solutes. Simulations with Car-Parrinello molecular dynamics and an empirical valence bond
model for water show that the one-dimensionally ordered water chains spanning the nanotube pores provide
excellent "proton wires" with forty-fold higher single-proton mobilities than bulk water. These results
have important implications for the transfer of water and protons through proteins and across membranes in
biological systems. In particular, they lead to a detailed molecular model of the proton-pumping mechanism of
cytochrome c oxidase that explains how this biological "fuel cell" can power aerobic life by exploiting the
unique properties of confined water.

March 15

(Note special
date, time, and location)

**MONDAY**
2:30PM, Physics 1201

Gershon Kurizki, Weizmann Institute of Science

Dynamical Control of Entanglement and Decoherence: From Nano-to Macro-
Systems
Joint Atomic Molecular and Optical Physics and Quantum Coherence and Information Seminar

Universal strategies are outlined for the entanglement of both continuous and
discrete variables in quantum systems with many degrees of freedom and for the
protection of this entanglement from environment- induced decoherence . These
strategies may considerably facilitate the processing of large amounts of
quantum information. Applications of these strategies are discussed for multi -
atom, multi-photon, molecular, superconducting and BEC systems.

Christopher Kulp, Department of Physics, College of William & Mary

Dissipative Controllers and Integrable Hamiltonian Systems

Controlling nonlinear systems is an important problem in nonlinear dynamics. Unlike linear systems, nonlinear
systems can have a wide range of complex behaviors. Hence, one all-encompassing theory for the control of nonlinear
systems may not exist. Typically, one approaches the problem of controlling nonlinear systems by studying the control
of a particular class of nonlinear system. The class of nonlinear systems that we are interested in are those
whose dynamics in some limit are related to integrable Hamiltonian models. In particular, we are interested
in controlling the integrable Hamiltonian system to one of the solutions of the uncontrolled (open loop) system.
The controller (drive term) will target a particular solution using both dissipative and resonant conservative
terms. It is well known that a resonant conservative controller will open an island around the control target.
However, it is the dissipative term which is of interest here. I will show that the dissipative term causes
a highly degenerate bifurcation to occur at the target. This bifurcation is known as a Takens-Bogdanov bifurcation.
The presence of a Takens-Bogdanov bifurcation implies that the control is very susceptible to noise. I will
illustrate these results using integrable Hamiltonian systems based on the nonlinear Schrodinger equation.

One-dimensional plane waves in an elastic material can be
modeled by a hyperbolic system of partial differential
equations. In a homogeneous material, a nonlinear stress-strain relation leads to the formation
of shock waves. Instead consider a laminated medium that consists of alternating layers of two different nonlinear materials. In this case the wave is partially reflected at each material interface, leading to dispersion and more complicated wave behavior. This dispersion, coupled with the nonlinearity, sets the stage for the appearance of solitary
waves that behave like solitons in many respects. I will present some computational results based on solving the hyperbolic system directly in the layered medium, and briefly discuss the high-resolution numerical methods used for such computations. I will also present some nonlinear homogenization results of Darryl Yong that yield an accurate effective equation containing dispersive terms.

Anne Chaka, Computational Chemistry at National Institute of Standards and Technology

How Scientific Computing, Knowledge Management, and Databases can Enable
Advances and New Insights in Chemical Technology

Today global competition is driving industry to reduce the time and cost to develop and
manufacture new products in the chemical, materials, and biotechnology sectors. Discovery and
process optimization are limited by a lack of property data as well as the lack of insight
into mechanisms and factors that determine performance. For the vast majority of applications,
particularly mixtures and complex systems, the evaluated property data simply do not exist and
are difficult, time-consuming, or expensive to obtain. Hence there is an explosion of
combinatorial, data mining, and sophisticated simulation technologies to supplement and guide
experimentation. The greatly increased pace of science and engineering, however, is already
beginning to outstrip our ability to produce needed data. The key questions are: How do we
develop the capability to supply massive amounts of evaluated data in a timely manner? How do
we generate models and simulation methods for predicting properties and physical phenomena with
quantitative measures of accuracy and reliability expressed as uncertainties? How do we transform
data and information into knowledge and wisdom to enable better technical decision-making?
Recent advances in computing and scientific algorithms have created the opportunity to begin
to address these issues.

De Witt Sumners, Department of Mathematics at Florida State University

DNA Topology: Experiments and Analysis

Cellular DNA is a long, thread-like molecule with
remarkably complex topology. Enzymes which manipulate the geometry and topology
of cellular DNA perform many important cellular processes (including segregation
of daughter chromosomes, gene regulation, DNA repair, and generation of antibody
diversity). Some enzymes pass DNA through itself via enzyme-bridged transient
breaks in the DNA; other enzymes break the DNA apart and reconnect it to
different ends. In the topological approach to enzymology, circular DNA is
incubated with an enzyme, producing an enzyme signature in the form of DNA knots
and links. By observing the changes in DNA geometry (supercoiling) and topology
(knotting and linking) due to enzyme action, the enzyme binding and mechanism
can often be characterized. This expository lecture will discuss topological
models for DNA strand passage and exchange, and using the spectrum of DNA knots
to infer bacteriophage DNA packing in viral capsids.

Chun Liu, Department of Mathematics at Penn State University

On Hydrodynamics of Viscoelastic Materials

I will discuss different models for viscoelastic materials. We will study these systems in the
unified framework of energetic variational procedures, focusing on the coupling between the special
transport and induced elastic stress. In particular, I will present the recent results on the existence
of classical solutions for systems with no postulated damping mechanism.

Crystal Surface Relaxation below the Roughening Transition: Shape Evolution and Scaling

Joint CSCAMM/Condensed Matter Physics
Seminar

The evolution of crystal surfaces below the roughening
transition is studied via a continuum approach that accounts for step energy (g1) and
step-step interaction energy (g3). In particular, we first contrast the dynamics with the more familiar case of surface
evolution above the roughening temperature. We then consider the relaxation of an axisymmetric crystal surface with
a single facet below the roughening transition. For diffusion-limited kinetics, free-boundary, and boundary-layer
theories are used for self-similar shapes close to the growing facet. For long times and g3 /g1<<1, a) a universal equation
is derived for the shape profile, b) the layer thickness varies as (g3 /g1)^1/3, c) distinct solutions are
found for different g3 /g1, and d) for conical shapes, the profile peak scales as (g3 /g1)^1/6. These results compare
favorably with kinetic simulations. The extension of these ideas to treat the grooving of bicrystals is also summarized.

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