In the talk we will survey some recent results in adaptive multivariate approximation. Some of the topics we hope to cover are:

• New estimates for polynomial approximation over multivariate domains
• Multivariate piecewise polynomial approximation. When is it better than wavelets?
• Geometric multivariate smoothness spaces beyond BV and Besov
• "Taming the problem I": Binary space partitions and geometric wavelets
• "Taming the problem II": On the approximation of the K~ functional
• Applications in image represnetation and low bit-rate image coding

This is joint work with Dany Leviatan. Preprints can be downloaded from: https://turbo.turboimage.com/~sd/default.html

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.

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.

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.

(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.

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.

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.

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.

(Note special date, time, and location)

 **MONDAY** 2:30PM, Physics 1201

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.

ABSTRACT

ABSTRACT

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.

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.

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.

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.

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.