Level set method uses a level set function, usually an approximate signed distance function, Phi, to represent the interface as the zero set of Phi. When Phi is advanced to the next time level by a transportation equation, its new zero level set will represent the new interface position. We update the level set function Phi forward in time and then backward to get another copy of the level set function, say Phi_1. Phi_1 and Phi should have been equal if there were no numerical error. Therefore Phi-Phi_1 provides us the information of error and this information can be used to compensate Phi before updating Phi forward again in time. One nice property is that it has the convenience of possibly improving the temporal and spatial order of an odd order scheme simultaneously. We found that when applying this idea to semi-Lagrangian schemes, e.g., CIR scheme (which has no CFL restriction, a nice feature for local refinement), the property is still valid (while MacCormack scheme having similar property may not be easily applied here). This technique coupled with a simple yet less diffusive redistancing technique produces a very efficient algorithm even for unstructured triangle meshes. Numerical results for interface movements with level set equation computed by the new methods will be presented in the talk. Also we would like show some interesting theoretical results for applying this idea to a general linear scheme.

I investigate how the topology of a network affects its dynamical properties. To this end, I will study the nature of the phase transition from an ordered to a disordered state that occurs in a family of neural network models with noise. These models are closely related to the Majority Voter Model, where a ferromagnetic-like interaction prevails. Each member of the family is distinguished by the network topology, which is determined by the distribution of the number of incoming links. I will show that for homogeneous random topologies, the phase transition always belongs to the mean-field universality class. However, scale-free networks depart from this universality class in the sense that phase transition exponents ranging from 1/2 to infinity are obtained. Furthermore, the scale-free topology provides the first example of a phase transition at finite temperature in networks with infinite average connectivity.

It is often difficult to decide whether we can reliably model the behavior of a physical system with a particular program. I will discuss some of the possible applications of PDE-constrained optimization in such situations. There are interesting numerical analytic questions to be addressed. I will summarize some new, very positive, results in the thesis of Andrei Draganescu about the amount of work involved. However, it is not at all clear how to cast the questions we want to answer so that optimal control gives us valuable insight.

In 1979, Hofstadter introduced and briefly discussed a chaotic, recursively-defined function which he called Q:

Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))

Over the past 25 years, a number of mathematicians have studied this function and a handful of variants of it, and though some statistical details about Q's type of chaos have been observed, no one has managed to prove a single fact about Q -- not even that it -is- a function! In my talk, I'll describe a new avenue of (mostly empirical) which looks at a family of variants of Q, and which borrows some ideas from other disciplines. It turns out that this family of functions, on the collective level, exhibits an amazing degree of regularity, and yet within the framework of that family-level regularity, there are very weird irregular patterns unlike any seen before. I'll discuss what is currently known and unknown about this new family of functions, presenting the ideas mainly through a series of computer-generated graphs which display the tantalizing eccentricities that have been recently uncovered.

Rather than discuss a particular computation, this talk will look at how the producers and consumers of scientific computation communicate, and how these interactions affect the scientific process, e.g. productivity, the quality of results, etc.

We can roughly divide software used in research efforts into three groups. In the first group, researchers purchase or download software created by programmers with whom they will never communicate. This software may be very general productivity software like Microsoft Excel, or it may be intended for a very specific use, such as supporting instrumentation. In the second group, a single researcher (or possibly a very small group) writes his/her own software, seeking to produce a novel, computational result. The software is not written for anyone but that researcher (although it may come to be used by others). In the third group, the scientific program involves a larger group or community of researchers, and part of the research requires completion of a set of data and/or computation intensive tasks. In this case, the users and the programmers are not the same people, and communication becomes a critical factor for the atmosphere and success of the project.

After completing a physics PhD involving a fair amount of programming at the University of Chicago, my professional background has been in both custom business software development and database bioinformatics in an academic setting. Rather than bore you with endless details about my experiences, however, I hope that this talk will be interactive. I will present a set of ideas and topics, and then will solicit experiences from the audience for discussion, examining as a group the way we all work.

People who write research code for themselves, who write code for others or who have code written for them are encouraged to attend.

Fluctuations of macroscopic quantities in systems at thermal equilibrium have well known properties. Much less is known for dissipative systems out of equilibrium. We first present several experimental or numerical results on a few examples: fluctuations of the total heat flux in turbulent convection, or of the power needed to drive a turbulent flow in a statistically stationnary regime, fluctuations of the kinetic energy or of the injected or dissipated power in a granular gas, etc. We then propose different methods to characterize these fluctuations and show that some of their statistical properties do not depend on the particular system under consideration.

The generation of a magnetic field by a flow of liquid sodium has been observed in recent experiments (Karlsruhe, Riga). We emphasize two very interesting features displayed by these experiments.

• The observed dynamo threshold is in good agreement with the one computed from the mean flow alone, i. e. neglecting turbulent fluctuations although the kinematic Reynolds number is of order 105 to 106.
• On the contrary, the mean magnetic field measured above dynamo threshold is 1000 times larger than the one predicted from a laminar weakly nonlinear calculation.

We first understand these two observations and give the expected scaling law for the magnetic energy above dynamo threshold. We then consider magnetic fluctuations above dynamo threshold or in MHD turbulence and report a Kolmogorov type spectrum in the inertial range and 1/f noise at low frequency. Finally, we discuss the effect of turbulence and rotation on dynamo onset and saturation in flows without strong geometrical constraints.

To manipulate suspensions of cells and other micron- sized particles in bioengineering or lab-on-a-chip applications, strong hydrodynamic forces on small scales are necessary. Ideally, these forces should both ensure rapid transport of the particles and be strong enough at well-defined locations to porate or rupture cell membranes in order to achieve transfection of large molecules such as drugs or DNA into the cell. We show that ultrasound is an efficient driving mechanism for such microfluidic flows, if the sound energy is focused onto small scales through oscillating microbubbles. The bubbles excite a streaming flow that can be used to aggregate, deform, and rupture lipid vesicles and cells, making it a promising tool for both membrane studies and drug delivery applications. When combining bubbles with passive flow elements, other modes of streaming are excited, allowing for directional microfluidic transport of cells and particles without microchannels. All observed flows are in quantitative agreement with Stokes flow singularity theory.

Combinatorial enumeration problems arise naturally in many problems of statistical physics. The number of partitions of an integer is one such enumeration problem with a history dating back to Euler. Partitions of an integer is the number of ways a positive integer can be decomposed into sum of smaller parts. Depending on the dimension of the lattice on which these parts are arranged, the partitions are called linear partitions, plane partitions, solid partitions and so on. Very little is known about solid partitions and higher dimensions. In this talk I will present numerical results for the asymptotic behavior of solid partitions. A simple proof for the MacMahon formula for the plane partitions of an integer will also be discussed.

Large ensembles of small particles display fascinating collective behavior when they acquire an electric charge and respond to competing long-range electromagnetic and short-range contact forces. We conduct experimental and theoretical studies of the dynamics of conducting microparticles in strong electric field in the air or in poorly conducting liquids. We show that granular media consisting of metallic microparticles immersed in a poorly conducting liquid in strong DC electric field self-assemble a rich variety of novel phases. These phases include static precipitate: periodic honeycombs and Wigner crystals; and novel dynamic condensate: toroidal vortices and pulsating rings. The observed structures are explained by the interplay between charged granular gas and electrohydrodynamic convective flows in the liquid. We developed continuum theory of self-assembly and pattern formation in this system. The theory is formulated in terms of two conservation laws for the densities of immobile particles (precipitate) and bouncing particles (granular gas) coupled to the Navier-Stokes equation for the liquid. This theory successfully reproduces correct topology of the phase diagram and primary patterns observed in the experiment: static crystals and honeycombs and dynamic pulsating rings and rotating multi-petal vortices.

The evolution of living organisms required developing special strategies for synthesizing complex molecules and creating order and asymmetrical distributions of molecules within their interior. Many of these operations are performed by nanometer-scale "molecular machines" that use chemical energy to drive unidirectional processes, produce linear motion and/or generate mechanical work. I will discuss the mechanism of kinesins, which are motor proteins that use ATP energy to move along microtubules. In humans, there are 45 different kinesin motor proteins that are involved in a myriad of different biological functions including organelle, protein, mRNA transport, mitosis/meiosis, and control of microtubule dynamics. We have used a combination of single molecule motility assays, cryo-electron microscopy, spectroscopy, x-ray crystallography and mutagenesis to identify a new type of mechanical amplification process for kinesin. Recently, we have tested our model by "watching" changes in the structure of an individual kinesin motor protein as it undergoes motility. This work, which involves a combination of protein engineering and low light level fluorescence microscopy, provides new insight into how chemical energy causes the step-wise motion of kinesin along its track.

A general review on kinesin (as well as another motor- myosin): Vale, R. D. and Milligan, R. A. 2000. The way things move: looking under the hood of molecular motor proteins. Science 288: 88-95.

A brief review is given of how convective and stably stratified turbulence occurs in geophysical flows, especially boundary layers. General aspects of the steady/unsteady (plume/puff) eddy structure, as a function of thermal/momentum surface boundary conditions are described using results from lab and field experiments, theory and numerical simulations. These results are important for determining the levels of temperature and velocity fluctuations at the surfaces bounding the convection. In stably stratified turbulent shear layers, perturbation theory and DNS results show how the fluctuations are strongly distorted by the buoyancy forces in such a way as to reduce the transfer of energy from the shear to the turbulence-a more universal and different mechanism to the GI Taylor instability or LF Richardson energy damping mechanisms. Because of its local structure, stably stratified turbulence, except when it is very inhomogeneous, has certain general characteristics found in a variety of different types of flow, except in highly inhomogeneous layers when the results are rather unpredictable (with consequences for weather forecasts). Forecasts are also difficult when convective turbulence is perturbed by weak shear; so that the mean flow can be amplified. In the resulting mean profile jets tend to appear.

In 1942, Jacques Monod discovered that E. coli bacteria can discriminate between glucose and lactose and decide to consume the glucose first. The talk will present a standard molecular biology description of how the E. coli cell makes this decision. The computational process involves sequence-specific DNA-binding proteins, protein phosphorylation, action of membrane proteins and soluble cytoplasmic enzymes, protein-protein interactions, small molecule messengers and the construction of highly specific nucleoprotein complexes. Because all compartments of the cell are involved in information processing, there is no Cartesian separation into dedicated "informational" and "operational" molecules. Because the DNA participates as a physical component of the nucleoprotein regulatory and transcriptional complexes, it is not simply a software "tape" in the sense of a Turing machine. Our inability to make basic Cartesian or Turing distinctions indicates that biological (i.e. cellular) computation follows a novel computing paradigm.

By 1572 the last legitimate heir to the Inca crown had been executed, and many of the important shrines had been desecrated, or destroyed. A great deal of information on Inca social structure, ceremonial activity, and belief is lost. Still, through ethnohistoric accounts, and archaeological fieldwork, it is possible to piece together the sky watching practices of the Inca, and understand some of its use in organizing their empire. From such work we know that as children of the sun god, Inti, the Inca ruled their empire, Tawintinsuyu. This elite position was reinforced through ceremonies honoring the sun, and involving a system of solar markers around the horizon of Cuzco. The remains of such solar markers have now been found at Titicaca, giving flesh to early Spanish accounts. However, this archaeological find demonstrated that the system required the support of other observations. Important clues defining these supporting observations were then found at Machu Picchu.

Every couple of years, a celestial body impacts the earth with energy near that of the Hiroshima bomb. On much longer timescales, impacts will occur with the potential to destroy regions, or whole civilizations. This lecture will present an overview on efforts to define the impact threat, followed by a systematic development of the requirements to divert an object on an earth-impacting course. We then examine today's technologies for achieving perturbation magnitudes necessary to protect the planet.

Liquids do not mix easily in microfluidic systems, which are being developed into "labs-on-a-chip" that promise revolutionary applications in biotechnology, chemistry and medicine. Recent studies have suggested that microfluidic stirring via chaotic advection can achieve the efficient mixing required in typical uses. For devices based on continuous flow through microchannels, strategies for inducing chaotic mixing by altering device geometries have been proposed. I will describe a general methodology for introducing chaotic mixing in discrete volume (microdroplet) systems, which allow miniaturization of many standard laboratory protocols that are difficult to realize with continuous flow. The mixing properties of the flows in microdroplets are governed by their symmetries, which give rise to invariant surfaces serving as barriers to transport. Complete three- dimensional mixing by chaotic advection requires destruction of all flow invariants. As an illustration of this idea, I will demonstrate that complete mixing can be obtained in a time-dependent flow produced by motion of a microdroplet along a two-dimensional path and describe the experiments that optically manipulate and mix microdroplets.

Glassy dynamics occur in a large variety of systems, such as supercooled liquids, foams and granular matter. They are characterized by an exponentially rapid increase of relaxation times, as a control parameter such as temperature or density in tuned, and by a non-exponential decay of time-dependent correlation functions indicating a broad distribution of time scales. In this talk, I will present an exact solution of a Landau model of an order-disorder transition with activated critical dynamics. The model describes a funnel-shaped topography of the order parameter space in which the number of energy lowering trajectories rapidly diminishes as the ordered ground state is approached. This leads to an asymmetry in the effective transition rates, which results in a non-exponential relaxation of the order-parameter fluctuations and a Vogel-Fulcher-Tammann divergence of the relaxation times, typical of a glass transition. I will discuss a lattice model where this class of critical dynamics is realized and I will argue that the Landau model provides a general framework for studying glassy dynamics in a variety of systems.

Shannon sampling is a special case of the general problem of reconstruction of a function from its values at a discrete set of points. The talk with deal with age-old algorithms for solving this problem and new estimates for their error and efficiency.

DNA chips are novel experimental tools that have revolutionized research in molecular biology and generated considerable excitement. A single chip allows simultaneous measurement of the level at which thousands of genes are expressed. A typical experiment uses a few tens of such chips, each devoted to one sample - such as material extracted from a tumor. Hence the results of such an experiment consist of a table, of several thousand rows (one for each gene) and 50 - 100 columns (one for each sample). Extracting relevant information from such a large, complex and noisy data set requires development of novel methods of analysis.

In this talk I will briefly explain how gene expression is measured by DNA chips, and demonstrate how we combine standard statistical analysis with these novel unsupervised methods to mine expression data obtained from leukemia samples. If time permits, I will demonstrate how some intriguing design principles can be obtained from recent experiments on stem cells.

DNA chips are novel experimental tools that have revolutionized research in molecular biology and generated considerable excitement. A single chip allows simultaneous measurement of the level at which thousands of genes are expressed. A typical experiment uses a few tens of such chips, each devoted to one sample - such as material extracted from a tumor. Hence the results of such an experiment consist of a table, of several thousand rows (one for each gene) and 50 - 100 columns (one for each sample). Extracting relevant information from such a large, complex and noisy data set requires development of novel methods of analysis.

This talk will provide a very basic introduction, with no prior knowledge of any biology assumed. I will explain what genes are, what is gene expression and how it is measured by DNA chips. I will also explain what is meant by clustering and sorting, and demonstrate how standard statistical methods and novel, unsupervised techniques are used to analyse data on various forms of cancer.

A major challenge in structural biology is to derive a quantitative understanding of the relationship between the structure of a protein and its function, and use this to extract general rules on protein structure-function relationships. While thousands of protein crystal structures have been determined, many proteins have been studied using mutagenesis, and computational approaches have been applied to functional aspects of proteins, this remains a central open problem in biochemistry.

I will present work on the use of photoactive yellow protein (PYP) as a model system to identify general principles in protein structure-function relationships. PYP is a photoreceptor protein found in photosynthetic bacteria. Activation of PYP by blue light causes this protein to generate a transient signal within the bacterial cell. This is initiated by the photoisomerization of the covalently attached chromophore buried within PYP. The following findings will be discussed. (i) We have found that photoactivation of PYP results in its transient partial unfolding, challenging the notion that signal transduction only occurs between fully folded proteins. (ii) We have developed a model in which this transient unfolding event is caused by light-triggered proton transfer, which generates a buried charge within PYP that functions as an "electrostatic epicenter" for the "protein quake" that activates PYP. (iii) We found that removal of the chromophore from PYP causes its partial unfolding. This partially unfolded state catalyzed the covalent attachment of the chromophore to itself. This presents the second example in the PYP system of a biological function for a partially unfolded state, challenging the influential notion that only fully folded proteins are functionally active. (iv) We recently initiated a high-throughput biophysics approach to systematically probe structure-function relationships in PYP. Computational methods will be essential in analyzing the experimental data obtained in this project.

The initial phase of sparking is determined by so-called streamers. These are weakly ionized channels during their growth period. The growth is characterized by a self-induced enhancement of the electric field at the tip of the discharge channel. Streamers propagate with velocities of the order of 1000 km/sec; recent ultrafast photography gives a new view on their dynamics. Streamer concepts are also being applied to recently discovered high altitude lightning, so-called red sprites.

I will review recent observations and then explain the state of microscopic modelling, computations and theoretical concepts. Basically, already a single discharge channel has a multiscale structure with a thin ionization front surrounding a rather inert body. I will present computational results with adaptive grids, and I will discuss the properties of ionization fronts, moving boundary approximations for these fronts, and solutions of the moving boundary problem with conformal mapping methods. The result is the prediction that streamers in a sufficiently high potential can branch spontaneously due to a Laplacian instability as is also observed in computations. This quantitative prediction has to be confronted with phenomenological models for spark branching of the type of diffusion limited aggregation.

We study the placement of charged particles on a two-dimensional conductor. Particularly, we focus on the placement of N equal discrete charges in the asymptotic limit as N goes to infinity, for both the case of a smooth conductor and also the situation in which the conductor contains cusp-like points.

This problem is closely related to diffusion limited aggregation (DLA), and to the two-dimensional motion of viscous fluids (Hele-Shaw). A similar problem has been originally introduced under the notion of "Fekete points". However, in that context, results about the asymptotic placement of the charges as well as the (first order) asymptotic energies have been derived only for domains bounded by analytic curves. In contrast, we are trying to study how non-analyticities of the conductor boundaries affect the charge placements and expansions of equilibrium energies in the number of charges.

Furthermore, systems with an intrinsic symmetry of the conductor show a breaking of this symmetry in the placement of the charges, depending on the number of charges and the local curvature of the boundary.

Copepods are micro-crustaceans of 1 to 5 mm in length. They create feeding currents to find food, use hydro-dynamical disturbances to distinguish predators from mates, and 1.347x10exp21 of them populate the vast 3D environments of our fresh and marine waters. I will show (in short videos) results of 30 years of detailed observations of the interaction between these animals and their surrounding water.

We study the impact of a jet of a viscous fluid in a bath of the same liquid. We measure the radius of curvature of the liquid air-interface where the impact occurs. We show that it decreases exponentially with the capillary number. Above a threshold speed, capillary can not overtake this high confinement and a thin sheet of air is dragged into the bath by the jet, in a trumpet-like form. We measure the threshold speed for which a film of air is dragged into the pool and the thickness of the film.

Schramm-Loewner evolution (SLE) describes the statistics of single random curves in 2d critical systems. It relates questions about these curves to simple problems in 1d Brownian motion. Many old and new results can be derived using these methods. I present a generalisation to $N$ curves. The corresponding 1d problem is Dyson's Brownian motion, which describes the statistics of the eigenvalues of random matrices. It is also related to the quantum Calogero-Sutherland hamiltonian. I show that this connection arises also in conformal field theory (and could have been discovered 20 years ago.) The values of bulk critical exponents of 2d systems are given by the spectrum of this hamiltonian.

Given a large but finite region, filled with a medium which is a composite of conducting and insulating components, can current flow between two contacts attached to separate parts of the boundary? This is a problem in percolation. Above the percolation threshold $p_c$ current always flows, while below $p_c$ it never does so. But at $p_c$ there is a finite probability, between zero and one, of a connection, and this depends on the shape of the region.

In quantum Hall systems, the value of the quantised conductance depends not just on whether there is a connection, but on the number of independent such ones.

These problems share the property of conformal invariance. In two dimensions, there are exact results for them. I shall present these, and discuss the variety of methods by which they have been obtained.

In a turbulent highly conducting fluid, magnetic fields are amplified since the field lines are generally stretched by randomly moving fluid elements in which these lines are frozen. Such a mechanism of turbulent dynamo is expected to work in a variety of astrophysical systems (galaxy clusters, interstellar medium, stars, planets), is confirmed numerically, and is consistent with simple analytical models.

Recently, there appeared the number of high-resolution numerical simulations of MHD turbulence with small magnetic Prandtl numbers [Pm=fluid viscosity/fluid resistivity], where magnetic fluctuations were not amplified. This revived old claims that dynamo does not exist in the Kolmogorov turbulence with Pm << 1. However, astrophysical observations show that magnetic fields are generated by turbulent motion rather effectively in planets and stars where magnetic Prandtl numbers are small (e.g., in the geo-dynamo, Pm=10^-5, in stars, Pm= 0.01). The talk will address this apparent contradiction.