Several years ago John Conway asked whether there could exist a polyhedron that "had a hole in every face," and coined the name "holyhedron" for them, if they should exist. We answer this question by constructing a holyhedron with 78,585,627 faces and genus 60,380,421. This is a polyhedron so that the interior of every face is connected but not simply connected.

Continuum models of the flow of granular materials in a hopper admit so-called radial solutions. These describe steady flows that appear realistic, and have been used extensively to design commercial hoppers. However, numerical results demonstrate that these solutions may not be robust to perturbation. Moreover, the time dependent equations are (notoriously) ill-posed. In this talk, I describe preliminary research designed to investigate the extent to which steady solutions may be used to represent granular flow. Using a combination of analysis and numerical experiments, we have explored simple models that are linearly ill posed. While there may be a stable steady state, it is a solution of a discretized continuum model, rather than the original equations. Moreover, the survival time of transients is inversely related to the mesh width, suggesting that the continuum limit is meaningless. While these results are not intended to invalidate the radial solutions, they do raise serious concerns about continuum modeling, and the possibility of designing a robust code that can be used to simulate a variety of granular flows.

In spite of the fact that symmetries play one of the major roles in physics, the ir usage in finance is relatively new and, to the best of our knowledge, can be traced to 1995 when Kholodnyi introduced the beliefs-preferences gauge symmetry. In this talk we present another symmetry, foreign exchange option symmetry, int roduced by Kholodnyi and Price in 1996. Foreign exchange option symmetry associa tes financially equivalent options on opposite sides of the foreign exchange mar ket. In a two-currency market, the foreign exchange option symmetry is formalized in terms of the one-dimensional Kelvin transform. In a multiple-currency market the foreign exchange option symmetry is formalized in terms of differential geometr y on graphs, that is, in terms of vector lattice bundles on graphs and connectio ns on these bundles. Foreign exchange option symmetry requires no assumptions on the nature of a prob ability distribution for exchange rates. In fact, it does not even require the a ssumptions of the existence of such a distribution. Furthermore, the symmetry is applicable not only to a foreign exchange market but to any financial market as well. The practical applications of the foreign exchange option symmetry range from th e detection of a new type of true arbitrage to the detection of inconsistent mod els of foreign exchange option markets and the development of algorithms and sof tware to value and analyze portfolios of foreign exchange options.

Previously, we developed a ``minimal'' dynamic model for the tubuloglomerular feedback (TGF) system in a single, short-looped nephron of the mammalian kidney. In that model, a semilinear hyperbolic partial differential equation was used to represent two fundamental processes of mass transport in the nephron's thick ascending limb (TAL): chloride advection by fluid flow through the TAL lumen and transepithelial chloride transport from the lumen to the interstitium. An empirical function and a time delay were used to relate nephron glomerular filtration rate to the chloride concentration at the macula densa of the TAL. Analysis of the model equations indicated that limit-cycle oscillations (LCO) in nephron fluid flow and chloride concentration can emerge for suffficiently large feedback gain and time delay. In this study, the single-nephron model has been extended to two nephrons, which are coupled through their filtration rates. Explicit analytical conditions were obtained for bifurcation loci corresponding to two special cases: (1) identical time-delays, but differing gains, and (2) identical feedback gain magnitudes, but differing time delays. Similar to the case of a single nephron, the analysis indicates that LCO can emerge in coupled nephrons for sufficiently large gains and delays. However, these LCO may emerge at lower values of the feedback gain, relative to a single (i.e., uncoupled) nephron, or at shorter delays, provided the delays are sufficiently close. These results suggest that, in vivo, if two nephrons are sufficiently similar, then coupling will tend to increase the likelihood of LCO. (In collaboration with Roman M. Zaritski, Leon C. Moore and H. E. Layton)

The object of our study is the set of equations of thermo-elasticity with viscosity and heat conduction. These equations include, as a special case, the compressible Navier-Stokes equation familiar from gas dynamics, but in addition allow for solid-like materials. We seek to understand the temporal asymptotic fate of large initial data under a variety of boundary conditions. The realm of phase changes, such as occur in Van-der-Waals gases and martensitic transformations, are of especial interest. Now, obtaining point-wise a priori bounds on the density which are time independent is a major analytical obstacle to resolving this question. We present two new results on this issue. First, for specified-stress boundary conditions we give a positive result applicable to a general class of materials. Second, for Dirichlet boundary conditions we derive the estimates for a special class of gaseous materials; p'th power gases. We conclude with a discussion on the relation between asymptotic states and minimization principles of associated free energies. Numerical simulations will highlight some surprising features of the dynamics. In particular, the limiting states are not necessarily strong minimizers, in the sense of the calculus of variations, of the free energy.

We consider a process of one-dimensional fluid flow through a soil packed tube in which a contaminant is initially distributed. The contaminant concentration, as a function of location in the tube and time after flushing begins, is classically modeled as the solution of a linear second order partial differential equation. Here, we consider the related issues of how contaminant concentration measured at some location-time combinations can be used to approximate concentration at other locations and times (ie., exprimental design). The method is demonstrated for the case in which initial concentrations are approximated based on data collected only at the downstream end of the tube. Finally, the effect of misspecifying one of the model parameters is discussed, and alternative designs are developed for instances in which that parameter must be estimated from the data.

The assimilation of a simultaneous momentum and heat source into aligned uniform and non-uniform streams is considered.The governing boundary layer equations are transformed utilising aggregate properties of the flow field with respect to the excesses of heat and momentum flux at the source relative to the external stream.A non-dimensional downstream coordinate Þ reflecting the local relative importance of jet to external stream characteristics provides a unified framework within which to investigate the entire semi-infinite flow field downstream of the source.The problems examined devolve down to transitional flow in Þ between acknowledged strong jet and weak jet regimes in the immediate vicinity of and downstream of the source respectively.Perturbation solutions are developed in the two regimes.The downstream asymptotic velocity and temperature profiles are shown to be associated with new solutions of the Falkner-Skan equation subject to the boundary condition of symmetry,as opposed to no slip, at the axis of symmetry.A stability analysis of the new solutions and comprehensive numerical solutions over the full flow field confirm that there may be physically realisable flows in which a residual jet identity remains distinguishable within the downstream flow.

In the past two decades a new type of chaotic dynamics has been noticed in the three and four body problems which has not been understood. In 1986, using a numerical algorithm, an interesting region supporting chaotic motion was discovered about the moon, under the perturbation of the earth. This region is now termed the weak stability boundary. New types of dynamics were subsequently discovered near this boundary. These dynamics have the property that they give rise to very low energy trajectories with many important applications. In 1991, a new type of low energy trajectory to the moon was discovered which was used to place a Japanese spacecraft, Hiten, in orbit about the moon in October of that year. This was the first application of this type of dynamics to space travel. These low energy trajectories, so called WSB transfers, are now being planned by NASA, Europe and Japan for several new missions to the moon, Europa, Mars. Motion near this boundary also gives rise to an interesting resonance transition dynamics, and work by the speaker with Brian Marsden at Harvard is discussed in its relevance to short period comets, and Kuiper belt objects. An analytic representation for this boundary is also presented and its connections with heteroclinic intersections of hyperbolic invariant manifolds is discussed. If there is time, a new type of periodic motion for Hill's problem is looked at.

We describe a general Bayesian approach for using computer codes for a complex physical system to assist in forecasting actual system outcomes. Our approach is based on expert judgements and experiments on fast versions of the computer code. These are combined to construct models for the relationships between the code's inputs and outputs, respecting the natural space/time features of the physical system. The resulting beliefs are systematically updated as we make evaluations of the code for varying input sets and calibrate the input space against past data on the system. The updated beliefs are then used to construct forecasts for future system outcomes. While the approach is quite general, it has been developed particularly to handle problems with high-dimensional input and output spaces, for which each run of the computer code is expensive. The methodology will be applied to problems in uncertainty analysis for hydrocarbon reservoirs.

One of the classical examples of hydrodynamic instability occurs when two fluids are separated by a free surface across which the tangential velocity has a jump discontinuity. This is called Kelvin-Helmholtz Instability. Kelvin-Helmholtz instability is a fundamental instability of incompressible fluid flow at high Reynolds number. The idealization of a shear layered flow as a vortex sheet separating two regions of potential flow has often been used as a model to study mixing properties, boundary layers and coherent structures of fluids. In a joint work with G. Hu and P. Zhang, we study the singularity of 3-D vortex sheets using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contribution of the integral equation. Moreover, after applying a transformation to the physical variables, we found that this leading order 3-D vortex sheet equation de-generates into a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This rather surprising result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. It also shows that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Detailed numerical study will be provided to support the analytical results, and to reveal the generic form and the three-dimensional nature of the vortex sheet singularity.

A major problem of Petroleum engineering si the prediction of future oil and water production from a reservoir whose properties are inferred from measurements along well paths, and from observations of pressure, production, and fluid saturations at well locations. If the properties of the porous material were known at all locations, and all boundary conditions were specified, the production rates of fluids would be computed from the numerical solution of a set of partial differential equations governing mass conservation and flow. Rock properties are known to be heterogeneous on many scales, however, and the measurements are always insufficient to determine the properties throughout the reservoir. In the petroleum and groundwater fields, rock properties (permeability and porosity) are modeled as spatial random fields, whose auto-covariance and cross-covariances are known from ovservations of outcrops and cores. Uncertainty in future production is characterized by the empirical distribution from the suite of realizations of rock properties. The problem is assessing uncertainty in reservoir production or groundwater remediation predictions is that while valid prodecures for sampling the posterior pdf are available, the computational cost of generating the necessary number of samples from such procedures is prohibitive. An increase in computer speed is unlikely to solve this problem as the trend has been to build more complex numerical models of the reservoir as computer capability increases. Most recent effort has gone in to approximate methods of sampling. In this talk, I will describe our experience with the use of Markov Chain Monte Carlo methods and with approximate sampling methods.

The PLANET collaboration has monitored nearly 100 microlensing events of which more than 20 have the sensitivity required to detect perturbations due to a planetary companion to the primary lens. No planets have been detected. These null results indicate that Jupiter mass planets with separations from 1-5 AU are not common -- the first such limits for extrasolar planets at these separations by any technique. While interpretation of null results is not trivial, interpretation of future detections will be substantially more difficult, due to degeneracies among the planetary fit parameters and degeneracies with perturbations due to other, non-planetary phenomena. The analysis is further complicated by the unusual situation that observational strategies are altered real-time when perturbations are detected. I discuss these difficulties and present methods to cope with them. Finally, I discuss future prospects for microlensing planet searches.

We discuss a new class of cubic interpolating and approximating "L₁ splines" that preserve the shape both of smooth data and of data with abrupt changes in magnitude or spacing. The coefficients of these splines are calculated by minimizing the L₁ norm of the second derivatives. These splines do not require constraints, penalties, a posteriori filtering or interaction with the user. Univariate and multivariate cases are treated in one and the same framework. L₁ splines are implemented using efficient interior-point methods for linear programs.

Experiments on knotted DNA molecules suggest that certain physical properties of DNA knots can be predicted from a corresponding ideal shape. Intuitively, when a given knot in a piece of string is pulled tight, it always achieves roughly the same geometrical configuration, with a minimum length of string within the knot. Such a configuration is called an ideal shape for the knot, and approximations of ideal shapes in this sense have been found via a series of computer experiments. These shapes have intriguing physical features and have been shown to capture average properties of knotted polymers. But when does a shape satisfy the intuitive geometrical definition for ideality? In this talk I show that ideal shapes can be understood using only elementary (but new!) mathematics. In particular, I show that global curvature, a very natural and simple generalization of the classic concept of local curvature, leads to a simple characterization of an ideal shape and to a necessary condition for ideality. Another application of global curvature can be found in characterizing the equilibria of knotted curves or rods, which may exhibit self-contact after sufficient twisting. Here global curvature provides a simple way to formulate the constraint that prevents a rod from passing through itself.

After the sun, the oceans are the most significant contributor to our climate. Oceanic surface gravity waves are thought to have no influence on the global circulation of the oceans. However, oceanic surface gravity waves have a mean Lagrangian motion, the Stokes drift. This talk will present preliminary results that suggest that the dynamics of basin-scale oceanic currents are modified by the presence of the Stokes drift. In places where the Stokes drift is significant, it is possible that the ocean circulation, and hence climate, is not entirely well captured by present day models of the general circulation.

Special day: Thursday, 4pm, Room 120 Physics

A plethora of biological and chemical pattern formation problems are modeled using coupled reaction-diffusion equations of activator-inhibitor type. In 1993, the new phenomenon of self- -replicating spots and pulses was discovered in the Gray-Scott model and in a ferrocyanide reaction it models, by John Pearson and Harry Swinney and collaborators. In this talk, we present an analysis of pulse splitting. Furthermore, it turns out that the perturbation theory developed for the Gray-Scott analysis can be extended in a natural way to analyze a general class of coupled activator-inhibitor systems, including the Gierer-Meinhart and Schnakenberg models, in order to determine whether pulses attract or repel each other, and if they repel, whether they also the undergo self-replication. The work may be classified as a treatment of the moderately strong and strong pulse interaction problem. This work is part of a larger collaborative project with Arjen Doelman, Wiktor Eckhaus, Rob Gardner and my student Dave Morgan. We will conclude with some open questions.

One of the pressing problems in the analysis of reaction-diffusion equations is obtaining accurate and reliable estimates of the error of numerical solutions. Recently, we made significant progress using a new approach that at the heart is computational rather than analytical. I will describe a framework for deriving and analyzing a posteriori error estimates, discuss practical details of the implementation of the theory, and illustrate the error estimation using a variety of well-known models. I will also briefly describe an application of the theory to the class of problems that admit invariant rectangles and discuss the preservation of invariant rectangles under discretization.

Adaptive methods for solving systems of partial differential equations have become widespread. Robust adaptive software for solving parabolic systems in one and two space dimensions is now widely available. Three spatial adaptive strategies and combinations thereof are frequently employed: mesh refinement (h-refinement); mesh motion (r-refinement); and order variation (p-refinement). These adaptive strategies are driven by a priori and a posteriori error estimates. I will present an adaptive h-refinement finite element code in three dimensions on structured grids. These structured grids contain irregular nodes. Solution values at these nodes are determined by continuity requirements across element boundaries rather than by the differential equations. The differential-algebraic system resulting from the spatial discretization is integrated using Linda Petzold's multistep DAE code DASPK. The large linear systems resulting from Newton's method applied to nonlinear system of differential algebraic equations is solved using preconditioned GMRES. In DASPK the matrix-vector products needed by GMRES are approximated by a ``directional derivative''. Thus, the Jacobian matrix need not be assembled. However, this approach is inefficient. I have modified DASPK to compute the matrix-vector product using stored Jacobian matrix. As in the earlier version of DASPK, DASSL, this matrix is kept for several time steps before being updated. I will discuss appropriate preconditioning strategies, including fast-banded preconditioners. In three dimensions when using multistep methods for time integration it is crucial to use a ``warm restart'', that is, to restart the dae solver at the current time step and order. This requires interpolation of the history information. The interpolation must be done in such a way that mode irregularity is enforced on the new grid. A posteriori error estimates on uniform grids can easily be generalized from two-dimensional results (Babuska and Yu showed that in the case of odd order elements, jumps across elemental boundaries give accurate estimates, and in the case of even order elements, local parabolic systems must be solved to obtain accurate estimates). Babuska's work can even be generalized to meshes with irregular modes but now they no longer converge to the true error (in the case of odd order elements). I have developed a new set of estimates that extend the work of Babuska to irregular meshes and finite difference methods. These estimates provide a posteriori error indicators in the finite element context. Several examples that demonstrate the effectiveness of the code will be given.

Most dangerous cardiac arrhythmia, ventricular fibrillation (VF), is characterized by chaotic electrical behavior of the tissue. At the onset of the first, more organized stage of VF waves of electric activity in the heart become reentrant leading to fast irregular contraction. Better understanding of the mechanisms underlying early VF events will lead to more efficient treatment. Reentry induction has been performed in several experiments. We devise a model of cardiac tissue and use it to obtain a close match to the experimental results. The model combines macroscopic and microscopic properties of cardiac tissue.

A general consumer--resource model assuming discrete consumers and a continuously structured resource is examined. We study two foraging behaviors, which lead to fixed and flexible patch residence times, in conjunction with a simple consumer energetics model linking resource consumption, foraging behavior, and metabolic costs. Results indicate a single evolutionarily stable foraging strategy for fixed and flexible foraging in a nonspatial environment, but flexible foraging in a spatial environment leads to consumer grouping, which affects the resource distribution such that no single foraging strategy can exclude all other strategies. This evolutionarily stable coexistence of multiple foraging strategies may help explain a dichotomous pattern observed in a wide variety of natural systems.

We investigate the effect of finite horizontal boundary properties on the critical Rayleigh and wave numbers for controlled Rayleigh-Benard convection in an infinite horizontal domain. Specifically, we examine boundary thickness, thermal diffusivity and thermal conductivity. Our control method is through perturbation of the lower boundary heat flux. A linear differential-proportional control law uses the local amplitude of a shadowgraph to actively distribute the lower boundary heat flux. Realistic boundary conditions for laboratory experiments are selected. Through linear stability analysis and experiment we examine the important boundary properties and make predictions of the properties necessary for successful control experiments. A surprising finding of this work is that for certain realistic parameter ranges, one may find an isola to time-dependent convection as the primary bifurcation.

Many physical problems involve interfaces. Examples include phase transition problems where the interface separates the solid and liquid regions, bubble simulation, Hele-Shaw flow, composite materials, and many other important physical phenomena. Mathematically, interface problems usually lead to differential equations whose input data and solutions have discontinuities or non-smoothness across interfaces. As a result, many standard numerical schemes do not work or work poorly for interface problems. This is an introductory talk about the interface problems and our immersed interface method. Through some simple examples, I will try to explain the problems of our interest and related background information. Then I will present our method for some typical model problems in two dimensions. Our method can handle both discontinuous coefficients and singular sources. The main idea is to incorporate the known jumps in the solution and its derivatives into the finite difference scheme, obtaining a modified scheme on the uniform grid for quite arbitrary interfaces. The second part of the talk will focus on applications of the methods combined with the the level set method for moving interface problems: including the Stokes flow with different surface tension, the simulation of Hele-Shaw flow, and computation of crystal growth.

Sea ice undergoes a marked transition in its transport properties at a critical temperature of around -5 C. Above this temperature, the sea ice is porous, allowing percolation of brine, sea water, nutrients, biomass, and heat through the ice. In the Antarctic, this critical behavior plays a particularly important role in air-sea-ice interactions, mixing in the upper ocean, in the life cycles of algae living in the sea ice, and in the interpretation of remote sensing data on the sea ice pack. Recently we have applied percolation theory to model the transition in the transport properties of sea ice. We give an overview of these results, and how they explain data we have taken in the Antarctic. We will also describe recent work in developing inverse algorithms for recovering the physical properties of sea ice remotely through electromagnetic means, and how percolation processes come into play. At the conclusion of the talk, we will show a short video on a recent winter expedition into the Antarctic sea ice pack.

I will explain some basic concepts for fluid flow in porous media, go through some details about the 3D streamline simulator and SGSIM (Sequential Gaussian SIMulation) for generating random permeability fields under multiGaussian assumption.