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.

A celebrated theorem of Carleson shows that the Fourier series of an L^2 function converges pointwise almost everywhere. At the heart of this work lies an L^2 estimate for a particular type of maximal singular integral operator, which has since become known as a Carleson operator. In the past 40 years, a number of important results have been proved for generalizations of the original Carleson operator. In this talk we will introduce the Carleson operator and survey its generalizations, and then describe new joint work with Po Lam Yung on Carleson operators with a certain type of polynomial phase that also incorporate the behavior of Radon transforms.

Functional information from the brain is available from a variety of modalities including functional magnetic resonance imaging (fMRI) and positron emission tomography (PET). Individual variability in the size, shape and extent of the folding patterns of the the human brain makes it difficult to compare functional activation differences across subjects. I will discuss a method that attempts to address this problem by creating flat maps of the cortical surface. These maps are produced using a novel computer realization of the Riemann Mapping Theorem that uses circle packings. These maps exhibit conformal behavior in that angular distortion is controlled. They are mathematically unique and canonical coordinate systems can be imposed on these maps. Some of the maps of the cortical surface that I have created in the Euclidean and hyperbolic planes and on a sphere will be presented.

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.

Abstract: One of the major themes of modern differential geometry is the relationship between the curvature and topology of Riemannian manifolds. In this lecture, I will describe some links between curvature and SMOOTH topological invariants --- i.e. invariants which can distinguish between different smooth structures on a given topological manifold. The specific invariants I will discuss are the Seiberg-Witten invariants of 4-manifolds, and I will describe the impact these have on the existence problem for Einstein metrics and some related Riemannian variational problems.

Dynamic singularities are ubiquitous. They arise in mathematical models of phenomena as grand as star formation or as familiar as the breakup of a thread of honey as it is being added to tea. Drop breakup allows one to study dynamics close to a singularity in a simple context which is also accessible to experiments. Recent works have revealed that a viscous liquid drop close to breakup looks self-similar---the drop profile looks the same if the length scales are rescaled appropriately. A new numerical strategy is developed to capture the drop breakup dynamics and show good agreement with experimental measurements. Surprisingly, the presence of even small amounts of viscous dissipation in the surrounding can dramatically alter the self-similar profile. In particular, when no exterior viscous dissipation is present, the thread profile is symmetric about the point of pinch-off. When small amounts of exterior viscous dissipation are present, the thread profile becomes severely asymmetric. An understanding of the final breakup process is crucial in elucidating the mechanisms underlying the formation of satellite drops, an issue relevant to the development of ink-jet printing technologies and emulsification processes.

A limit of a suitably-chosen string theory compactified on an elliptic Calabi-Yau threefold is believed to be equivalent to a Yang-Mills field theory with gravity in six dimensions. The Lie group associated to this Yang-Mills theory is encoded in the geometry of the Calabi-Yau space --- the Cartan subalgebra is generated by ruled surfaces and the weights of representations of fields appear as rational curves living inside the threefold. Cancellation of anomalies in the six-dimensional field theory predicts rather peculiar constraints on the configurations of such ruled surfaces and rational curves.