## Lillian Pierce : Carleson operators of Radon type

- Other Meetings and Events ( 40 Views )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.

## Monica Hurdal : Approximating Conformal (Angle-Preserving) Flat Maps of Cortical Surfaces

- Other Meetings and Events ( 39 Views )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.

## Claude LeBrun : Einstein Metrics, Variational Problems, and Seiberg-Witten Theory

- Other Meetings and Events ( 37 Views )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.

## Christian Hasse : Perles at Bings House -- Facet Subgraphs of Simple Polytopes

- Other Meetings and Events ( 34 Views )The combinatorial structure of a d-dimensional simple convex polytope -- as given, for example, by the set of the (d-1)-regular subgraphs of facets -- can be reconstructed from its abstract graph [Blind & Mani 1988, Kalai 1988]. However, no polynomial/efficient algorithm is known for this task, although a polynomially checkable certificate for the correct reconstruction exists [Kaibel & Koerner 2000]. A much stronger certificate would be given by the following characterization of the facet subgraphs, conjectured by M. Perles: ``The facet subgraphs of a simple d-polytope are exactly all the (d-1)-regular, connected, induced, non-separating subgraphs'' [Perles 1970,1984]. We give examples for the validity of Perles conjecture: In particular, it holds for the duals of cyclic polytopes, and for the duals of stacked polytopes. On the other hand, we observe that for any counterexample, the boundary of the (simplicial) dual polytope P^* contains a 2-complex without a free edge, and without 2-dimensional homology. Examples of such complexes are known; we use a simple modification of ``Bing's house'' (two walls removed) to construct explicit 4-dimensional counterexamples to Perles' conjecture.

## Paul Aspinwall : Lie Groups, Calabi-Yau Threefolds and Anomalies

- Other Meetings and Events ( 32 Views )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.

## Zhuoxin Bi : tracer flow in random porous media

- Other Meetings and Events ( 29 Views )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.

## Dean Oliver : Sampling the Posterior Distribution for Reservoir Properties Conditional to Production Data

- Other Meetings and Events ( 29 Views )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.

## Phil Hanlon : The Combinatorial Laplacian

- Other Meetings and Events ( 28 Views )The combinatorial laplacian is a method for computing the rational homology of an algebraic or simplicial complex. Although the method is quite old, it has recently been applied with success to a number of problems in algebraic combinatorics. We will survey the method and describe some of these recent applications.