MSI Weekly Bulletin - Week starting Monday 30 October, 2006

In this talk, I will focus on a general class of statistical problems where the underlying objective is to detect items belonging to a rare class from a very large database. I will introduce an efficient computational algorithm called LAGO. In theory, I will show that LAGO can be justified as an adaptive-bandwidth kernel density estimate of the rare class density that is then adjusted locally by a factor which approximates the background class density to the first order. I will also argue that LAGO is a highly efficient way to construct a radial basis function network (RBFnet) for the rare target detection problem.

All knowledge of the interior of the Earth is based on indirect inference. Even apparently simple tasks such as the location of seismic events are actually highly non-linear inverse problems with data inputs of various types and quality. Many problems involve either data dependency on multiple classes of parameters or many different sources of data associated with the same description of an Earth model. The result is that there has been a strong independent tradition of innovation in geophysical inverse problems, since conventional tools do not directly translate to the problems at hand. A major problem is the description of the 3-dimensional interior structure of the Earth using observations of seismograms at the Earth's surface, which can rapidly lead to large numbers of parameter and data inputs. The dominant structure depends on radius and so progress has been made by developing reference models for the average radial structure of the seismic wavespeed in the Earth and then seeking the 3-D variations about this state. I will illustrate the successes and problems associated with the generation of such reference models and the current state of imaging for 3-D structure on both global and regional scales.

The category of small covariant functors from simplicial sets to simplicial sets supports the projective model structure. In this talk we will describe various localizations of the projective model structure and also give a variant for functors from simplicial sets to spectra. We apply these model categories in the study of calculus of functors, namely for classification of polynomial and homogeneous functors. Finally we show that the $n$-th derivative induces a Quillen map between the $n$-homogeneous model structure on small functors from pointed simplicial sets to spectra and the category of spectra with $\Sigma_n$-action. We consider also a finitary version of the $n$-homogeneous model structure and the $n$-homogeneous model structure on functors from pointed finite simplicial sets to spectra. In these two cases the above Quillen map becomes a Quillen equivalence. This improves the classification of finitary homogeneous functors by T. G. Goodwillie. (Joint work with G.Biederman and O.Roendigs.)

This talk concerns attractors of hyperbolic iterated function systems (IFSs). An IFS is a finite set of strictly contractive transformations acting on a complete metric space. Its attractor is the unique nonempty compact set which is the same as the union of its images under all of the transformations in the IFS. Examples of IFS attractors include such familiar fractals such as Sierpinski triangles and Julia sets. We will show that any IFS attractor has associated with it a set of shift spaces which arise in a natural manner. Each of these shift spaces provides what we call an address structure for the IFS attractor. This address structure provides a complete invariant for the IFS attractor: two IFS attractors are homeomorphic if and only if they possess the same address structure. We will also define and disuss some related invariants including the entropy of an IFS attractor. We will also discuss transformations between arbitrary pairs of IFS attractors.

In this talk we discuss geometric properties of positive solutions to Neumann problems for semi-linear elliptic equations. The issues we are concerned are the existence and the profile of solutions. When a parameter is very small, the solutions become concentrated around certain points and their shapes look like spikes. In a conformal metric these spikes are asymptotically spheres or bubbles.