The Einstein equations have been a source of many interesting problems in Physics, Analysis and Geometry. Despite the great deal of work which has been devoted to them, with many success stories, several important questions remain open. One of the them is a satisfactory theory of isolated systems, such as stars, both from a perspective of the time development of the space-time, as well as from the point of view of the geometry induced on a space-like three surface. This talk will focus on the former situation. More specifically, we shall discuss relativistic fluids with and without viscosity, and prove a well-posedness result for the Cauchy problem. The viscous case, in particular, is of significant interest in light of recent developments in astrophysics.

In 1973, Richard Thompson considered the question of whether his newly defined group $F$ was amenable. The motivation for this problem stemed from his observation --- later rediscovered by Brin and Squire --- that $F$ did not contain a free group on two generators, thus making it a candidate for a counterexample to the von Neumann-Day problem. While the von Neumann-Day problem was solved by Ol'shanskii in the class of finitely generated groups and Ol'shanskii and Sapir in the class of finitely presented groups, the question of $F$'s amenability was sufficiently basic so as to become of interest in its own right. In this talk, I will analyze this problem from a Ramsey-theoretic perspective. In particular, the problem is related to generalizations of Ellis's Lemma and Hindman's Theorem to the setting of nonassociative binary systems. The amenability of $F$ is itself equivalent to the existence of certain finite Ramsey numbers. I will also discuss the growth rate of the F\olner function for $F$ (if it exists). Tea at 3:30 pm in SC 1425.

This is a joint work with Denis Osin. We show that every group $H$ of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group $G$ such that $G$ is amenable (respectively, solvable, satisfies a non-trivial identity, elementary amenable, of finite decomposition complexity, etc.) whenever $H$ is. We will briefly discuss some applications to subgroup distortion, compression functions of Lipschitz embeddings into uniformly convex Banach spaces, F\o lner functions, and elementary classes of amenable groups.

We propose a method for the 3D-rigid motion invariant texture discrimination for discrete 3D-textures that are spatially homogeneous. We model these textures as stationary Gaussian random fields. We formally develop the concept of 3D-texture rotations in the 3D-digital domain. We use this novel concept to define a `distance' between 3D-textures that remains invariant under all 3D-rigid motions of the texture. This concept of `distance' can be used for a monoscale or a multiscale setting to test the 3D-rigid motion invariant statistical similarity of stochastic 3D-textures. To extract this novel texture `distance' we use the Isotropic Mutliresolution Analysis. We also show how to construct wavelets associated with this structure by means of extension principles and we discuss some very recent results by Atreas, Melas and Stavropoulos on the geometric structure underlying the various extension principles. The 3D-texture `distance' is used to define a set of 3D-rigid motion invariant texture features. We experimentally establish that when they are combined with mean attenuation intensity differences the new augmented features are capable of discriminating normal from abnormal liver tissue in arterial phase contrast enhanced X-ray CT-scans with high sensitivity and specificity. To extract these features CT-scans are processed in their native dimensionality. We experimentally observe that the 3D-rotational invariance of the proposed features improves the clustering of the feature vectors extracted from normal liver tissue samples. This work is joint with R. Azencott, S. Jain, S. Upadhyay, I.A. Kakadiaris and G. Gladish, MD.

The local properties of non-negative weak solutions to the singular parabolic equation $u_t-\Delta \ln u = 0$ are largely unclear though some research has been done for the Cauchy problem of such an equation.  In this talk, we address the local positivity of this equation in the form of a Harnack-type inequality. Under the assumption $\ln u$ is sufficiently integrable, we show if $u$ does not vanish identically in a space neighborhood of $x_0$ and on some time level $t_0$ then $u$ is positive in a neighborhood of $(x_0,t_0)$.

Over the last two decades our understanding of small index subfactors has improved substantially. We have discovered a slew of examples, some related to finite groups or quantum groups, and other `sporadic' examples. At present we have a complete classification of (hyperfinite) subfactors with index at most 5, and a few results that push past 5. I'll explain the main techniques behind these classification results, and also spend a little time describing how we construct the sporadic examples. (Joint work with many people!) Tea at 3:30 pm in SC 1425.

I'll introduce Khovanov homology, a `categorical' extension of the Jones polynomial. Since the discovery of Khovanov homology, representation theorists have opened up a new world of categorical quantum groups. In this talk, I'll head in a different direction, explaining how the 4-dimensional nature of Khovanov homology makes it ideally suited for building a new 4-manifold invariant. I'll explain the construction, then discuss its present limitations and how we hope to get past them. (Joint work with Kevin Walker)

Compressed sensing is a recent development in the field of sampling Based on the notion of sparsity, it provides a theory and techniques for the recovery of images and signals from only a relatively small number of measurements. The key ingredients that permit this so-called subsampling are (i) sparsity of the signal in a particular basis and (ii) mutual incoherence between such basis and the sampling system. Provided the corresponding coherence parameter is sufficiently small, one can recover a sparse signal using a number of measurements that is, up to a log factor, on the order of the sparsity. Unfortunately, many problems that one encounters in practice are not incoherent. For example, Fourier sampling, the type of sampling encountered in Magnetic Resonance Imaging (MRI), is typically not incoherent with wavelet or polynomials bases. To overcome this `coherence barrier' we introduce a new theory of compressed sensing, based on so-called asymptotic incoherence and asymptotic sparsity. When combined with a semi-random sampling strategy, this allows for significant subsampling in problems for which standard compressed sensing tools are limited by the lack of incoherence. Moreover, we demonstrate how the amount of subsampling possible with this new approach actually increases with resolution. In other words, this technique is particularly well suited to higher resolution problems. This is joint work with Anders Hansen and Bogdan Roman (University of Cambridge).

There are only two known examples of positively curved compact (orientable) Einstein 4-manifolds: The round metric on the 4-sphere, and the Fubini-Study submersion metric on the complex projective plane. It is an open question whether or not this is the complete list. In this talk, we will prove that if we in addition assume that the metric is compatible with a complex structure on the manifold, then it has to be the Fubini-Study metric.