Events for the week of October 26 2009

Abstract. Given a family of p-divisible groups, the Newton polygon defines a stratification of the base. Over each stratum one may define a p-adic monodromy representation. I'll present some work which indicates that, in a neighborhood of the boundary of a Newton stratum, the local monodromy is already quite large. (Joint with Peter Norman.)

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Abstract. In the past decades the G(n,p) model of random graphs, introduced by Erdös and Renyi in the 60's, has led to numerous beautiful and deep results. A key feature that is used in basically all proofs is that edges in G(n,p) appear independently. This situation changes dramatically if one considers graph classes with structural side constraints. For example, in a random planar graph Pn (a graph drawn uniformly at random from the class of all labeled planar graphs on n vertices) the edges are obviously far from being independent. In this talk we show that recent progress in the construction of Boltzmann samplers can be used to reduce the study of properties of combinatorial objects to properties of sequences of independent and identically distributed random variables -- to which we can then again apply well known machinery from random graph theory.

Abstract. Shape analysis plays an important role in brain mapping as it has the potential of computing informative biomarkers for the early diagnosis and tracking of neurological diseases. In this talk, I will present our recent work on 3D brain shape analysis using the eigenfunctions of the Laplace-Beltrami (LB) operator. The spectra of the LB operator provides an intrinsic way of characterizing 3D shapes. By computing the Reeb graph of LB eigenfunctions, we can construct robust skeletal features of 3D surfaces. Using global features extracted from the LB eigenfunctions, we can compute robust surface maps that enable the localized analysis of morphometric changes. Experimental results on the automated analysis of thousands of shapes will be presented.

Abstract. An asymptotic theory for randomly-forced discrete heat equations Davar Khoshnevisan Summary. Stochastic heat equations are a family of fundamental stochastic processes that arise in a variety of scientific settings. In this talk, I describe these equations in a completely-discrete setting, derive them from a simple particle picture, and say some things about their moments' Liapounov exponent[s], if and when a solution exists. This is joint work with M. Foondun

Abstract. By a singular manifold we understand a stratified space, iteratively modeled on cones or wedges with bases of lower singular order. Examples are manifolds with conical or edge singularities, or manifolds with (smooth or singular) boundary. Configurations of that kind appear in diverse applications, and a typical problem is to describe regularity and asymptotics of solutions near the geometric singularities. We consider the case of elliptic operators, i.e., when the components of a certain principal symbolic hierarchy are invertible. In special cases the meaning of the components is standard (e.g., in conical singularities we have conormal symbols, in boundary value problems boundary symbols, etc.) However, for more general singularities (even in the edge case) the existence and the role of such symbols is much less evident. Asymptotics of solutions along edges may be variable and branching; the involved data are determined by variable and branching poles of families of meromorphic operator functions (the conormal symbols subordinate to the edge symbols). We develop a general framework to understand their nature in terms of vector and operator-valued analytic functionals in the complex plane of the Mellin covariable belonging to the distance variable to the edge, and we estabish parametrices that are sensitive enough to produce such asymptotics in the process of characterizing the corresponding elliptic regularity.

Abstract. Borel equivalence relations arising from recursion theoretic reducibilities seem to exhibit a stubborn resistance to complete classification. One reason may be seen in the fact that they relate to deep recursion theoretic problems, first and foremost Martin's Conjecture on degree invariant functions. I will present some recent progress on the classification of several recursion theoretic equivalence relations.

Abstract. The McK--V system is a non--linear diffusion equation with a non-- local non--linearity provided by convolution. Recently popular in a variety of applications, it enjoys an ancient heritage as a basis for understanding equilibrium and near equilibrium fluids. The model is discussed in finite volume where, on the basis of the physical considerations, the correct scaling (for the model itself) is identified. For dimension two and above and in large volume, the phase structure of the model is completely elucidated in (somewhat disturbing) contrast to dynamical results. This seminar represents joint work with V. Panferov.