Georgia Institute of Technology College of Sciences

With recent advances in experimental imaging, computational methods, and dynamics insights it is now possible to start charting out the terra incognita explored by turbulence in strongly nonlinear classical field theories, such as fluid flows. In presence of continuous symmetries these solutions sweep out 2- and higher-dimensional manifolds (group orbits) of physically equivalent states, interconnected by a web of still higher-dimensional stable/unstable manifolds, all embedded in the PDE infinite-dimensional state spaces. In order to chart such invariant manifolds, one must first quotient the symmetries, i.e. replace the dynamics on M by an equivalent, symmetry reduced flow on M/G, in which each group orbit of symmetry-related states is replaced by a single representative.Happy news: The problem has been solved often, first by Jacobi (1846), then by Hilbert and Weyl (1921), then by Cartan (1924), then by [...], then in this week's arXiv [...]. Turns out, it's not as easy as it looks.Still, every unhappy family is unhappy in its own way: The Hilbert's solution (invariant polynomial bases) is useless. The way we do this in quantum field theory (gauge fixing) is not right either. Currently only the "method of slices" does the job: it slices the state space by a set of hyperplanes in such a way that each group orbit manifold of symmetry-equivalent points is represented by a single point, but as slices are only local, tangent charts, an atlas comprised from a set of charts is needed to capture the flow globally. Lots of work and not a pretty sight (Nature does not like symmetries), but one is rewarded by much deeper insights into turbulent dynamics; without this atlas you will not get anywhere.This is not a fluid dynamics talk. If you care about atomic, nuclear or celestial physics, general relativity or quantum field theory you might be interested and perhaps help us do this better.You can take part in this seminar from wherever you are by clicking onevo.caltech.edu/evoNext/koala.jnlp?meeting=M2MvMB2M2IDsDs9I9lDM92

The correlation of gaps in dimer systems was introduced in 1963 by Fisher and Stephenson, who looked at the interaction of two monomers generated by the rigid exclusion of dimers on the closely packed square lattice. In previous work we considered the analogous problem on the hexagonal lattice, and we extended the set-up to include the correlation of any finite number of monomer clusters. For fairly general classes of monomer clusters we proved that the asymptotics of their correlation is given, for large separations between the clusters, by a multiplicative version of Coulomb's law for 2D electrostatics. However, our previous results required that the monomer clusters consist (with possibly one exception) of an even number of monomers. In this talk we determine the asymptotics of general defect clusters along a lattice diagonal in the square lattice (involving an arbitrary, even or odd number of monomers), and find that it is given by the same Coulomb law. Of special interest is that one obtains a conceptual interpretation for the multiplicative constant, as the product of the correlations of the individual clusters. In addition, we present several applications of the explicit correlation formulas that we obtain.

In the talk we demonstrate the usefulness of the so-called Beveridge-Nelson decomposition in asymptotic analysis of sums of values of linear processes and fields. We consider several generalizations of this decomposition and discuss advantages and shortcomings of this approach which can be considered as one of possible methods to deal with sums of dependent random variables. This decomposition is derived for linear processes and fields with the continuous time (space) argument. The talk is based on several papers, among them [V. Paulauskas, J. Multivar. Anal. 101, (2010), 621-639] and [Yu. Davydov and V. Paulauskas, Teor. Verojat. Primenen., (2012), to appear]