Georgia Institute of Technology College of Sciences

The lecture will outline how the method of characteristics can be used in the context of solutions to hyperbolic conservation laws that are merely continuous functions. The Hunter-Saxton equation will be used as a vehicle for explaining the approach.

Much research in modern, quantitative seismology is motivated -- on the one hand -- by the need to understand subsurface structures and processes on a wide range of length scales, and -- on the other hand -- by the availability of ever growing volumes of high fidelity digital data from modern seismograph networks or multicomponent acquisition systems developed for hydro-carbon exploration, and access to increasingly powerful computational facilities. We discuss (elastic-wave) inverse scattering of reflection seismic data, wave-equation tomography, and their interconnection using techniques from microlocal analysis and applied harmonic analysis. We introduce a multi-scale approach and present a framework of partial reconstruction in connection with limited boundary acquisition geometry. The formation of caustics leads to one of the complications which will be discussed. We illustrate various aspects of this research program with examples from global seismology and mineral physics coupled to thermo-chemical convection.

The Pentagram map is a projectively natural iteration on plane polygons. Computer experiments show that the Pentagram map has quasi-periodic behavior. I shall explain that the Pentagram map is a completely integrable system whose continuous limit is the Boussinesq equation, a well known integrable system of soliton type. As a by-product, I shall demonstrate new configuration theorems of classical projective geometry.

Real life networks are usually large and have a very complicated structure. It is tempting therefore to simplify or reduce the associated graph of interactions in a network while maintaining its basic structure as well as some characteristic(s) of the original graph. A key question is which characteristic(s) to conserve while reducing a graph. Studies of dynamical networks reveal that an important characteristic of a network's structure is a spectrum of its adjacency matrix. In this talk we present an approach which allows for the reduction of a general weighted graph in such a way that the spectrum of the graph's (weighted) adjacency matrix is maintained up to some finite set that is known in advance. (Here, the possible weights belong to the set of complex rational functions, i.e. to a very general class of weights). A graph can be isospectrally reduced to a graph on any subset of its nodes, which could be an important property for various applications. It is also possible to introduce a new equivalence relation in the set of all networks. Namely, two networks are spectrally equivalent if each of them can be isospectrally reduced onto one and the same (smaller) graph. This result should also be useful for analysis of real networks. As the first application of the isospectral graph reduction we considered a problem of estimation of spectra of matrices. It happens that our procedure allows for improvements of the estimates obtained by all three classical methods given by Gershgorin, Brauer and Brualdi. (Joint work with B.Webb) A talk will be readily accessible to undergraduates familiar with matrices and complex functions.