Inverse spectral theory for a class of non-compact Hankel operators

We characterize all bounded Hankel operators $\Gamma$ such that $\Gamma^*\Gamma$ has finite spectrum. We identify spectral data corresponding to such operators and construct inverse spectral theory including the characterization of these spectral data

Weighted model spaces and Schmidt subspaces of Hankel operators

For a bounded Hankel matrix $\Gamma$, we describe the structure of the Schmidt subspaces of $\Gamma$, namely the eigenspaces of $\Gamma^* \Gamma$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term "weighted model space" to describe the range of an isometric multiplier acting on a model space. Further, we obtain similar results for Hankel operators acting in the Hardy space on the real line. Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using the language of weighted model spaces.Comment: Final version, to appear in Journal of the London Mathematical Societ

Biased random walks on random graphs

These notes cover one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. The aim is to review recent mathematical developments in the field of random walks in random environment. Our main focus will be on directionally transient and reversible random walks on different types of underlying graph structures, such as $\mathbb{Z}$, trees and $\mathbb{Z}^d$ for $d\geq 2$.Comment: Survey based one of the topics programmed for the St Petersburg School in Probability and Statistical Physics of June 2012. 64 pages, 16 figure

Response of Marine‐Terminating Glaciers to Forcing: Time Scales, Sensitivities, Instabilities, and Stochastic Dynamics

Recent observations indicate that many marine‐terminating glaciers in Greenland and Antarctica are currently retreating and thinning, potentially due to long‐term trends in climate forcing. In this study, we describe a simple two‐stage model that accurately emulates the response to external forcing of marine‐terminating glaciers simulated in a spatially extended model. The simplicity of the model permits derivation of analytical expressions describing the marine‐terminating glacier response to forcing. We find that there are two time scales that characterize the stable glacier response to external forcing, a fast time scale of decades to centuries, and a slow time scale of millennia. These two time scales become unstable at different thresholds of bed slope, indicating that there are distinct slow and fast forms of the marine ice sheet instability. We derive simple expressions for the approximate magnitude and transient evolution of the stable glacier response to external forcing, which depend on the equilibrium glacier state and the strength of nonlinearity in forcing processes. The slow response rate of marine‐terminating glaciers indicates that current changes at some glaciers are set to continue and accelerate in coming centuries in response to past climate forcing and that the current extent of change at these glaciers is likely a small fraction of the future committed change caused by past climate forcing. Finally, we find that changing the amplitude of natural fluctuations in some nonlinear forcing processes, such as ice shelf calving, changes the equilibrium glacier state

The cubic Szego equation on the real line: explicit formula and well-posedness on the Hardy class

We establish an explicit formula for the solution of the cubic Szego equation on the real line. Using this formula, we prove that the evolution flow of this equation can be continuously extended to the whole Hardy class $H^2$ on the real line. <br/