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Analysis/PDE Reading Seminar

Date Time	Location	Speaker	Title – click for abstract
01/31 4:00pm	MILN317	Constanze Liaw TAMU	*Dynamical properties of perturbations* We study the properties of the spectral measure of self-adjoint rank one perturbations via two methods, singular integral operators and model theory. A further investigation of our approach using singular integral operators gives rise to a new interpretation of such operator in which we, in particular, include Calderon-Zygmund kernels and allow measures without any growth condition. The material is presented at the level of a Colloquium talk.
02/14 4:00pm	MILN317	Andrew Comech TAMU	*Spectral stability of nonlinear Dirac equation. On bifurcations of eigenvalues from the essential spectrum.* The nonlinear Dirac equation has solitary wave solutions of the form \phi_\omega(x)e^{-i\omega t}, with \omega from certain interval. We consider the linearization of the nonlinear Dirac equation at one of its solitary waves. The solitary wave is called spectrally stable if the spectrum of such a linearization has no eigenvalues with positive real part. We prove that point eigenvalues could only emerge from the eigenvalues embedded into the essential spectrum, and then prove the absence of such embedded eigenvalues beyond the "threshold" points. The goal is to prove that unstable eigenvalues could ONLY bifurcate from the origin, as a result of collision of imaginary eigenvalues (as it happens for the NLS). The approach is based on Agmon-type estimates in the weighted spaces near the essential spectrum. We will consider how this analysis works in 1D. This is a joint work with N. Boussaid, Universite-Besancon.
02/21 4:00pm	MILN317	Roman Bessonov Chebyshev Laboratory, St. Petersburg, Russia	*Truncated Toeplitz operators* Truncated Toeplitz operators are Toeplitz operators in the star-invariant subspaces ${K_\theta = H^2 \ominus \theta H^2}$ of the Hardy class $H^2$, where $\theta$ is an inner function. They can be regarded as generalizations of truncated Wiener-Hopf operators. In the talk we discuss the following related topics: \begin{itemize} \item[1)] Truncated Toeplitz operators; \item[2)] Factorizations of functions from $K_\theta$; \item[3)] Carleson measures for $K_\theta$. \end{itemize} The talk is based on the joint work with A.Baranov and V.Kapustin.
03/06 4:00pm	MILN317	Alexei Poltoratski TAMU	*Oscillations of Fourier Integrals with a Spectral Gap* I will discuss properties of measures whose Fourier transform vanishes on an interval. I will start by reviewing solutions to the Gap and Type problems in Harmonic Analysis, where such measures appear. The main new result of the talk is an extension of a theorem by Eremenko and Novikov on the density of the sequence of sign changes of a measure with a spectral gap. The theorem confirmed a conjecture by Gurevich from 1965 and solved one of the problems from Arnold's list (2000). Some of the results in the talk are joint with M. Mitkovski.
04/10 4:00pm	MILN317	Gregory Berkolaiko TAMU	*Colin de Verdiere's proof of the magnetic nodal count theorem* I will discuss the proof by Colin de Verdiere of a theorem which links the number of sign flips in an eigenfunction of a discrete Laplacian (on a graph) with the Morse index associated to deformations of the Laplacian by magnetic fields. The proof is in the preprint arXiv:1201.1110. Original theorem is in arXiv:1110.5373.

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