Georgia Institute of Technology College of Sciences

We shall discuss the quantum dynamics associated with ergodic
Schroedinger operators with singular continuous spectrum. Upper bounds
on the transport moments have been obtained for several classes of
one-dimensional operators, particularly, by Damanik--Tcheremchantsev,
Jitomirskaya--Liu, Jitomirskaya--Powell. We shall present a new method
which allows to recover most of the previous results and also to
obtain new results in one and higher dimensions. The input required to
apply the method is a large-deviation estimate on the Green function
at a single energy. Based on joint work with S. Sodin.

The talk will be online at https://gatech.zoom.us/j/96285037913

We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disk almost as large as the optimal disk for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim1/(e\Delta)$). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge-sizes strictly greater than $2$. We conjecture that for $k\geq 3$, there exist families of $k$-uniform linear hypergraphs that have a much larger zero-free disk of radius $\Omega(\Delta^{-1/(k-1)})$. We establish this in the case of linear hypertrees. Joint work with David Galvin, Gwen McKinley, Will Perkins and Prasad Tetali.

A meromorphic inner function is a bounded analytic function on the upper half plane with unit modulus almost everywhere on the real line and a meromorphic continuation to the complex plane. Meromorphic inner functions and equivalently meromorphic Herglotz functions play a central role in inverse spectral theory of differential operators. In this talk, I will discuss some uniqueness problems for meromorphic inner functions and their applications to inverse spectral theory of canonical Hamiltonian systems as Borg-Marchenko type results.

I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.