Mathematics Colloquium Series

Mathematics Department Colloquium Series 

The Mathematics Department established the Mathematics Department Colloquium Series through a Faculty Development Grant in 1999 to provide a place to discuss new research in Mathematics. Speakers for the series come from both inside the UTC Department of Mathematics as well as guest speakers from other institutions. The Colloquium Series meets periodically throughout the year.

Upcoming Colloquium Talks

Brian Choi

• Date/Time: Fri., Nov. 7 at 2:30 p.m.
• Location: Lupton 393
• Title: Excitation thresholds and breather dynamics in multi-dimensional nonlocal lattices
• Abstract: This talk surveys new results on discrete breathers in multi-dimensional lattices with long-range (nonlocal) coupling. Using a variational framework and concentration–compactness, I obtain an exact threshold condition that ties together the lattice dimension, the strength of the nonlinearity, and the nonlocality, yielding a sharp mass-threshold dichotomy: no positive threshold in the mass-subcritical regime and a strictly positive threshold at and above criticality. In the anti-continuum limit, I show uniqueness of ground states and derive analytic formulas that quantify the excitation threshold. Then, I establish sharp spatial decay of ground states and identify a qualitative change in dispersive ime 
  decay at α=1, corroborated by numerics. The results extend the nearest-neighbor DNLS theory to genuinely nonlocal lattices and clarify when localization persists versus when dispersive dynamics dominate.

Wilhelm Treschow

• Date/Time: Fri., Nov. 14 at 3:30 p.m.
• Location: Lupton 393
• Title: Inverse scattering theory for embedded eigenvalues: A Marchenko approach
• Abstract: In this talk I will explore an approach to apply inverse scattering theory of Schrödinger operators to reconstruct the potential from the scattering data at embedded eigenvalues. I begin with an overview of embedded eigenvalues and why they are delicate (lying inside the continuous spectrum), as well as my previous results on the persistence of such eigenvalues under perturbations of the potential. This previous work led to a natural inverse question: Given certain information of the embedded eigenvalue, what can we say about the potential? In an attempt to answer this question, I will proceed with providing a motivating example and then outline a simpler one-dimensional setting and review the essentials of inverse scattering on the line, deriving the Levitan-Gelfand-Marchenko integral equation. I then explain the additional difficulties associated with the presence of embedded eigenvalues in a slightly more general setting and how we are aiming to resolve those issues