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Current Colloquia & Seminars

  1. Learning seminar on stability conditions: Quadratic differentials as stability conditions III

    Location: MSRI: Baker Board Room
    Speakers: Dylan Allegretti (MSRI - Mathematical Sciences Research Institute)

    This lecture is part 3 of an introduction to the work of Bridgeland and Smith. I will formulate the main theorem identifying stability conditions and quadratic differentials and talk about the relationship between stable objects and finite-length trajectories of quadratic differentials.

    Updated on Sep 19, 2019 12:26 PM PDT
  2. MLA-Graduate Student Seminar: Wave invariants and inverse spectral theory

    Location: MSRI: Simons Auditorium
    Speakers: Amir Vig (University of California, Irvine)

    The wave trace is a distribution on $\mathbb{R}$ given by $\sum_{j = 1}^\infty e^{it \lambda_j}$, where $\lambda_j^2$ are the (positive) eigenvalues of the Laplacian on a compact domain. In general, two linear waves can be superimposed to give another solution to the wave equation. When we add up a bunch of waves at different frequencies, the peak singularities appear at points with substantial constructive interference. On a manifold, the famous ``propagation of singularities” tells us that waves propagate along geodesics, so the constructive interference is most pronounced along orbits which are traversed infinitely often (i.e. periodic orbits). On the trace side of things, this phenomenon is reflected in the Poisson relation, which says that the singular support of the wave trace is contained in the length spectrum (the collection of lengths of all periodic orbits). For planar domains, the geodesic flow is replaced by the billiard (or broken bicharacteristic) flow and we see an interesting connection between geometric, dynamical and spectral properties of the domain. In this talk, we introduce some simple cases of wave trace formulas before discussing recent work on explicit formulas for wave invariants associated to periodic orbits of small rotation number. This involves proving a dynamical theorem on the structure of such orbits and then constructing an explicit oscillatory integral representation, which microlocally approximates the wave propagator in the interior.

    Updated on Sep 20, 2019 02:15 PM PDT