Geometry & Analysis Seminar Spring 2020

THURSDAY, May 7, 2020

Gabriel Martins (CSU - Sacramento) 

Semiclassical mechanics on manifolds 

In this talk I will survey a number of results which show that dynamical properties of a classical mechanical system on a Riemannian manifold \((M,g)\) like completeness, integrability and ergodicity can manifest themselves in the spectral properties of the associated quantum system. We will start by discussing a simple quantization procedure and some of the issues that arise in the process. Then we will introduce the relevant results and analyze many examples where they can be more easily understood, while mentioning some open questions. 

THURSDAY, May 14, 2020

Joey Zou (Stanford)

Streak artifacts from non-convex metal objects in X-ray tomography

In X-ray CT scans with metallic objects, streak artifacts in the computed image may arise due to beam hardening effects, where the attenuation coefficient of metallic objects vary strongly with energy. A mathematical description of these artifacts using the notion of wavefront sets was given by Choi, Park, and Seo in 2014, followed by the work of Palacios, Uhlmann, and Wang, who gave quantitative descriptions of the artifacts that recovered qualitative observations from CT scans when the metallic objects are strictly convex. In this talk, I will discuss joint work with Yiran Wang which builds on the previous work by using microlocal analysis to study artifacts generated by non-convex metallic objects, as well as artifacts associated to a broader class of attenuation variations than was considered before. The problem relies on the analytic behavior of a nonlinear function composed with the image of the X-ray transform applied to certain functions, for which we use the work of Melrose, Ritter, Sa Barreto et al. on semilinear wave equations via the usage of iterated regularity spaces in which both the X-ray transform image and its nonlinear composition live.

THURSDAY, May 21, 2020

Steven Flynn (UCSC)

Quantizing the Fourier Slice Theorem

The Fourier Slice Theorem (or Projection-Slice Theorem) is a fundamental result exhibiting a duality of the Fourier transform in \(\mathbb{R}^n\)---that projecting a function onto a line in the direction \(\vec{\omega}\) in space is the same as restricting its Fourier modes to those parallel to \(\vec{\omega}\). It is used to invert Radon transforms in \(\mathbb{R}^n\), where one wishes to recover a function from knowledge of its integrals over hyperplanes. We present the analogous theorem on the Heisenberg group, where the object of interest is the X-ray transform for the sub-Riemannian metric. We then apply the Fourier Slice Theorem to prove that the Heisenberg X-ray transform is injective.

THURSDAY, May 28, 2020

Sean Gasiorek (University of Sydney)

Minkowski Billiards on the Hyperboloid of One Sheet

We give a review of Euclidean and pseudo-Euclidean billiards in the plane and in d-dimensional space. If the billiard table is bounded by confocal quadrics, periodic trajectories can be expressed in algebro-geometric terms based on work of Poncelet, Cayley, and others. In particular, we consider a billiard problem for compact domains on a hyperboloid of one sheet bounded by confocal quadrics using the pseudo-Euclidean metric. Using a matrix factorization technique of Moser and Veselov, the billiard is shown to be integrable in the sense of Liouville. Further, we derive a Cayley condition for the billiards under consideration and explore geometric consequences.

THURSDAY, June 4, 2020

Amir Vig (UC Irvine)

Wave invariants and inverse spectral theory

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 in a smooth, strictly convex bounded planar domain. 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.