To join us for dinner after the talks, please RSVP here by Thursday, August 24 for reservation count.

Getting to the roots: From X-ray crystallography to factoring polynomials
Bernhard Bodmann
University of Houston

This talk concerns a problem in non-linear signal reconstruction which has a
long history, unsolved problems and many modern applications: signal
recovery from intensity measurements. A notorious example is X-ray
crystallography, the determination of a function from the magnitude of its
Fourier transform. After a brief overview of the history of this inverse
problem, we study a toy model, determining a complex polynomial from its
magnitudes on the unit circle. This simple problem already exhibits the main
difficulties that need to be overcome in X-ray crystallography and points to
methods from harmonic analysis and real algebraic geometry that resolve the
underdetermined nature of intensity measurements. The talk will conclude
with an alternative to a construction by Cynthia Vinzant, addressing the
minimal number of quantities that are needed for recovery in the cubic case,
and an open problem.

Spikes, Graphs and Modulations: Phase Retrieval for Finitely Supported Complex Measures
Bernhard Bodmann
University of Houston

This talk continues the discussion of mathematical models for X-ray
crystallography. Here, we consider the task of recovering a finitely
supported complex measure from observing the magnitude of its Fourier
transform or the magnitude of differences of its Fourier transform at
several locations. Following a strategy by Alexeev and others, the structure
of the locations used for these intensity measurements is encoded in a
graph. More precisely, a vertex in the graph represents a magnitude
measurement of the Fourier transform at a given frequency, and the edge
represents the magnitude of a (modulated) difference between the values of
the Fourier transform at two points. We show that a measurement chosen in
accordance with a Ramanujan graph of degree at least 3 and a sufficiently
large number of vertices is sufficient for identifying the complex measure
up to an overall multiplicative constant.

The material presented in this
talk is joint work with Ahmed Abouserie.