Computing Spectra of Quasicrystals: Case Studies in Complexity and Linear Algebra | Richard M. Karp Distinguished Lecture

Quasicrystals are fascinating materials that sit between the extremes of highly ordered crystals and disordered matter. Their discovery, which earned Dan Shechtman the 2011 Nobel Prize in Chemistry, has fueled interest in the mathematics of aperiodic order. For a computational spectral theorist, the field is ripe with challenging problems. Even the simplest models exhibit exotic behavior: the spectrum of the one-dimensional Fibonacci Hamiltonian is a Cantor set; the spectrum of its two-dimensional analogue is the sum of Cantor sets. Can one estimate the fractal dimension of such spectra? Other two-dimensional models derive from aperiodic tilings of the plane, such as Roger Penrose’s famous kite-and-dart construction. The graph Laplacians for such models contain high-multiplicity eigenvalues having compactly supported eigenvectors, for which one seeks a tidy description. This talk will survey a variety of such problems: describing the computational challenges, acknowledging a debt to excellent mathematical software, and highlighting the need for continued algorithm development.

This talk describes collaborative work with James Chok, Matthew Colbrook, David Damanik, Jake Fillman, Anton Gorodetski, May Mei, and Charles Puelz.

Mark Embree is a professor of mathematics at Virginia Tech, where he led the undergraduate major in computational modeling and data analytics from 2015 to 2025. He served on the faculty of the Department of Computational and Applied Mathematics at Rice University from 2002 to 2013, following graduate work with Andy Wathen and a postdoc with Nick Trefethen, both at Oxford. With Trefethen, he coauthored Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. His research interests include matrix computations, non-self-adjoint operators, and spectral theory.

Refreshments will be served at 3 p.m., before the event.

The Richard M. Karp Distinguished Lectures were created in Fall 2019 to celebrate the role of Simons Institute Founding Director Dick Karp in establishing the field of theoretical computer science, formulating its central problems, and contributing stunning results in the areas of computational complexity and algorithms. Formerly known as the Simons Institute Open Lectures, the series features visionary leaders in the field of theoretical computer science and is geared toward a broad scientific audience.

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