## Department Colloquium

### Friday, November 13th, 2020

**Time:** 2:30 p.m. **Place:** Online (via Zoom)

**Speaker:** Yaiza Canzani (University of North Carolina)

**Title:** Eigenfunction concentration via geodesic beams.

**Abstract: ** A vast array of physical phenomena, ranging from the propagation of waves to the location of quantum particles, is dictated by the behavior of Laplace eigenfunctions. Because of this, it is crucial to understand how various measures of eigenfunction concentration respond to the background dynamics of the geodesic flow. In collaboration with J. Galkowski, we developed a framework to approach this problem that hinges on decomposing eigenfunctions into geodesic beams. In this talk, I will present these techniques and explain how to use them to obtain quantitative improvements on the standard estimates for the eigenfunction's pointwise behavior, Lp norms, and for both pointwise and integrated Weyl Laws. One consequence of this method is a quantitatively improved Weyl Law for the eigenvalue counting function on all product manifolds.

**Yaiza Canzani** is an Assistant Professor in the Department of Mathematics at the University of North Carolina at Chapel Hill. She was awarded a Sloan Research Fellowship in 2018. Before joining UNC, Prof. Canzani was a Benjamin Peirce Fellow at Harvard University and a member of the Institute for Advanced Study. She obtained her Ph.D. from McGill University in 2013 under the supervision of Dmitry Jakobson and John Toth. She works on geometric analysis and spectral theory.