Department of Mathematics

Co-organizers:

Angela Gibney (angela.gibney {at} gmail [dot] com)

Alex Kontorovich (alex.kontorovich {at} rutgers [dot] edu)

Danny Krashen (daniel.krashen {at} gmail [dot] com)

Chris Woodward (ctw {at} math [dot] rutgers [dot] edu)

October 21: Kasso Okoudjou (Tufts)

October 28: Inna Zakharevich (Cornell)

November 4: Chelsea Walton (Rice)

November 11: Rob Ghrist (University of Pennsylvania)

November 18: Rob Lazarsfeld (Stony Brook)

December 4: Wei Ho (Michigan)

Time-frequency analysis seeks to study functions or signals in both space and frequency. In this talk, I will focus on one time-frequency tool: the Weyl-Heisenberg or Gabor systems. The origin of these systems can be partially attributed to Dennis Gabor, who, in 1946, claimed that any square Lebesgue integrable function can be written as an infinite linear combination of time and frequency shifts of the standard Gaussian. Since then, decomposition methods for larger classes of functions or distributions in terms of various elementary building blocks have led to an impressive body of work in harmonic analysis. In this talk, I will give a brief introduction to the theory of Gabor frames and will survey some related easily stated yet unresolved open problems.

Hilbert's 13th Problem (H13) is a fundamental open problem about polynomials in one variable. It is part of a beautiful (but mostly forgotten) story going back 3 thousand years. In this talk I will explain how H13 (and related problems) fits into a wider framework that includes problems in enumerative algebraic geometry and the theory of modular functions. I will then report on some recent progress, joint with Mark Kisin and Jesse Wolfson. While some fancy objects will appear in this talk, much of it should (I hope) be understandable to undergraduate math majors.

This talk will offer information on K-12 math education in the US, how this connects with the job of a university math department, and roles that individual concerned mathematicians can play. The speaker, whose primary activity is math research, is also Vice Chair of the US National Commission on Math Instruction (NAS).

I will discuss geometric and dynamical properties of actions of discrete groups on Riemannian symmetric spaces. I will highlight some aspects of the interplay between geometry and dynamics, and present some recent results which generalize theorems of Sullivan, Bridgeman-Taylor, McMullen for convex cocompact subgroups acting on hyperbolic space in the framework of discrete subgroups of Lie groups of higher rank.

We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. The linearization of this problems involves the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, joint with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary.

No previous knowledge of Riemannian geometry will be assumed.

The first meeting of the department colloquium is tomorrow at 3:30; please see the Zoom link below. The first few weeks will be short talks by new postdocs, to get to know and welcome them to the department (as best we can virtually). Tomorrow's speakers are: Abid Ali, Yunbai Cao, and Chris Lutsko. We invite all to meet at "tea" (gather.town) as usual at 3, and head over to zoom at 3:30. The schedule is then: 10 min lecture, 5 min questions/discussion, 5 mins break (if people like, they can go back into the "lounge", gather.town), repeating for each of the 3 speakers.

Best,

Alex Kontorovich, on behalf of co-organizers Angela Gibney, Danny Krashen, and Chris Woodward.

This page is maintained by Chris Lutsko.