Tuesday, November 12, 2019
Time  Items 

All day 

4:00pm 
11/12/2019  4:15pm We will discuss some of the diverse historical motivations for the study of dynamics on moduli spaces of surfaces, and survey a selection of exciting recent developments by different authors, aiming to convey the breadth of the field. We will end by discussing joint work in progress with Paul Apisa on the classification problem for orbit closures of the GL(2,R) action. Location:
LOM 206
11/12/2019  4:15pm Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that $$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(xn))}{\pi(x n)}. $$ This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers. We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$ In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero. Location:
LOM 215
