Analysis & Applied Math

Event Information New Interactions between Analysis and Number Theory
14:10 on Friday November 25, 2016
15:00 on Friday November 25, 2016
BA1180, Bahen Center, 40 St. George St.
Stefan Steinerberger

Yale University

I will tell three unrelated stories describing new mysteries occurring somewhere in between Analysis and Number Theory. (1) The Poincare inequality is a cornerstone of mathematical physics (and related to the behavior of vibrating plates). I will present a curious improvement on the Torus that has a strong number theoretical flavor (even Fibonacci numbers appear). (2) If a function f(x) has the property that its Hardy-Littlewood maximal function is pretty easy to compute, then the function is sin(x). This gives a pretty weird definition of sin(x) and has applications in delay-differential equations. One would think that any characterization of sin(x) is easy to prove - the only proof I could find relies on a fantastic miracle to occur (and transcendental number theory). (3) An old integer sequence (1,2,3,4,6,8...) defined by Stanislaw Ulam in the 1960s turns out to have very strange properties that may hint towards a new mechanism in additive combinatorics (and creates very surprising pictures).