Stefan Steinerberger, Yale University, New Haven, CT , Analysis meets Number Theory, two theorems and a mystery

Description

Analysis and Number Theory have always enjoyed a fruitful interaction; we discuss three new (and independent) chapters in the story (with focus on the Analysis part). (1) Improvements of the Poincare inequality obtained by replacing the gradient with a gradient in a fixed direction - amusing things happen (KAM-type estimates, Thue-Siegel-Roth theorem and Fibonacci numbers). (2) The Hardy-Littlewood maximal function is a cornerstone in real analysis -- we describe a new rigidity phenomenon: if the maximal function is easy to compute, then the function is, up to symmetries, $\sin(x)$. The proof requires nontrivial input from transcendental number theory (the Lindemann-Weierstrass theorem). (3) The Mystery: some experimental (but truly elementary) observations connecting Ulam's mysterious integer sequence from additive combinatorics $(1,2,3,4,6,8,11,\dots)$, about which still not a single statement is known (except infinitude), to curious phenomena in measure theory (and a quick remark about zeros of the Riemann zeta function).

Period	29 Dec 2015
Event type	Guest talk
Location	AustriaShow on map