BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Simon Bortz (University of Alabama) DTSTART:20210201T170000Z DTEND:20210201T175000Z DTSTAMP:20260423T005852Z UID:anpdews/48 DESCRIPTION:Title: New Developments in Parabolic Uniform Rectifiability\nby Simon Bortz (University of Alabama) as part of HA-GMT-PDE Seminar\n\n\nAbstract\nIn th e 1980’s the $L^2$ boundedness of the Cauchy integral was established (b y Coifman\, McIntosh and Meyer) and this $L^2$ boundedness was quickly gen eralized to `nice’ singular integral operators (Coifman\, David and Meye r / David) on Lipschitz graphs. David and Semmes then asked the natural qu estion: For what sets are all `nice’ singular integral operators $L^2$ b ounded? They were remarkably successful in this endeavor\, providing more than 15 equivalent notions and called these sets “uniformly rectifiable ” or UR. These sets are still studied extensively today\, most recently in their connection to elliptic partial differential equations.\n\n \nAmon g the characterizations of UR sets provided by David and Semmes is a quadr atic estimate on the so-called $\\beta$-numbers\, which measure the flatne ss of the set at a particular location and scale. In a paper of Hofmann\, Lewis and Nyström a notion of “parabolic uniform rectictifiable sets” was introduced by taking the definition as a “quadratic estimate on the parabolic $\\beta$-numbers”. There are no correct proofs of ANY of the analogues of the David Semmes theory\; for instance\, it is not known if $ L^2$ boundedness of parabolic singular integral operators characterizes pa rabolic uniformly rectifiable sets.\n\n \nIn this talk I will discuss some recent progress in the direction of establishing the parabolic David-Semm es theory and some open problems that remain. On one hand\, we have made s ignificant progress and provided some useful characterizations of paraboli c uniform rectifiability. On the other hand\, we have also discovered that many of the `elliptic’ characterizations do not hold in this parabolic setting. This is joint work with J. Hoffman\, S. Hofmann\, J.L. Luna and K . Nyström.\n LOCATION:https://researchseminars.org/talk/anpdews/48/ END:VEVENT END:VCALENDAR