BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Armin Rainer (Universität Wien) DTSTART:20210524T160000Z DTEND:20210524T170000Z DTSTAMP:20260418T130317Z UID:ScReGeFor/4 DESCRIPTION:Title: From Ultra-differentiable to Quasi-Analytic Analysis\, Lecture 1\nby Armin Rainer (Universität Wien) as part of School of Real Geometry in Fo rtaleza - ScReGeFor\n\n\nAbstract\nThe mini-course is intended as an intro duction to ultradifferential analysis with special emphasis on ultra-diffe rentiable extension theorems. The development of differential analysis in the last century was decisively influenced by Whitney’s work on the exte nsion of differentiable functions from closed sets. We shall be interested in quantitative versions of Whitney’s extension theorem. The quantitati ve aspect is implemented by uniform growth properties of the multisequence of partial derivatives which measure the deviation from the Cauchy estima tes and hence from analyticity. These growth conditions determine ultradif ferentiable function classes which form a scale of regularity classes betw een the real analytic and the smooth class.\n\nLecture 1. We will recall W hitney’s classical extension theorem and formulate the quantitative prob lem. This will lead us to Denjoy–Carleman classes which are the ultradif ferentiable classes of main interest in this series of lectures. After dis cussing inclusion and stability properties we shall begin with the study o f the Borel map (i.e. infinite Taylor expansion) on Denjoy–Carleman clas ses. We will investigate when the Borel map is injective and\, in the cour se of this\, prove the Denjoy–Carleman theorem which discriminates betwe en quasianalytic and non-quasianalytic classes.\n\n--\n\nUltra-differentia ble Analysis concerns sub-algebras of smooth functions with constrained gr owth of the Taylor series coefficients. Besides its importance in the anal ysis of partial differential equations the development of this theory was influenced by the classical Whitney extension problem and the composition problem. Both problems followed a path that often meets sub-analytic geome try\, and later o-minimal geometry. Of special interest among the ultra-di fferentiable classes are the quasi-analytic classes which possess a (quasi -)analytic continuation property similar to the real analytic class. This property make these classes interesting from an analytic viewpoint and als o suitable for questions of tame geometry. In relation with asymptotic exp ansions of solutions of analytic ODEs they also naturally appear when deal ing with non-oscillation problems of the solutions. Armin Rainer will intr oduce us to ultra-differentiable classes\, in particular to quasi-analytic ones.\n LOCATION:https://researchseminars.org/talk/ScReGeFor/4/ END:VEVENT END:VCALENDAR