BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Armin Rainer (Universität Wien) DTSTART:20210525T160000Z DTEND:20210525T170000Z DTSTAMP:20260418T131010Z UID:ScReGeFor/8 DESCRIPTION:Title: From Ultra-differentiable to Quasi-Analytic Analysis\, Lecture 2\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 2 (Tue\, May 25\, 1 3:00). Next we will study the image of the Borel map on Denjoy–Carleman classes which naturally sits in the sequence space defined by the characte ristic bounds. The quasianalytic and the non-quasianalytic case are fundam entally different. While in the non-quasianalytic case we will give necess ary and sufficient conditions for the Borel map being onto the sequence sp ace\, we shall see that on quasianalytic classes\, strictly containing the real analytic class\, the Borel map is never onto. This will be complemen ted by some results on the description of the Borel image and a discussion of further intricacies of the quasianalytic setting.\n\n--\n\nUltra-diffe rentiable Analysis concerns sub-algebras of smooth functions with constrai ned growth of the Taylor series coefficients. Besides its importance in th e analysis of partial differential equations the development of this theor y was influenced by the classical Whitney extension problem and the compos ition problem. Both problems followed a path that often meets sub-analytic geometry\, and later o-minimal geometry. Of special interest among the ul tra-differentiable 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 a nd also suitable for questions of tame geometry. In relation with asymptot ic expansions of solutions of analytic ODEs they also naturally appear whe n dealing with non-oscillation problems of the solutions. Armin Rainer wil l introduce us to ultra-differentiable classes\, in particular to quasi-an alytic ones.\n LOCATION:https://researchseminars.org/talk/ScReGeFor/8/ END:VEVENT END:VCALENDAR