Extra Session

There will be an extra talk by an invited speaker:

Professor Tobias Kaiser (Passau)

will speak on

Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analytic functions with value group of finite archimedean rank

  • Time: Friday, 06.03., 10:30 -- 12:00
  • Room: 25.22.O3.73

Abstract: We work in a model of the theory of the real field with restricted analytic functions such that its value group has finite archimedean rank. An example is given by the field of Puiseux series over the reals. We show how one can extend the restricted logarithm to a global logarithm with values in the polynomial ring over the model with dimension the archimedean rank. The logarithms are determined by algebraic data from the model, namely by a section of the model and by an embedding of the value group into its Hahn group. If the archimedean rank of the value group coincides with the rational rank the logarithms are equivalent. We illustrate how one can embed such a logarithm into a model of the real field with restricted analytic functions and exponentiation. This allows us to define constructible functions with good lifting properties. As an application we establish a full Lebesgue measure and integration theory with values in the polynomial ring.

Termine

Date Speaker Subject
April 2 Braun Overview
April 9 Rohrschneider Criteria for Regularity
April 16 no seminar  
April 23 Adams Construction of a Fundamental Solution
April 30 Bradley-Williams The Curve Selection Lemma
May 7 Halupczok Localizations at Infinity
May 14 Balkenhol Description of the Wave Front Set
May 21 Braun Hypoelliptic Operators
May 28 Kısakürek Introduction to O-Minimality
June 4 Severin Why the class of subanalytic sets is (not) o-minimal
June 11 no seminar  
June 18 no seminar  
June 25 no seminar  
July 2 Khalilian From model completeness to o-minimality
July 9 no seminar  

There will be an extra session with a guest somewhen in the winter term.

Literatur

  • van den Dries, Lou: Tame Topology and O-minimal Structures
  • Hörmander, Lars: The Analysis of Linear Partial Differential Operators I
  • Hörmander, Lars: The Analysis of Linear Partial Differential Operators II

Vorträge

Anwendungen der Modelltheorie auf Partielle Differentialgleichungen

Themen

Einführung

  • Ziel des ganzen
  • Grundlegende Begriffe: Distribution, Fouriertransformation, Fundamentallösung, Wellenfront

Regularitätskriterien

  • \(B_{p,k}\)-Räume

10.1.1.-10.1.15., hauptsächlich Definitionen und Lemmata analytischer Natur

Konstruktion einer Fundamentallösung

  • Existenzbeweis via Fouriertransformation
  • Regularität der konstruierten Lösung

7.3.10-7.3.12 und 10.2.1.10.2.3., technisch analytisch

Kurvenauswahlsatz

  • Puiseuxreihe
  • Satz von Tarski-Seidenberg zitieren
  • Kurvenauswahlsatz

Anhang A oder eine andere Quelle, modelltheoretisch

Lokalisierungen im Unendlichen

  • Definition
  • Ein Approximationsergebnis

10.2.4.-10.2.10., (naiv) geometrisch und modelltheoretisch

Beschreibung der Wellenfront

  • Verwendung der Approximation aus dem vorigen Vortrag, um eine obere Abschätzung der Wellenfront zu geben

10.2.11.-10.2.13, analytisch diffizil

Hypoelliptische Operatoren

  • Begriff des hypoelliptischen Operators
  • Regularität von Nulllösungen

11.1.1.-11.1.4, zusammenfassen