Prof. Dr. Thomas Kappeler

Institut für Mathematik
Universität Zürich
Winterthurerstrasse 190
CH-8057 Zürich

E-Mail:
Tel.: +41 44 635 58 85
Fax: +41 44 635 57 06
Büro: Y27K24
Arbeitsfelder:
Analysis
Sekretariat:
E-Mail:
Tel.
Anna Koller

+41-(0)44-63 55860

Vita

Education

  • ETH Zürich, PhD in mathematics
  • University of St. Gallen, Master in economics
     

Positions

  • Visiting positions at the Mathematics Department of the University of California at Berkeley, University of Pennsylvania, Brandeis University, Brown University
  • Professor of Mathematics, Ohio State University, 1990-1996
  • Professor of Mathematics, University of Zurich, 1996 - present
     

Professional Activities

  • Monographs in Mathematics, EMS Publishing House, Editor
  • Zurich Lectures in Advanced Mathematics, EMS Publishing House, Editor
     

Research Interest

  • Global Analysis
  • Dynamical systems (of infinite dimension)

Forschung

Research project on Hamiltonian systems of infinite dimension

KdV & KAM, Springer, 2003 (with Jürgen Pöschel)
Reprint: Higher Education Press, Beijing, to appear in 2010
Russian translation: KdV & KAM, Regular & Chaotic Dynamics, Moscow, 2008
Abstract: In this book we consider the Korteweg-de Vries equation u_t=u_xxx+6uu_x with periodic boundary conditions. Derived as a model equation for long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general. Being an infinite dimensional Hamiltonian system, we construct global action-angle coordinates, which make evident that all solutions are periodic, quasi-periodic or almost-periodic, and which also lead to some new results along the way. Subsequently, these coordinates allow us to apply a general KAM theorem for Hamiltonian pdes, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations. The pertinent nondegeneracy conditions are verified by calculating the first few terms in the Birkhoff normal forms - an essentially elementary calculation.

The defocussing NLS equation and its normal form, preliminary version 2009, (with Benoît Grébert, Jürgen Pöschel)
Abstract: In this book we consider the defocusing Nonlinear Schrödinger equation (NLS), iut=-uxx+2|u|^2u with periodic boundary conditions. This equation is a nonlinear perturbation of the Schrödinger equation for the wave function of a free particle in one space dimension. Actually, the equation is a model for slowly varying wave envelopes in dispersive media. Important applications are in the area of nonlinear optics. In this book we present a concise treatment of the NLS equation viewed as an integrable PDE. In particular, based on a general construction of action and angle variables, we show that this equation admits global Birkhoff coordinates in various weighted Sobolev spaces. As an application we derive wellposedness results for the NLS equation in these spaces and prove various properties of its solutions.


Research project on regularized determinants and Witten-Laplacians

Witten-Deformation of the de Rham complex, book in preparation (with Dan Bughelea and Leonid Friedlander)
Abstract: In this book we consider Witten's deformation of the de Rham complex of a closed Riemannian manifold (M,g), obtained by deforming the exterior differential d by means of a Morse function h : M → R. In a first part we study the generalized triangulation τ of M defined by the unstable manifolds of the gradient flow associated with the Morse function h and the Riemannian metric g. In a second part we show that the deformed de Rham complex splits into two subcomplexes, one of them, referred to as the small complex, being finite dimensional. It turns out that up to scaling, this complex is isomorphic to the Morse complex induced by the generalized triangulation τ. The Witten deformation of the de Rham complex can be used to compare analytic invariants of M, defined in terms of geometric data such as a Riemannian metric, with corresponding combinatorial invariants of M, defined in terms of a (generalized) triangulation. An example of such invariants is the analytic torsion and the Riemannian torsion, both in the classical as well as in the L2-setting.

Further information on current research projects: Please contact Thomas Kappeler

Vorlesungen & Seminare

Forschungsseminare
Analysis Seminar
Di
15.15-17.00
Alessandro Carlotto, Francesca Da Lio, Alessio Figalli, Norbert Hungerbühler, Tom Ilmanen, Thomas Kappeler, Tristan Rivière, Michael Struwe
 
PDE and Mathematical Physics
Do
18.10-19.00
Camillo De Lellis, Thomas Kappeler, Benjamin Schlein
 
Studentenseminare
Seminar Analysis
Fr
13.00-14.45
Thomas Kappeler
 

Vorlesungen
Analysis für die Chemie
Mi
10.15-12.00
Fr
10.15-11.00

Übungen Analysis für die Chemie
Fr 11:15-12, Räume: 13-L-11, 21-F-70, 22-F-62, 22-F-68, 27-H-46

Publikationen

Books

T. Kappeler, J. Pöschel
KdV & KAM, Springer, 2003
Reprint: Higher education Press, Beijing, to appear in 2010
Russian translation: KdV & KAM, Regular and Chaotic Dynamics, Moscow, 2008

B. Grébert, T. Kappeler, J. Pöschel
Normal form theory for the NLS equation, preliminary version, arXiv

D. Burghelea, L. Friedlander, T. Kappeler
Witten deformation of the de Rham complex, in preparation

Articles
University of Zürich - open access archive See MathSciNet and arXiv. For a complete list of preprints and recent publications and further information please contact Thomas Kappeler

Gäste


20182017201620152014201320122011201020092008200720062005200220012000
GästeAufenthalt
Dr. Alberto Maspero, SISSA (Trieste)
Talk: Long time stability of small finite gap solutions of the cubic Nonlinear Schrödinger equation on the two dimensional torus
30.05.18 - 01.06.18Kappeler, Thomas
Massimiliano Esposito, Department of Mathematics, Imperial College (London)
02.03.18 - 31.03.18Kappeler, Thomas
Prof. Dr. Alfonso Sorrentino, (Rome)
Talk: On the integrability of mathematical billiards
28.02.18 - 02.03.18Kappeler, Thomas
Prof. Dr. Alfonso Sorrentino, (Rome)
Talk: On the integrability of mathematical billiards
28.02.18 - 02.03.18Kappeler, Thomas