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Janko Latschev

Forschungsschwerpunkte

Meine aktuelle Forschung hat ihren Schwerpunkt in der symplektischen Geometrie. Oft spielen dabei holomorphe Kurven als Werkzeug eine wichtige Rolle. Darüber hinaus gibt es diverse Querverbindungen zu Themen in der Differentialtopologie, der homologischen Algebra und in der Theorie der dynamischen Systeme.

Drittmittelprojekte

  • Graduiertenkolleg 1670 "Mathematics inspired by string theory and Quantum field theory"
    1. Phase, 4/2011-9/2015, Sprecher: Prof. Dr. B. Siebert:
    Teilprojekt String topology and holomorphic curves (gemeinsam mit Prof. Dr. B. Richter)
    2. Phase, 10/2015-3/2020, Sprecher: Prof. Dr. C. Schweigert:
    Teilprojekt String topology, holomorphic curves and mirrow symmetry (gemeinsam mit Prof. Dr. B. Richter und Prof. Dr. B. Siebert)
  • DFG-Sachbeihilfe, Projekt "Algebraic structures on symplectic homology and their applications"
    (in Zusammenarbeit mit Prof. Dr. K. Cieliebak, Augsburg, bewilligt 12/2012 für 3 Jahre)

Letzte Veröffentlichungen - Recent publications

K. Cieliebak, T. Ekholm, J. Latschev, L. Ng Knot contact homology, string topology and the cord algebra, J. Éc. polytech. Math. 4 (2017), 661-780
J. Latschev, A. Oancea (eds.) Free Loops spaces in geometry and topology, EMS Publishing House, IRMA Lectures in Mathematics and Theoretical Physics Vol. 24, 2015
K. Cieliebak, K. Fukaya, J. Latschev, Homological algebra related to surfaces with boundary, preprint, arxiv:1508.02741
J. Latschev, D. McDuff, F. Schlenk, The Gromov width of 4-dimensional tori, Geometry & Topology 17 (2013), 2813-2853
J. Latschev, C. Wendl, Algebraic torsion in contact manifolds (mit einem Anhang von M. Hutchings), Geom. Funct. Anal. 21 (2011), no. 5, 1144-1195
K. Cieliebak, T. Ekholm, J. Latschev, Compactness for holomorphic curves with switching Lagrangian boundary conditions, J. Symplectic Geom. 8 (2010), no.3, 267-298
K. Cieliebak, J. Latschev The role of string topology in symplectic field theory, in: New perspectives and challenges in symplectic field theory, 113-146, CRM Proc. Lecture Notes, 49, Amer. Math. Soc., Providence, RI, 2009.
K. Cieliebak, H. Hofer, J. Latschev, F. Schlenk Quantitative symplectic geometry, in: Dynamics, ergodic theory, and geometry, 1-44, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007.

Einige offene Fragen - Some open questions

I believe the following problems are open. If you think you can solve one or more of them, then please contact me. Some of these are probably hard. The collection is somewhat (but not completely) random, and severely restricted by the desire to formulate each problem in a single sentence. Not all of these problems are equally interesting, and their order has no significance that I am aware of:
  • Compute the Hofer-Zehnder capacity of the standard product tori of all even dimensions 2n>2!
  • Prove or disprove: The Hofer-Zehnder capacity of every compact Liouville domain is finite.
    (The question seems open even for general disk cotangent bundles.)
  • Prove or disprove: If a connected sum of manifolds admits a Lagrangian embedding into R^2n, then at least one of the summands does, too.
    (Note in higher dimensions this is slightly ambiguous, since the direct sum decomposition is not unique.)
  • What is the "correct" lower bound for the number of closed Reeb orbits in a quantitative Weinstein conjecture in arbitrary dimensions?
  • Does there exist a contact form on a closed manifold admitting a fixed point free "anti-contact" involution (sending the form to its negative) with only finitely many closed Reeb orbits?
  • Find interesting obstructions to the existence of exact fillings of strongly fillable manifolds! Here "interesting" means that there should be (preferably higher-dimensional) examples where one can actually compute them...

 
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