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
- Teilprojekt String topology and holomorphic curves (gemeinsam mit Prof. Dr. B. Richter) im Graduiertenkolleg 1670 "Mathematics inspired by string theory and Quantum field theory" (Sprecher: Prof. Dr. B. Siebert)
(bewilligt 4/2011-9/2015)
- 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
J. Latschev, D. McDuff, F. Schlenk, The Gromov width of 4-dimensional tori, preprint, November 2011
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J. Latschev, C. Wendl, Algebraic torsion in contact manifolds (mit einem Anhang von M. Hutchings), Geom. Funct. Anal. 21 (2011), no. 5, 1144-1195
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J. Latschev, Fukaya's work on Lagrangian embeddings, preprint, last changed January 2011, (written as chapter for book project on free loop spaces)
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K. Cieliebak, T. Ekholm, J. Latschev, Compactness for holomorphic curves with switching Lagrangian boundary conditions, J. Symplectic Geom. 8 (2010), no.3, 267-298
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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.
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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.
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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:
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Prove or disprove: Every linear symplectic structure on a torus of dimension 2n, n>2, admits a full packing by one symplectic ball (full here really means up to arbitrarily small epsilon)!
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Compute the Hofer-Zehnder capacity of the standard product tori of all even dimensions 2n>2!
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Prove or disprove: The Hofer-Zehnder capacity of every compact Liouville domain is finite.
(The question seems open even for general disk cotangent bundles.)
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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.)
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What is the "correct" lower bound for the number of closed Reeb orbits in a quantitative Weinstein conjecture in arbitrary dimensions?
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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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