Prof. Dr. Klaus Mohnke

Profil

• Punctured Holomorphic Curves in Symplectic Geometry (I)

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  Förderer: DFG Sachbeihilfe Zeitraum: 07/2003 - 06/2005 Projektleitung: Prof. Dr. Klaus Mohnke

• Punctured Holomorphic Curves in Symplectic Geometry (II)

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  Förderer: DFG Sachbeihilfe Zeitraum: 08/2005 - 07/2007 Projektleitung: Prof. Dr. Klaus Mohnke

• Punctured Holomorphic Curves in Symplectic Geometry (III)

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  Förderer: DFG Sachbeihilfe Zeitraum: 04/2008 - 03/2010 Projektleitung: Prof. Dr. Klaus Mohnke

• VA: Internationale wissenschaftliche Veranstaltung: "Symplectic Field Theory VIII", Berlin. 06.08.-12.08.16

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  Förderer: DFG sonstige Programme Zeitraum: 08/2016 - 08/2016 Projektleitung: Prof. Dr. Klaus Mohnke

• Economic and Social Researc Council

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  1 Treffer55.7%
  • VA: The Future and Promises of International Assessment (15.09.16 - 16.09.16)
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    55.7%

• RISC Software GmbH

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  13 Treffer52.9%
  • EU: Scattering Amplitudes: From Geometry to EXperiment (SAGEX)
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    52.9%

• Johannesstift Diakonie Proclusio gGmbH

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  4 Treffer52.7%
  • Professionalisierung in der Deutsch-als-Zweitsprache-Förderung für geflüchtete Menschen mit Lernschwierigkeiten
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    52.7%

• SoftBank Robotics

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  5 Treffer51.9%
  • Embodied Audition for RobotS
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    51.9%

• InnovationLab GmbH

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  6 Treffer51.8%
  • Interfaces in opto-electronic thin film multilayer devices
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    51.8%

• International Business Machines Corporation Research GmbH

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  6 Treffer51.5%
  • EU: Simulation in Multiscale Physical and Biological Systems (STIMULATE)
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    51.5%

• Magwel NV

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  6 Treffer51.5%
  • EU: Simulation in Multiscale Physical and Biological Systems (STIMULATE)
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    51.5%

• NVIDIA GmbH

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  6 Treffer51.5%
  • EU: Simulation in Multiscale Physical and Biological Systems (STIMULATE)
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    51.5%

• Bernstein Center for Computational Neuroscience Berlin

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  14 Treffer51.3%
  • SFB 1315/2: Mechanismen und Störungen der Gedächtniskonsolidierung: Von Synapsen zur Systemebene
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    51.3%
  • DFG-Sachbeihilfe: Aufmerksamkeit und sensorische Integration im aktiven Sehen von bewegten Objekten
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    49.6%

• Science & Startups Berlin

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  4 Treffer50.6%
  • Zuwendung im Rahmen des Programms „exist – Existenzgründungen aus der Wissenschaft“ aus dem Bundeshaushalt, Einzelplan 09, Kapitel 02, Titel 68607, Haushaltsjahr 2026, sowie aus Mitteln des Europäischen Strukturfonds (hier Euro-päischer Sozialfonds Plus – ESF Plus) Förderperiode 2021-2027 – Kofinanzierung für das Vorhaben: „exist Women“
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    50.6%

• Symplectic hypersurfaces and transversality in Gromov-Witten theory

  2007

  Journal of Symplectic Geometry · 127 Zitationen · DOI

  We present a new method to prove transversality for holomorphic curves in symplectic manifolds, and show how it leads to a definition of genus zero Gromov-Witten invariants. The main idea is to introduce additional marked points that are mapped to a symplectic hypersurface of high degree in order to stabilize the domains of holomorphic maps.

• Coherent orientations in symplectic field theory

  2004

  Mathematische Zeitschrift · 120 Zitationen · DOI

• Compactness for punctured holomorphic curves

  2005

  Journal of Symplectic Geometry · 85 Zitationen · DOI

  a general compactness result for moduli spaces of punctured holomorphic curves arising in symplectic field theory. In this paper we present an alternative proof of this result. The main idea is to determine a priori the levels at which holomorphic curves split, thus reducing the proof to two separate cases: long cylinders of small area, and regions with compact image. The second case requires a generalization of Gromov compactness for holomorphic curves with free boundary.

• Holomorphic Disks and the Chord Conjecture

  2001

  Annals of Mathematics · 68 Zitationen · DOI

  We prove ( a weak version of) Arnold's Chord Conjecture in [2] using Gromov's "classical" idea in [9] to produce holomorphic disks with boundary on a Lagrangian submanifold.Arnold's Chord Conjecture.In this paper we prove the following theorem which was conjectured by Arnold [2]:

• Punctured holomorphic curves and Lagrangian embeddings

  2017

  Inventiones mathematicae · 64 Zitationen · DOI

• Punctured holomorphic curves and Lagrangian embeddings

  2014

  arXiv (Cornell University) · 15 Zitationen · DOI

  We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions.

• How to (symplectically) thread the eye of a (Lagrangian) needle

  2001

  ArXiv.org · 11 Zitationen · DOI

  We show that there exists no Lagrangian embeddings of the Klein bottle into $\CC^{2}$. Using the same techniques we also give a new proof that any Lagrangian torus in $\CC^2$ is smoothly isotopic to the Clifford torus.

• Symplectic hypersurfaces and transversality in Gromov-Witten theory

  2007

  ArXiv.org · 8 Zitationen · DOI

  We use Donaldson hypersurfaces to construct pseudo-cycles which define Gromov-Witten invariants for any symplectic manifold which agree with the invariants in the cases where transversality could be achieved by perturbing the almost complex structure.

• Legendrian links of topological unknots

  2001

  Contemporary mathematics - American Mathematical Society · 7 Zitationen · DOI

  We use an estimate on the Thurston–Bennequin invariant of a Legendrian link in terms of its Kauffman–polynomial to show that links of topological unknots, e.g. the Borromean rings or the Whithead link, may not be represented by Legendrian links of Legendrian unknots. In [2] Eliashberg classified all Legendre knots representing the unknot in terms of their Thurston– Bennequin number, tb, and their rotation, r. Bennequin’s inequality in these cases reads as tb + |r | ≤ −1. Thus the Legendre knot given by the wavefront of the ’eye ’ which has tb = −1, r = 0 is referred to as the trivial Legendrian knot. Figure 1: A front projection of the Legendrian unknot Back then there was no other obvious obstruction for Legendrian links consisting of (topological) unknots and Eliashberg asked the question: ”Given a link of topological unknots, can it be realized as a link of [...] Legendrian unknots?” The answer is negative in general and the new obstructions are given by a sharper inequality on the Thurston–Bennequin number governed by the Kauffman polynomial K(x, t) which was independently found by Chmutov and Goryunov [1] and Fuks and Tabachnikov [3]. It simply states that the Thurston–Bennequin number is not bigger than the minus the maximal degree in the variable x of the Kauffman polynomial: tb ≤ − max-deg x K. The contribution of the author is to apply this to links of topological unknots. We had to be careful because the two groups of authors [1, 3] used different Kauffman polynomials and thus obtain slightly different inequalities: here we work with the version Chmutov and Goryunov used. Let us first recall the definition of the Thurston–Bennequin number: Definition 1 Let L be an oriented Legendrian link given by a wave front projection. Then the Thurston–Bennequin number of L, tb(L), is the number of sideward crossings minus the number of up – or downward crossings minus half the number of cusps. Figure 2: Combinatorial definition of the Thurston–Bennequin number in terms of the front projection From that the following observation is immediate

• Lagrangian embeddings in the complement of symplectic hypersurfaces

  2001

  Israel Journal of Mathematics · 6 Zitationen · DOI

• Holomorphic Disks and the Chord Conjecture

  2000

  ArXiv.org · 2 Zitationen · DOI

  We prove (a weak version of) Arnold's Chord Conjecture using Gromov's ``classical'' idea in to produce holomorphic disks with boundary on a Lagrangian submanifold.

• Lagrangian Embeddings and Holomorphic Discs in the Complement of Complex Hypersurfaces

  1998

  ArXiv.org · 2 Zitationen · DOI

  For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

• ON SEIBERG–WITTEN EQUATIONS ON SYMPLECTIC 4–MANIFOLDS

  2008

  CERN Document Server (European Organization for Nuclear Research) · 1 Zitationen

  We discuss Taubes' idea to perturb the monopole equations on symplectic manifolds to compute the Seiberg-Witten invariants in the light of Witten's symmetry trick in the Kaehler case. The article thought as a supplement to a series of lectures given in Lindow and Berlin on that subject. (orig.)

• The Classification of Surfaces via Normal Curves

  2023

  Jahresbericht der Deutschen Mathematiker-Vereinigung · DOI

  Abstract We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser’s and Milnor’s proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not need any invariants from algebraic topology to distinguish surfaces.

• The Classification of Surfaces via Normal Curves

  2022

  arXiv (Cornell University) · DOI

  We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser's and Milnor's proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not need any invariants from algebraic topology to distinguish surfaces.

• Coherent Orientations in Symplectic Field Theory

  2001

  arXiv (Cornell University) · DOI

  We study the coherent orientations of the moduli spaces of holomorphic curves in Symplectic Field Theory, generalizing a construction due to Floer and Hofer. In particular we examine their behavior at multiple closed Reeb orbits under change of the asymptotic direction. The orientations are determined by a certain choice of orientation at each closed Reeb orbit, that is similar to the orientation of the unstable tangent spaces of critical points in finite--dimensional Morse theory.

• Symplectic threading through Lagrangians

  2001

  arXiv (Cornell University)

  Abstract. We show that there exists no Lagrangian embeddings of the Klein bottle into C 2. Lagrangian Embeddings in C 2. The topology of closed Lagrangian embeddings into C n is still an elusive problem in symplectic topology. Before Gromov invented the techniques of pseudo–holomorphic curves it was almost intractable and the only known obstructions came from the fact that such a submanifold has to be totally real. Then in [3] he showed that for any such closed, compact, embedded Lagrangian there exists a holomorphic disk with boundary on it. Hence the integral of a primitive over the boundary is different from zero and the first Betti number of the Lagrangian submanifold cannot vanish, excluding the possibility that a three–sphere can be embedded into C 3 as a Lagrangian. A further analysis of these techniques led to more obstructions for the topology of such embeddings in [9] and [11]. For C 2 the classical obstructions restrict the classes of possible closed, compact surfaces which admit Lagrangian embeddings into C 2 to the torus and connected sums of the Klein bottle with oriented surfaces. There are obvious Lagrangian embeddings of the torus (e.g. of the form S 1 × S 1 ⊂ C × C) and not so obvious ones for the connected sums except for the Klein bottle (see [1]). One may further ask which topological types of embeddings may be realized as a Lagrangian embedding. There has been a partial answer to that [6] and an announcement in [5] stating that all Lagrangian tori are (topogically) unknotted. Here we show that the same circle of ideas solves the question for the Klein bottle, namely Theorem 1. There is no Lagrangian embedding of the Klein bottle into C 2. The theorem follows from the main result of this paper: Theorem 2. Let L ⊂ CP 2 \\ H be any closed, compact Lagrangian embedding of the torus or the Klein bottle into the complement of a projective line H. Then there exists an almost complex structure on CP 2 which is standard near H and a pseudo–holomorphic sphere G in the complement of L nontrivially linked with it: The homomorphism lk CP 2 \\H(G \\ G ∩ H),.) : H1(L) − → Z. is non-trivial. From that we can deduce the theorem on the Klein bottle. Proof of Theorem 1. For convenience we rescale the Lagrangian submanifold such that it lies in the open unit ball. Hence we may consider it as a Lagrangian embedding into CP 2 in the

• On fillings by holomorphic discs of the levels of a Morse function

  1999

  Differential Geometry and its Applications · DOI

• Legendrian links of topological unknots

  1998

  arXiv (Cornell University) · DOI

  We use an estimate on the Thurston--Bennequin invariant of a Legendrian link in terms of its Kauffman-polynomial to show that links of topological unknots, e.g. the Borromean rings or the Whithead link, may not be represented by Legendrian links of Legendrian unknots.

• On Seiberg-Witten equationsοn symplectic 4-manifolds

  1997

  Banach Center Publications · DOI

  We discuss Taubes' idea to perturb the monopole equations on symplectic manifolds to compute the Seiberg-Witten invariants in the light of Witten's symmetry trick in the Khler case.

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Name

Prof. Dr. Klaus Mohnke

Titel

Prof. Dr.

Fakultät

Mathematisch-Naturwissenschaftliche Fakultät

Institut

Institut für Mathematik

Arbeitsgruppe

Analysis II

Telefon

+49 30 2093-45433

HU-FIS-Profil

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Zuletzt gescrapt

26.4.2026, 01:09:31