Georgios Dimitroglou Rizell

Talks & Slides

Here you can find slides and videos from selected talks.

Slides:
  • Lagrangian Poincaré Recurrence via pseudoholomorphic foliations.
    Slides: & Video:
    IBS Center for Geometry and Physics, Pohang, Korea.
    Zoom, 26 April 2021.
    Abstract: or any Hamiltonian displaceable closed curve inside a closed symplectic surface, there is a bound on the number of pairwise disjoint Hamiltonian isotopic copies of the curve that one can produce. This phenomenon is called Lagrangian Poincaré Recurrence, and it was only shown very recently by Polterovich and Shelukhin that there exist displaceable Lagrangians in higher dimension that satisfy the analogous property. In this work in progress joint with E. Opshtein, we use the technique of pseudoholomorphic foliations to show that the bound on the number of disjoint copies in the surface persists after increasing the dimension by the following stabilisation: take the cartesian product of the symplectic surface with a sufficiently small symplectic annulus, and take the product of the curve with the with the core of the annulus to produce a Lagrangian torus.
  • Lagrangian classification and recurrence via pseudoholomorphic foliations.
    Slides: & Video:
    41th Winter School Geometry and Physics (Czech Republic, Srni, January 2021).
    Zoom, 19 January 2021.
    Abstract: We give an overview of how the techniques of pseudoholomorphic foliations and neck stretching can be used to obtain classification results for Lagrangian submanifolds in dimension four. We also report on work in progress with E. Opshtein where we use the same techniques to establish a case where Lagrangian reccurrence holds: for any k>1 there exist Lagrangian tori which can be mapped to k pairwise disjoint copies by Hamiltonian diffeomorphisms, while k+1 such copies must intersect.
  • Hamiltonian classification and unlinkedness of fibres in cotangent bundles of Riemann surfaces.
    Slides: & Video:
    Symplectic Zoominar (CRM-Montreal, Princeton/IAS, Tel Aviv, and Paris).
    Zoom, 4 September 2020.
    Abstract: In a joint work with Laurent Côté we show the following result. Any Lagrangian plane in the cotangent bundle of an open Riemann surface which coincides with a cotangent fibre outside of some compact subset, is compactly supported Hamiltonian isotopic to that fibre. This result implies Hamiltonian unlinkedness for Lagrangian links in the cotangent bundle of a (possibly closed Riemann surface whose components are Hamiltonian isotopic to fibres.
  • The classification of monotone Lagrangian tori in a four-dimensional symplectic vectorspace.
    AMS-EMS-SPM International Meeting 2015.
    Porto, 12 June 2015.
    Abstract: We prove that there are exactly two monotone Lagrangian tori in a four-dimensional symplectic vectorspace up to Hamiltonian isotopy and rescaling: the Clifford and the Chekanov torus. This is shown by, first, Hamiltonian isotoping the torus to a homotopically non-trivial torus inside C*×C* and, second, applying a classification result for homotopically non-trivial Lagrangian tori inside T*T2. The latter result follows using methods due to Ivrii.
  • Lagrangian unknottedness in the cotangent bundle of a torus and applications.
    IX Workshop on Symplectic Geometry, Contact Geometry, and Interactions.
    ENS Lyon, 31 January 2015.
    Abstract: The homotopically non-trivial Lagrangian tori in the cotangent bundle of a two-torus are classified up to Hamiltonian isotopy. This is done by following ideas and techniques due to Ivrii, who treated the case when the torus is homologous to the zero-section. We apply this result to obtain answers concerning the classification of monotone Lagrangian tori in a symplectic vector space. Both questions are studied by considering the behaviour of pseudo-holomorphic foliations when stretching the neck around the Lagrangian submanifold.
  • The effect of ambient Legendrian surgery on the Chekanov-Eliashberg DGA.
    Workshop on Legendrian submanifolds, holomorphic curves and generating families.
    Brussels, 30 August 2014.
Videos:
  • Lagrangian Poincaré Recurrence via pseudoholomorphic foliations.
    IBS Center for Geometry and Physics, Pohang, Korea.
    Zoom, 26 April 2021.
    Abstract: or any Hamiltonian displaceable closed curve inside a closed symplectic surface, there is a bound on the number of pairwise disjoint Hamiltonian isotopic copies of the curve that one can produce. This phenomenon is called Lagrangian Poincaré Recurrence, and it was only shown very recently by Polterovich and Shelukhin that there exist displaceable Lagrangians in higher dimension that satisfy the analogous property. In this work in progress joint with E. Opshtein, we use the technique of pseudoholomorphic foliations to show that the bound on the number of disjoint copies in the surface persists after increasing the dimension by the following stabilisation: take the cartesian product of the symplectic surface with a sufficiently small symplectic annulus, and take the product of the curve with the with the core of the annulus to produce a Lagrangian torus.
  • Lagrangian classification and recurrence via pseudoholomorphic foliations.
    41th Winter School Geometry and Physics (Czech Republic, Srni, January 2021).
    Zoom, 19 January 2021.
    Abstract: We give an overview of how the techniques of pseudoholomorphic foliations and neck stretching can be used to obtain classification results for Lagrangian submanifolds in dimension four. We also report on work in progress with E. Opshtein where we use the same techniques to establish a case where Lagrangian reccurrence holds: for any k>1 there exist Lagrangian tori which can be mapped to k pairwise disjoint copies by Hamiltonian diffeomorphisms, while k+1 such copies must intersect.
  • Hamiltonian classification and unlinkedness of fibres in cotangent bundles of Riemann surfaces.
    Symplectic Zoominar (CRM-Montreal, Princeton/IAS, Tel Aviv, and Paris).
    Zoom, 4 September 2020.
    Abstract: In a joint work with Laurent Côté we show the following result. Any Lagrangian plane in the cotangent bundle of an open Riemann surface which coincides with a cotangent fibre outside of some compact subset, is compactly supported Hamiltonian isotopic to that fibre. This result implies Hamiltonian unlinkedness for Lagrangian links in the cotangent bundle of a (possibly closed Riemann surface whose components are Hamiltonian isotopic to fibres.
  • The wrapped Fukaya category of a Weinstein manifold is generated by the Lagrangian cocore discs.
    IMPA, Rio de Janeiro, 29 August 2019
  • Bulky Hamiltonian isotopies of Lagrangian tori.
    Bernoulli Center, Lausanne, 22 August 2019
  • Classification results for two-dimensional Lagrangian tori.
    Princeton/IAS Symplectic Geometry Seminar.
    Princeton, USA, 7 April 2016.
    Abstract: We present several classification results for Lagrangian tori, all proven using the splitting construction from symplectic field theory. Notably, we classify Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy; they are either product tori or rescalings of the Chekanov torus. The proof uses the following results established in a recent joint work with E. Goodman and A. Ivrii. First, there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, as well as the monotone S2×S2. Second, the nearby Lagrangian conjecture holds for the cotangent bundle of the torus.
  • Classification results for two-dimensional Lagrangian tori.
    IBS-CGP, Postech.
    Pohang, Korea, 3 March 2016.
    Abstract: We discuss recent classification results for two-dimensional Lagrangian tori in certain symplectic manifolds, all proven using the splitting construction from symplectic field theory. Notably, these techniques are used to give a complete classification result for Lagrangian tori in the symplectic vector space up to Hamiltonian isotopy. In addition, in joint work with E. Goodman and A. Ivrii, we also show that there is a unique torus up to Lagrangian isotopy inside the symplectic vector space, the projective plane, and the monotone S2×S2. Finally, the nearby Lagrangian conjecture for the cotangent bundle of the torus is established.
  • Floer homology for Lagrangian cobordisms.
    IBS-CGP, Postech.
    Pohang, Korea, 8 March 2016.
    Abstract: Legendrian contact homology (LCH) is a Legendrian isotopy invariant. We introduce a version of wrapped Floer homology for pairs of Lagrangian cobordisms having Legendrian ends that admit augmentations. This theory is used to establish long exact sequences involving the singular homology of a Lagrangian cobordism and the linearised LCH of its Legendrian ends. As an application we show that an exact Lagrangian cobordism from a Legendrian sphere to itself necessarily is a concordance in high dimensions. This is joint work with B. Chantraine, P. Ghiggini and R. Golovko.