To Japanese version
                                                             Kagurazaka Differential Geometry Seminar

Date : 18 May
  Place : Tokyo University of Science (Kagurazaka Campus)
    Speaker : Tsukasa Takeuchi (Tokyo University of Science)
    Title : On the construction of recursion operators and
        of symplectic-Haantjes manifolds

    Abstract : For a completely integrable system, the way of
    finding the first integrals is not formulated in general.
  In classical
    mechanics, a completely integrable systems in the sense of Liouville
    are called simply an integrable system.
  It is well known there are
    several approaches to find the first integrals such as the method
    of Lax, the Lie algebra adject from the soliton theory, etc.
    certain ways of characterizing integrable systems with (1, 1)-tensors
    is investigated, recently.
  Known examples of integrable systems
    with (1, 1)-tensors are recursion operators and symplectic-Haantjes
  For a dynamical system, the system is proved to be
    integrable if there exists a (1, 1)-tensor which satisfies certain
  In this talk, using their method we construct recursion
    operators and symplectic-Haantjes manifolds for several Hamiltonian
    systems of two degrees of freedom.
    This talk is based on a joint work with Akira Yoshioka and Kiyonori Hosokawa.

    Speaker : Tomoya Nakamura (Waseda University)
    Title :On the development of a theories parallel to
        Lie algebroids for Hom-Lie algebroids

    Abstract :

  Organizers : Naoyuki Koike, Makiko Tanaka,
Akifumi Sako,
         Yasufumi Nitta,
Kurando Baba, Homare Tadano,
         Hikaru Yamamoto

            Record of 2019

  Date : 2 March
  Place : Tokyo University of Science (Kagurazaka Campus)
    Speaker : Alexander Hock
(Mathematisches Institut der Westfälischen Wilhelms-Universität, Germany)
    Title : Exact solution of matricial \phi^3_2 model to all genus by
         topological recursion
     Abstract :
Developed methods will be extended to find exact solution for
\phi_2^3 matrix model with an external matrix E in the
        large N limit.This model can be understood as regularized
        Kontsevich model, where the
regularization ensures
        UV finiteness in 2 dimensions in case of linear eigenvalues.
The universal structure of topological recursion is used
        and slightly affected by the regularization for the one
        boundary (puncture) sector. All correlation functions of
        genus g with B boundary components are achieved by
        a differential operator A_X acting on the one boundary
        solution. The results are exact and describes the summation
        of all weighted 3 valent graphs on a B-punctured Riemann
        surface of genus g. Even though a \phi^3 model is unstable,
        it arises as a exactly solvable noncommutative qunatum
        field theory on a highly deformed Moyal space.

   Date : 27 April
   Place : Tokyo University of Science (Kagurazaka Campus)
    Speaker : Honoka Kobayashi (Tokyo University of Science)
    Title : Pseudo-hyperbolic Gauss maps of submanifolds with
        constant mean curvature in pseudo-hyperbolic space

    Abstract :
We investigated oriented surfaces of constant mean and

Gaussian curvatures and non-diagonalizable shape operator in pseudo-hyperbolic
space. It is known that such Lorentzian surfaces in 3-dimensional anti-de Sitter
space are
either a B-scroll or a complex circle.In this talk, we first state result
for the type numbers of the pseudo-hyperbolic Gauss maps of a B-scroll and
a complex circle
. Secondly, we state results for the type numbers of the
pseudo-hyperbolic Gauss maps of
generalized umbilical hypersurfaces which
are a natural generalizations of B-scrolls in the
general dimensional anti-de Sitter
space.Also, we state constructions of consider as surfaces can be generalized
umbilical hypersurfaces in the pseudo-hyperbolic space and
pseudo-sphere of
index 2 and B-scroll in 5-dimensional
pseudo-hyperbolic space of index 2.
This talk is based on the jointwork with Naoyuki Koike.

    Speaker : Atsufumi Honda (Yokohama National University)
    Title : Mixed type surfaces with bounded Gaussian curvature
        in three-dimensional Lorentzian manifolds

Abstract : A mixed type surface is a connected regular surface in
a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced
metric of a mixed type surface is a signature-changing metric, and their lightlike points

may be regarded as singular points of such metrics. In this talk, we investigate the behavior
of Gaussian curvature at a non-degenerate lightlike point of a mixed type surface.
To characterize the boundedness of Gaussian curvature at a non-degenerate lightlike
points, we introduce several fundamental invariants along non-degenerate lightlike points,
such as the lightlike singular curvature and the lightlike normal curvature. Moreover, using
the results by Pelletier and Steller, we obtain the Gauss-Bonnet type formula for mixed

ype surfaces with bounded Gaussian curvature. This talk is based on a joint work
(arXiv:1811.11392) with K. Saji (Kobe University) and K. Teramoto (Kyushu University).

             Record of 2018

  Date : 22 December
  Place : Tokyo University of Science (Kagurazaka Campus)
    Speaker : Keita Kunikawa (Tohoku University)
    Title : Convergence of mean curvature flow in hyperkähler manifolds
    Abstarct :

     Inspired by the work of Leung-Wan, we study the mean curvature flow in
      hyperkahler manifolds starting from hyper-Lagrangian submanifolds,
      a class of middle dimensional submanifolds, which contains the class of
      complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold,
      we define a new energy concept called the "twistor energy" by means of
      the associated twistor family. In this talk, we will show that the mean curvature
      flow starting at any hyper-Lagrangian submanifold with sufficiently small
      twistor energy will exist for all time and converge to a complex Lagrangian
      submanifold for one of the hyperkähler complex structure. In particular, our result
      implies some kind of energy gap theorem for hyperkä
hler manifolds which have
      no complex Lagrangian submanifolds.
      This talk is based on a joint work with Ryosuke Takahashi

    Speaker : Yasufumi Nitta (Tokyo University of Science)
    Title : Uniform relative stability and coercivity
         for polarized toric manifolds
    Abstract :
      We study a relation between algebro-geometric stability and the growth of
      the modified K-energy which characterizes the extremal K
ahler metric as
      a critical point. In this talk, we introduce uniform relative K-polystability
      for polarized toric manifolds and show that it implies the coercivity of the
      modified K-energy modulo the maximal torus action. If time allows, we will
      also discuss on the converse direction. 
      This talk is based on a joint work with Shunsuke Saito and Naoto Yotsutani.

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