## Symposiums & Workshops

### International workshop on Geometry of Foliated Spaces

2019.10.24 Thu up
 Date: 29 November – 1 December 2019 Place: Colloquium Room, West-Wing 6th floor, Biwako-Kusatsu Campus, Ritsumeikan University Access to Biwako-Kusatsu Campus Campus Map Campus Map(pdf) Address: Nojihigashi 1-1-1, Kusatsu, Shiga, Japan

Program

29 November (Fri)
13:30-14:20 Hitoshi Moriyoshi (Nagoya Univ.)
Geometry on the space of equi-centro-affine curves

14:40-15:30 Ramón Barral Lijó (Ritsumeikan Univ.)
Distinguishing colorings

15:50-16:40 Jesús Antonio Álvarez López  (Univ. of Santiago de Compostela)
Haefliger’s cohomology and tautness for matchbox manifolds

17:00-17:30 Subaru Nomoto (Ritsumeikan Univ.)
Some hierarchy in frames on curves on 4-dimensional Euclidean space

19:00- Banquet at Isaribi An near JR Kusatsu Station

30 Novermber (Sat)
9:00-9:50 Ryuma Orita (Tokyo Metropolitan Univ.)
Rigid fibers of spinning tops

10:10-11:00 Olga Lukina (Univ. of Vienna)
Cantor actions and Galois groups

11:20-12:10 Ramón Barral Lijó (Ritsumeikan Univ.)
Breaking symmetry

14:30-15:20 Naoki Kato (Chukyo Univ.)
Turbulizations of transversely affine foliations of codimension two

15:40-16:30 Noboru Ito (Univ. of Tokyo)
A categorified Vassiliev skein relation on Khovanov homology

16:50-17:20 Norihisa Takahashi (Ritsumeikan Univ.)
Dehn twist presentations of hyperelliptic periodic maps

1 December (Sun)
9:00-9:50 Olga Lukina  (Univ. of Vienna)
Stabilizers of points for Cantor group actions

10:00-10:50 Tetsuya Abe (Ritsumeikan Univ.)
On annulus presentations of knots

11:10-12:00 Jesús Antonio Álvarez López  (Univ. of Santiago de Compostela)
A distributional trace formula for foliated flows

Contact: Hiraku Nozawa (Ritsumeikan Univ.)
E-mail：hnozawa（at）fc.ritsumei.ac.jp

Nov. 29: Program changed.
Oct. 24: Page created. First-announcement.
Abstract

Hitoshi Moriyoshi (Nagoya Univ.)

Title: Geometry on the space of equi-centro-affine curves

Abstract: A curve $$c$$ on the Euclidean plane is called equi-centro-affine if it satisfies $$\mathrm{det} (c c’)=1$$. The space of all equi-centro-affine curves on plane turns out to be a manifold of infinite dimension, which admits an interesting action by the group of orientation-preserving diffeomorphisms on unit circle. In this talk, we exhibit invariant pre-symplectic forms on the infinite-dimensional manifold and momentum maps related to those forms, which are also involved with the celebrated Gelfand-Fuchs $$2$$-cocycle.

Ramón Barral Lijó (Ritsumeikan Univ.)

Title: Distinguishing colorings

Abstract: We will introduce the concepts or distinguishing and limit distinguishing colorings for graphs, as well as the corresponding indices. We will also present known results and bounds for these indices for different families of graphs.

Jesús Antonio Álvarez López  (Univ. of Santiago de Compostela)

Title: Haefliger’s cohomology and tautness for matchbox manifolds

Abstract: (joint work with Steven Hurder and Hiraku Nozawa) For a compact foliated space $$\mathfrak M$$ of dimension $$p$$, we introduce a cochain complex which combines the de~Rham complex along the leaves and the Alexander-Spanier complex in the transverse direction. This complex also has a filtration by differential spaces, giving rise to a version of the Leray spectral sequence $$(E_k^{u,v},d_k)$$ for $$\mathfrak M$$. The part $$E_2^{\cdot,p}$$ is a version of the Haefliger cohomology for $$\mathfrak M$$, which can be used to characterize a version of tautness like in the case of foliations on compact manifolds. In particular, all matchbox manifolds are taut in our sense.

Subaru Nomoto (Ritsumeikan Univ.)

Title: Some hierarchy in frames on curves on 4-dimensional Euclidean space

Abstract: Bishop introduced a frame on space curves different from the well known Frenet frame. It is characteristic that any regular space curve admit this frame. For curves on R^4, there are four different type of Bishop type frames including the Frenet frame and the Bishop frame. In this talk, we discuss the relation between these four types of frames from the viewpoints of geometric structures on normal bundle f curves and ordinary differential equations.

Ryuma Orita (Tokyo Metropolitan Univ.)

Title: Rigid fibers of spinning tops

Abstract: In the talk, we deal with fibers of classical Liouville integrable systems containing the Lagrangian top and the Kovalevskaya top. Especially, we find a non-displaceable fiber for each of them. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich. This is a joint work with Morimichi Kawasaki (RIMS).

Olga Lukina  (Univ. of Vienna)

Title: Cantor actions and Galois groups

Abstract: In this talk, we consider applications of dynamical invariants for Cantor group actions in Number theory, namely for the study of a certain type of representations of absolute Galois groups of number and function fields.

Ramón Barral Lijó (Ritsumeikan Univ.)

Title: Breaking symmetry

Abstract: We will contextualize the previous talk in the setting of the Gromov space of graphs, and will present some applications to other areas of mathematics.

Naoki Kato (Chukyo Univ.)

Title: Turbulizations of transversely affine foliations of codimension two

Abstract: In 1980, Seke gave a condition for turbulized foliations of codimension one transversely affine foliations to have transverse affine structures again. Recently, Mitsumatsu and Vogt defined the notion of turbulizations for higher codimensional foliations. In this talk, we give a condition for turbulized foliations of codimension two transversely affine foliations to have transverse affine structures again.

Noboru Ito (Univ. of Tokyo)

Title: A categorified Vassiliev skein relation on Khovanov homology

Abstract: This is a joint work with Jun Yoshida (The University of Tokyo). In this talk, we define a chain map that categorifies the operation “crossing change” on Khovanov homology using a canonical short exact sequence of a mapping cone at each crossing of a link diagram. The chain map is invariant under moves of singular knots. By the setting, taking graded Euler characteristics, a Vassiliev skein relation for a Jones polynomial is recovered.

Norihisa Takahashi  (Ritsumeikan Univ.)

Title: Dehn twist presentations of hyperelliptic periodic maps

Abstract: Ishizaka classified up to conjugation hyperelliptic periodic diffeomorphisms of surfaces and gave Dehn twist presentations in terms of Humphries generators. In this talk, we will give an explicit decomposition of surfaces into pentagonal fundamental domains of hyperelliptic periodic diffeomorphisms. As an application, we obtain the Dehn twist presentations of hyperelliptic periodic mapping classes, which are closely related to the ones obtained by Ishizaka.

Olga Lukina  (Univ. of Vienna)

Title: Stabilizers of points for Cantor group actions

Abstract: We study properties of stabilizers of points for Cantor group actions, such as the number of distinct stabilizers of points without holonomy, and the genericity of points with non-trivial holonomy in a measure-theoretical sense. We then apply our results to study the properties of invariant random subgroups, defined by such action. Joint work with Maik Groeger.

Tetsuya Abe (Ritsumeikan Univ.)

Title: On annulus presentations of knots

Abstract: In 2013, Jong, Omae, Takeuchi and I introduced the notion of annulus presentations of knots. In this talk, we discuss the following topics:

1. Why we are interested in annulus presentations of knots.
2. An obstruction to knots admitting annulus presentations in terms of the Jones polynomial.
3. Table of knots which admit annulus presentations up to 8 crossings.

This is joint work with Keiji Tagami. Note that Gen Suzuki (who is a student of Ritsumeikan University) partially made the table of knots which admit annulus presentations up to 8 crossings.

Jesús Antonio Álvarez López  (Univ. of Santiago de Compostela)

Title: A distributional trace formula for foliated flows

Abstract: Let $$\mathcal F$$ be a smooth codimension one foliation on a compact manifold $$M$$. A flow $$\phi^t$$ on $$M$$ is said to be foliated if it maps leaves to leaves. If moreover the closed orbits and preserved leaves are simple, then there are finitely many preserved leaves, which are compact, forming a compact subset $$M^0$$, and a precise description of the transverse structure of $$\mathcal F$$ can be given. A version of the reduced leafwise cohomology, $$\overline{H}I(\mathcal F)$$, is defined by using distributional leafwise differential forms conormal to $$M^0$$. The talk will be about our progress to define distributional traces of the induced action of $$\phi^t$$ on $$\overline{H}^rI(\mathcal F)$$, for every degree $$r$$, and to prove a corresponding Lefschetz trace formula involving the closed orbits and leaves preserved by $$\phi^t$$. The formula also involves a distributional version of the $$\eta$$-invariant of $$M^0$$, which might be zero. This kind of distributional trace formula was conjectured by Christopher Deninger, and it was proved by the first two authors when $$M^0=\emptyset$$.