Atelier de travail franco-japonais
sur la géométrie des groupes modulaires et des espaces de Teichmüller

Les 16-20 novembre 2015
Salle 002, Faculté de sciences mathématiques, Université de Tokyo

Organisateurs:
Ken'ichi Ohshika (Osaka), Takuya Sakasai et Nariya Kawazumi (Tokyo)

<<Le lundi 16 novembre>>

(10:30-12:00  Salle 128
(Séminaire sur l'analyse géométrique complexe:
(Hideki Miyachi (Université dfOsaka)

13:30-14:30
Athanase Papadopoulos (Université de Strasbourg/CNRS)
Timelike geometry

<<Le mardi 17 novembre>>

10:00-11:00
Athanase Papadopoulos (Université de Strasbourg/CNRS)
Spherical geometry I

11:30-12:30
Takahito Naito (Université de Tokyo)
Sullivan's coproduct on the relative loop homology

14:00-15:00 + 15:30-16:30
Gwénaël Massuyeau (Université de Strasbourg/CNRS)
Fox pairings in Hopf algebras and Poisson structures

(16:30-17:00 Salle commune <premier étage>
(pause café pour Séminaire du mardi sur la topologie

(17:00-18:30 Salle 056
(Séminaire du mardi sur la topologie
(Atsuko Katanaga (Université de Shinshu)

<<Le mercredi 18 novembre>>

10:00-11:00
Athanase Papadopoulos (Université de Strasbourg/CNRS)
Spherical geometry II

11:30-12:30
Yasuo Wakabayashi (Université de Tokyo)
A theory of dormant opers

<<Le jeudi 19 novembre>>

10:00-11:00
Masatoshi Sato (Tokyo Denki University)
On the cohomology ring of the handlebody mapping class group of genus two

11:30-12:30
Elena Frenkel (Université de Strasbourg)
Area formula for Hyperbolic Triangles and Lexell problem

14:00-15:00
Yuanyuan Bao (Université de Tokyo)
Heegaard Floer homology for transverse graphs with sinks and sources

15:30-16:30
Tadayuki Watanabe (Université de Shimane)
An invariant of fiberwise Morse functions on surface bundle over S1 by counting graphs

<<Le vendredi 20 novembre>>

09:30-10:30 + 11:00-12:00
Olivier Guichard (Université de Strasbourg)
Compactifications of certain locally symmetric spaces

(Les 20 à 22 novembre Salle 123
(Rigidity School, Tokyo 2015


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Résumés des exposés


Papadopoulos: Timelike geometry
Papadopoulos: Spherical geometry I


Naito: Sullivan's coproduct on the relative loop homology

Sullivan's coproduct is the coproduct on the relative homology of the
free loop space of a closed oriented manifold (called the relative
loop
homology). It is known that the relative loop homology is an
infinitesimal bialgebra with respect to this coproduct and the loop
product. In this talk, we will give a homotopical description of
Sullivan's coproduct and introduce its properties. Moreover, we will
compute the coalgebra structure of spheres over the rational number
field by using the description.


Massuyeau: Fox pairings in Hopf algebras and Poisson structures

We will present the general theory of Fox pairings in Hopf algebras,
which will be illustrated through several algebraic examples. Next, we
will recall how such operations naturally appear in  topology by
considering intersections of curves in surfaces, and sketch how this
generalizes to higher-dimensional manifolds. Finally, we will use Fox
pairings to construct some natural Poisson structures on the affine
scheme of representations of a cocommutative Hopf algebra in an
arbitrary group scheme.  (Based on joint works with Vladimir Turaev.)


Wakabayashi: A theory of dormant opers

A(n) (dormant) oper, being our central object of this talk, is a
certain
principal homogeneous space on an algebraic curve (in positive
characteristic) equipped with an integrable connection. The study of
dormant opers and their moduli may be linked to various fields of
mathematics, e.g., the p-adic Teichmuller theory developed by Shinichi
Mochizuki, Gromov-Witten theory, combinatorics of rational polytopes (
and spin networks), etc. In this talk, we would like to introduce the
definition of a dormant oper and to present some related results,
including an explicit formula for the generic number of dormant opers,
which was conjectured by Kirti Joshi.


Papadopoulos: Spherical geometry II


Sato: On the cohomology ring of the handlebody mapping class group of
genus two

The genus two handlebody mapping class group acts on a tree
constructed
by Kramer from the disk complex,
and decomposes into an amalgamated product of two subgroups.
We determine the integral cohomology ring of the genus two handlebody
mapping class group
by examining these subgroups and the Mayer-Vietoris exact sequence.
Using this result, e estimate the orders of low dimensional homology
groups
of the genus three handlebody mapping class group.


Frenkel: Area formula for Hyperbolic Triangles and Lexell problem

My talk will be about an area formula in terms of side lengths for
triangles in plane hyperbolic geometry and its geometrical
interpretation . The proof of this formula is analogous to a proof
given by Leonhard Euler in the spherical case. I will speak in particular
about the Lexell problem, that is, the problem of finding the locus of
vertices of triangle of fixed area and fixed base.


Bao: Heegaard Floer homology for transverse graphs with sinks and
sources

We defined the Heegaard Floer homology (HF) for balanced bipartite
graphs. Around the same time, Harvey and OfDonnol defined the
combinatorial HF for transverse graphs without sink and source (see
the definition in [arXiv:1506.04785v1]). In this talk, we compare these
two methods and consider the HF for transverse graphs with the same number
of sinks and sources in both analytic and combinatorial ways.


Watanabe: An invariant of fiberwise Morse functions on surface bundle
over S1 by counting graphs

We apply Lescopfs construction of Z-equivariant perturbative invariant
of knots and 3-manifolds to the explicit equivariant propagator of gZ-pathsh.
We obtain an invariant of certain equivalence classes of fiberwise Morse
functions on a 3-manifold fibered over S1, which can be
considered as a higher loop analogue of the Lefschetz zeta function
and whose construction will be applied to that of finite type invariants of
knots in such a 3-manifold.


Guichard: Compactifications of certain locally symmetric spaces