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
=======
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