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