Monday 25 June | Tuesday 26 June | Wednesday 27 June | Thursday 28 June | Friday 29 June | |||||
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9:00-09:30 | Welcome to you ! | ||||||||

09:30-10:30 | Margalit | 09:30-10:30 | Margalit | 09:30-10:30 | Margalit | 09:30-10:30 | Sorger | 09:30-10:30 | Habiro |

10:30-11:00 | Coffee break | 10:30-11:00 | Coffee break | 10:30-11:00 | Coffee break | 10:30-11:00 | Coffee break | 10:30-11:00 | Coffee break |

11:00-12:00 | Kawazumi | 11:00-12:00 | Kawazumi | 11:00-12:00 | Kawazumi | 11:00-12:00 | Marché | 11:00-12:00 | Kashaev |

14:30-15:30 | Putman | 14:30-15:30 | Sorger | 14:30-15:30 | Sorger | 14:30-15:30 | Kaufmann | ||

15:30-16:00 | Coffee break | 15:30-16:00 | Coffee break | 15:30-16:00 | Coffee break | 15:30-16:00 | Coffee break | ||

16:00-17:00 | Reid | 16:00-17:00 | Turaev | 16:00-17:00 | Luo | 16:00-17:00 | Andersen |

Nariya KAWAZUMI:
Johnson-Morita theory and the Goldman-Turaev Lie bialgebra

The fundamental group of a once-bordered surface of genus g>0 with basepoint on the boundary is a free group of rank 2g.
The mapping class group of the surface acts on the free group and its nilpotent tower in a natural way.
The Torelli group is, by definition, the kernel of the action on the abelianization of the free group.
In 1980's Dennis Johnson defined a decreasing filtration of the Torelli group induced by the action on the nilpotent tower,
and described the graded quotient of the filtration in terms of the free Lie algebra over the first homology group of the surface.
This description gives an injective Lie algebra homomorphism, and is called the Johnson homomorphisms nowadays.
It is a fundamental but open problem to determine the image of the Johnson homomorphism.
Unfortunately, Johnson quitted mathematics, but later Shigeyuki Morita and other people have elaborated Johnson's theory.
For example, an extension of the Johnson homomorphism gives
the MMM (Mumford-Morita-Miller) cohomology classes on the mapping class group.
In the first half of this mini-course, these will be explained in details.

In the second half we will explain the Goldman-Turaev Lie bialgebra and its application to Johnson-Morita theory.
In 1980's William Goldman introduced a Lie bracket on the free linear space over the free loops on an oriented surface.
Later Vladimir Turaev discovered a Lie bialgebra structure on Goldman's Lie algebra, which is called the Goldman-Turaev Lie bialgebra.
The origin of the algebra was in the moduli space of flat bundles on the surface, and so it has been rather unconnected to the Torelli groups.
But, very recently, Kuno and the speaker found that the Johnson homomorphism can be interpreted
as an embedding of the Torelli group into (a completion of) the Goldman-Turaev bialgebra.
The key to connecting these two theories is the notion of symplectic expansions discovered by Massuyeau.
This construction can be extended to the `smallest' Torelli group
in the sense of Putman of compact oriented surfaces with non-connected boundary,
and the Turaev cobracket provides us with a geometric constraint of the image of the Johnson homomorphism.

Slides:
**Lectures 1-2**
**Lectures 2-3**

Dan MARGALIT:
Perspectives and Problems on Mapping Class Groups

The mapping class group is the group of homotopy classes of homeomorphisms of a surface.
This group was first studied by Dehn and Nielsen in the early part of the 20th century.
More recently, the subject has experienced a renaissance, due to its deep connections
to 3-manifold theory, symplectic geometry, representation theory,
Teichmuller theory, dynamics, homotopy theory, algebraic geometry, and geometric group theory.
In this series of lectures, I will give an introduction to the theory of mapping class groups, accessible to graduate students.
The goal will be to highlight some of the many gems of the subject, while also concentrating on open questions and new directions.
My three lectures will center on the cohomology of the mapping class group, the group structure of the Torelli subgroup,
and the dynamics of pseudo-Anosov homeomorphisms.

Slides:
**Lecture 1**
**Lecture 2**
**Lecture 3**

Videos:
**Lecture 1**
**Lecture 2**
**Lecture 3** (recorded by Seonhwa Kim)

Christoph SORGER:
Algebraic conformal field theory and the mapping class group

The aim of these lectures is to give an introduction, accessible for graduate students, to algebraic conformal field theory as defined via generalized theta-functions or via the vacua spaces of Tsuchiya, Ueno and Yamada and their associated monodromy representations. These representations have in general elements of infinite order. This should have non trivial consequences for the associated p-curvatures once one is able to define conformal field theory in characteristic p. Part of my lectures will be devoted to explain these consequences through the general framework of the arithmetic theory of differential equations known as the Grothendieck-Katz conjectures.

Jørgen ANDERSEN:
The geometric construction of the Reshetikhin-Turaev Topological Quantum Field Theory

In this talk, we will discuss the geometric construction of the Reshetikhin-Turaev Topological Quantum Field Theory using the geometric quantization of the moduli spaces of flat connections on two dimensional surfaces. We will then discuss various results on the large level asymptotics of these theories.

Kazuo HABIRO:
On the structure of the mapping class category

The mapping class category is the category of surfaces and isotopy classes of embeddings. It contains as the subcategory of automorphisms the mapping class groupoid, which is the category of surfaces and isotopy classes of homeomorphisms. In this talk, I plan to discuss the structure of the mapping class category, and relate it to the 3-dimensional cobordism category and the category of 3-manifolds and isotopy classes of embeddings.

Rinat KASHAEV:
On non-compact realizations of tetrahedral symmetry relations

Tetrahedral symmetry relations (TSR) are satisfied by 6j-symbols coming from Psi-systems of linear monoidal categories. In particular, the multiplicity spaces in Psi-systems are always finite dimensional. In this talk, I will describe two realizations of TSR in infinite dimensional vector spaces, one corresponding to Teichmüller TQFT, and another one corresponding to a new but simpler TQFT associated with Gaussian integrals.

Ralph KAUFMANN:
Categorified correlators

We review our actions on Hochschild complexes by moduli spaces and the version leading to string topology. After a brief categorical interlude we then show how to categorify these correlators. One interpretation of the result is that it is a fully extended 3-2-1-0 version of Reshetikhin-Turaev theory. This latter part is work in progress.

Feng LUO:
Simplicial SL(2,R) Chern-Simons theory and
Boltzmann-Shannon entropy on 3-manifolds

We show that real solutions of Thurston's gluing equation on triangulated 3-manifolds are related to maximizers of the Boltzmann-Shannon entropy.

Julien MARCHÉ:
Ergodicity of the Torelli group on representation spaces

Given a compact oriented surface without boundary S and a group G,
the modular group Mod(S) acts on the space M(S,G) of representations of the fundamental group of S in G.
In the case when G=PSL_2(R), the group Mod(S) acts properly on the Teichmuller component
but in the case when G=SU_2, Goldman showed that the action was ergodic.
The aim of this talk is to show that the Torelli group (those elements of Mod(S)
which act trivially on H_1(S)) still acts ergodically. This is joint work with L. Funar.

Video:
**Lecture** (recorded by Seonhwa Kim)

Andrew PUTMAN:
Stability in the homology of congruence subgroups

The homology groups of many natural sequences of groups G_n (e.g.
general linear groups, mapping class groups, etc.) stabilize as n goes
to infinity. Indeed, there is a well-known machine for proving such
results that goes back to early work of Quillen. Church and Farb
discovered that many sequences of groups whose homology groups do not
stabilize in the classical sense actually stabilize in some sense as
representations. They called this phenomena representation stability. We
prove that the homology groups of congruence subgroups of GL(n,R) (for
almost any reasonable ring R) satisfy a strong version of representation
stability that we call central stability. The definition of central
stability is very different from Church-Farb's definition of
representation stability (it is defined via a universal property), but we
prove that it implies representation stability. Our main tool is a new
machine analogous to the classical homological stability machine for
proving central stability.

Video:
**Lecture** (recorded by Seonhwa Kim)

Alan REID:
All finite groups are involved in the Mapping Class group

We will discuss how the (projective) unitary repns arising
from SO(3)-TQFT can be used to show that on fixing the Mapping Class groug
Mod(g), given any finite group A, there exists a finite index subgroup G
of Mod(g) such that G surjects A. In addition we will describe explicit
finite quotients of Mod(g).

Video:
**Lecture** (recorded by Seonhwa Kim)

Vladimir TURAEV:
Quasi-Poisson structures on representation spaces of surfaces

Given an oriented surface S with base point * on the boundary,
we introduce for all N>0, a canonical quasi-Poisson bracket
on the space of N-dimensional linear representations of \pi_1(S,*).
Our bracket extends the well-known Poisson bracket on GL_N-invariant functions on this space.
Our main tool is a natural structure of a quasi-Poisson double algebra
(in the sense of M. Van den Bergh) on the group algebra of \pi_1(S,*).

Video:
**Lecture** (recorded by Seonhwa Kim)