During the first, the second and
the fourth weeks there will be mini-courses, while the third week will be
devoted to a workshop.
During the first two weeks, there will be research talks as well.
First week (26 - 30 May)
Aula
Magna (ground floor) at the Math Department
Mini-courses
·F. Bonahon (USC), The Hitchin component. Abstract.
The Hitchin component is a preferred component in
the character variety consisting of all homomorphisms from a
surface group to PSL_n(R), and corresponds to the Teichmüller
space when n=2. Hitchin showed that the Hitchin component is
diffeomorphic to the euclidean space of dimension
2(g-1)(n-1)(n+1). The goal of the minicourse is to provide
geometric proofs of this result, of increasing complexity and
applicability. We will also investigate geometric and dynamical
properties of the homomorphisms that correspond to points in the
Hitchin component.
Let X be an infinite area hyperbolic surface. The Teichmuller space T(X) of the surface X has been extensively studied using the quasiconformal deformations of X by various authors.
We study T(X) via the deformations of hyperbolic metrics on X. Our first tool in deforming the hyperbolic metrics are earthquakes. We characterize which earthquakes deform the base hyperbolic metric on X
in a bounded fashion such that the new metric is also in T(X). Furthermore, we establish a homeomorphism between the Teichmuller space T(X) and the space of earthquake measures. Using the Liouville
currents, we introduce Thurston's bordification of T(X) and prove that it equals to the space PML_{bdd}(X) of projective bounded measured laminations on X. The universal Teichmuller space
T(H) is the Teichmuller space of the hyperbolic plane H. We give a parametrization of the universal Teichmuller space T(H) in terms of shear coordinates on the Farey
tesselation of the hyperbolic plane H. Furthermore, we describe the tangent space and the complex structure of T(H) in terms of shear coordinates.
We also study the hyperbolic metrics on X via its length spectrum (as
started by Shiga, and further developed by Alessandrini, Liu, Papadopuolos
and Su, Basmajian, Kinjo, Kim etc). The Thurston's boundary to T(X) using
the length spectrum is introduced and compared to the Thurston's boundary
using the Liouville currents.
Research
talks:
·I. Chatterji (Orléans), Introduction to CAT(0) cube complexes. Abstract.
CAT(0) cube complexes are cellular complexes built using euclidean cubes and admitting a non-positively
curved metric. They are crucial in Agol's solution of the virtual Hacken conjecture. In this talk I will give
equivalent definitions of CAT(0) cube complexes, examples and discuss related questions. This talk will be accessible
to non-specialists.
·S. Kim (Kias), Simplicial volume of compact manifolds with amenable boundary. Abstract.
Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each
path component of the boundary of V has amenable fundamental group, then we prove that the simplicial volume of M is equal
to the relative simplicial volume of V and also to the Lipschitz simplicial volume of any Riemannian metric on
M whenever the latter is finite. As an application we establish the proportionality principle for the
simplicial volume of complete, pinched negatively curved manifolds of finite volume.
·B. Safnuk (Michigan), Operadic structures in the moduli space of curves Abstract.
In this talk, I will illustrate a new algebraic structure arising in the moduli space of Riemann surfaces. I will focus
primarily on an example construction that can be thought of as the mirror symmetric dual to the Catalan numbers, but will
also discuss related examples that involve the symplectic and hyperbolic structures on the moduli space of Riemann
surfaces. All of these rather different constructions, plus many others, are unified by the structure of a topological
recursion operad.
·A. Seppi (Pavia), Infinitesimal deformations of hyperbolic metrics on a surface and flat Lorentzian structures. Abstract.
A maximal globally hyperbolic flat Lorentzian manifold M is a 3-manifold endowed with a flat pseudo-Riemannian metric
(of signature 2,1) with some good properties related to causality. For example, M is homeomorphic to SxR, where S is a
closed surface of genus g>1. We will survey some classification results of such space-times, with a prescribed
topology, due to Mess. In particular, their moduli space can be identified to the tangent bundle of Teichmuller space
of the closed surface S. We will recover this identification in a different way by means of geometric differential
techniques, which enable to extend the previous results to the case of space-times and surfaces containing cone
singularities.
·A. D. Valdivia (Florida South. Coll.), Minimal asymptotic translation in the curve complex. Abstract.
Analogous to asymptotic bounds for minimal dilatations of psuedo-Anosov mapping classes, we find bounds for
the asymptotic translation in the curve complex.
Monday
Tuesday
Wednesday
Thursday
Friday
9.30 - 10.00
Registration and Opening
10.00 - 11.00
Kim
Saric (9.30 - 11.00)
Safnuk
Bonahon (9.30 - 11.00)
11.00 - 11.30
Coffee
Coffee
Coffee
Coffee
Coffee
11.30 - 12.30
Seppi
Bonahon
Bonahon
Saric
Chatterji
14.30 - 15.30
Bonahon
Saric
Valdivia
15.30 - 16.00
(14.30 - 16)
(14.30 - 16)
19.00 - 20.00
Cocktail (SNS inner court)
Second week (2 - 6 June)
Palazzo del Castelletto
Mini-courses
·T. Barbot
(Avignon), Globally hyperbolic spacetime of constant curvature. Abstract.
Globally hyperbolic spacetimes of constant curvature provide a nice parallel
in the lorentzian context of the classical study of Kleinian groups in
hyperbolic geometry. Nowadays, the classification of these spacetimes are
well understood, at least in the regular case, ie. without particles.
The goal of the minicourse is to provide the foundation of this theory, to
present the link with Anosov representations of Gromov hyperbolic groups in
SO(2,n) or SO(1,n) x R^{1,n}, the question of the geometric convergence
of level sets of time functions near the "initial singularity", and related
open questions in the field.
·J. Kahn (CUNY), The Surface Subgroup Theorem and the Ehrenpreis conjecture. Abstract.
I will present a careful outline of the proofs of the Surface
Subgroup Theorem and the Ehrenpreis conjecture, with an emphasis on the
proof of the Ehrenpreis conjecture. If time permits I will include one
lecture on some new applications of this theory.
As a generalisation of results for SL(3,R) (proved
independently by the author and Loftin) and for SL(2,R)xSL(2,R) (Schoen), I will explain that
for any rank 2 semisimple split real group G and any Hitchin
representation rho in G, there exists a unique rho-equivariant
minimal surface in
the corresponding symmetric space. As a corollary all the
corresponding Hitchin components are parametrised; in a mapping class
group invraiant way, by a pair (J,Q) where J is a
complex structure on the surface and Q a holomorphic differential. A
weaker result subsist for general cyclic Higgs bundles as introduced
and studied by Baraglia.
I will start by explaining what is a Higgs bundle, and I will also
explain basic Lie Theory. A basic knowledge of Lie groups (not
assuming root theory) will be appreciated.
Research
talks:
·J. Behrstock (Columbia), Higher dimensional isoperimetric and divergence functions for mapping class groups. Abstract.
We will discuss the higher dimensional filling functions for mapping class groups of surfaces. We will
establish bounds for these families of functions and show they exhibit phase transitions at the rank (as measured by 3
genus+number of punctures-3); this phase transition is analogous to a corresponding result for symmetric spaces which
results from the combined work of Brady--Farb, Hindawi, Leuzinger, and Wenger. This is joint work with Cornelia Drutu.
·G. Smith (Rio de Janeiro),
On ramified coverings of the sphere and immersed surfaces in hyperbolic space.Abstract.
We construct a bijection between the space of ramified coverings of the sphere on the one hand and the space of complete finite area locally strictly convex immersed surfaces in hyperbolic space of constant Gaussian curvature equal to k, for any fixed k between 0 and 1 on the other. Furthermore, this bijection restricts over each stratum to a homeomorphism.
·M. Burger
(ETH Zürich), Volumes, bounded cohomology and representations of three-manifold groups. Abstract.
In 1982 Gromov introduced bounded cohomology and showed that the bounded cohomology of a topological space is
isometrically isomorphic to the bounded cohomology of its fundamental group. This result is at the basis of most
rigidity results proven with cohomological methods. When dealing with manifolds with boundary, the use of bounded
cohomology relative to the boundary components is paramount.
In this minicourse we present some of the results in relative bounded cohomology parallel to the one in relative
cohomology and we show some applications to the study of the bounded cohomology of the fundamental group of appropriate
graphs of groups.
Further, we define the volume of a representation of a three-manifold group and we prove some of its properties. As a
corollary we obtain rigidity results for such representations into the isometry group of real hyperbolic space.
In a different direction, but for the same manifold groups, we define the volume of a representation into SL(n,C),
prove that it satisfies a Milnor-Wood type inequality and study the properties of its maximality.
·W. Goldman
(College Park), Flat affine and projective manifolds.
Abstract.
This minicourse will survey the few known results and open problems concerning
affine and projective structures. I will concentrate on examples, giving the details of their contruction.
·A. Goncharov
(Yale), Ideal webs and coordinates
on the moduli spaces of flat connections on surfaces.Abstract.
I will show that given an ideal web on a surface there is a cluster
coordinate system on the moduli space of
flat GL(n) connection on a surface. In some particular cases it
reduces to
the coordinate systems defined by Vladimir Fock and
the author a decade ago.
·R. Mazzeo
(Stanford), Deformation theory for conifolds.Abstract.
I will describe the some aspects of the geometry of constant curvature
surfaces with
conic singularities and constant curvature three-manifolds with branched
edge singularities along geodesic networks. These are the (constant
curvature)
conifolds in dimensions 2 and 3. My emphasis wiill be on various existence
results
(for surfaces these involve variational theory and/or Ricci flow), and the
deformation theory
of these objects, again using methods drawn from global analysis. The
two-dimensional
theory was developed in collaboration with H. Weiss, and also touches on
work of
Troyanov and others, while the part of the three dimensional theory I will
focus on is
from joint work with Montcouquiol, and other work of Weiss and
Montcouquiol-Weiss,
all of which generalizes older work of Hodgson and Kerckhoff.
·A. Neitzke
(Austin), Spectral networks and their uses.Abstract.
Spectral networks are certain networks of
codimension-1 "walls" on manifolds. Spectral networks on
2-manifolds have appeared in many places, e.g. in the theory of
cluster varieties, Teichmuller theory and its higher analogues,
Hitchin systems, wall-crossing in Donaldson-Thomas theory, and
4-dimensional supersymmetric quantum field theory. One key idea
is that the spectral network gives a way of reducing nonabelian
phenomena to abelian ones, e.g. replacing GL(K) connections over
some space by GL(1) connections over a K-fold covering space. I
will describe what a spectral network is, how they give rise to
nice "cluster-like" coordinate systems on moduli spaces of
complex flat connections over 2-manifolds (character varieties),
and how they can be used to study the solutions of Hitchin
equations. Time permitting, I will also briefly discuss an
extension of the story to 3-manifolds.
The main part of the story is joint work with Davide Gaiotto and
Greg Moore, motivated by work of many other people, especially
Fock-Goncharov, Kontsevich-Soibelman, Joyce-Song. The extension
to 3-manifolds is joint work in progress with Dan Freed, also
influenced by work of Thurston, Goncharov,
Garoufalidis-Thurston-Zickert.
