In 2005-2006, the seminar was organised by Javier Aramayona and Samuel Lelièvre.

Cliff Earle (Cornell University)

*The genus two Jacobians that are isomorphic to a product of elliptic
curves.*

**Abstract:**
Every compact Riemann surface $R$ has a Jacobi variety (often called its
Jacobian) $J(R)$, which is a complex torus with some extra structure that
makes it a principally polarized Abelian variety and allows us to
reconstruct $R$ from $J(R)$. I shall describe this briefly from scratch.
The dimension of $J(R)$ equals the genus of $R$. When the genus is two, we
can ask whether $J(R)$ is (isomorphic to) the product of two
one-dimensional tori (elliptic curves). The answer is no if the
isomorphism is required to respect the polarization.
But many examples are known where $J(R)$ is isomorphic as a complex
manifold to a product of two elliptic curves. This talk will list all
possible examples. The proof that our list is complete is surprisingly
elementary.

Kirill Krasnov (Nottingham)

*Equidistant foliations and volumes of co-compact hyperbolic
3-manifolds.*

**Abstract:**
We review some well-known results about foliations of constant
curvature 3-manifolds M by surfaces equidistant to a given one. Given the
first and second fundamental forms of a surface S, the 3-manifold metric
can be written down explicitly in the Gaussian coordinate system based on
S, at least in the neighbourhood of S. We apply this description to convex
co-compact hyperbolic 3-manifolds M and consider two cases: (i) S
is a minimal surface; (ii) S is a convex surface. In the first case we
are led to the notion of almost-fuchsian manifolds which contain
a unique minimal surface S and are foliated by surfaces equidistant to S.
We also show that not all quasi-fuchsian manifolds are almost-fuchsian.
In the second case we get an equidistant foliation of a hyperbolic end of
M. Given such a foliation for each hyperbolic end one can
define the so-called renormalized volume of M. We compute the
renormalized volume explicitly and show that it is related to certain
other volume. We show that this other volume is given by the so-called
Liouville action functional for the so-called horospherical metric built
from the first and second fundamental forms of S.

Samuel Lelièvre (Warwick)

*Square-tiled surfaces and moduli spaces of abelian
differentials*

**Abstract:**
We will introduce square-tiled surfaces and discuss how these
combinatorial objects
can give insight on the moduli spaces of abelian differentials in which
they sit.

There will be no seminar this week.

Marc Lackenby (Oxford)

*Counting covering spaces and subgroups in dimension three*

Thierry Monteil (IML, Marseille)

*Finite blocking property on polygonal billiards and translation
surfaces*

Hideki Miyachi (Tokio Denki)

*On the image of the asymptotic map*

Samuel Lelièvre (Warwick)

* Square-tiled surfaces: telling orbits apart and counting*

**Abstract:**
There is a natural action of the group SL(2,R) on strata of
moduli spaces of abelian differentials, whose orbits are
Teichmüller discs.
We will discuss how to enumerate the Teichmüller discs of
square-tiled surfaces and how to count their integer points
in the case of abelian differentials with a double zero on
Riemann surfaces of genus two.

Jens Marklof (Bristol)

*Geometric aspects of the distribution of n^2 alpha mod 1*

**Abstract:**
It is well known that for every irrational alpha, the sequence n^2
alpha is equidistributed modulo one: the number of n=1,...,N for
which the fractional part of n^2 alpha is contained in a subinterval
[a,b] of the unit interval is approximately N(b-a) for N large. I
will explain, how more subtle properties of the sequence that
characterize its randomness can be described in terms of geodesics on
the modular surface. One particular application is a limit law for
skew translations.

Bertrand Deroin (IHES)

*Random 1-dimensional dynamical systems*

**Abstract:**
This is a joint work with Victor Kleptsyn. We study the Brownian motion
along the leaves of a foliation. This theory were originated by L.
Garnett. Here we restrict our attention to the foliations that are
transversely conformal, for instance real codimension $1$ foliations or
complex codimension $1$ holomorphic foliations. We prove that either there
is a transversely invariant measure, or there is a transverse contraction
phenomenom and a unique invariant measure by the leafwise Brownian motion.
The proof deals with the Lyapunov exponents of such invariant measures.

Cormac Long (Southampton)

*Some dodecahedral tessellations associated
with the Coxeter group $[5,3,5]$*

**Abstract:**
We describe some dodecahedral tessellations
of hyperbolic $3$-manifolds arising from
subgroups of the Coxeter group $[5,3,5]$.
These include a chiral pair of copies of
the Weber-Seifert space, their homology
covers and their smallest common cover.
We also discuss some other covering
manifolds that arose during this study.

Bas Lemmens (Warwick)

*How many points at the same distance?*

**Abstract:**
A set in a metric space is called equilateral
(or equidistant) if any two distinct elements are at the
same distance from each other. It is well known that
an equilateral set in the n-dimensional Euclidean space
has at most n+1 elements. But how large can such a
set be if the norm is changed to another norm say an
$\ell_p$ norm? In this talk I will discuss several results
and conjectures concerning this problem.

Moon Duchin (UC Davis)

*Curvature and dynamics in Teichmuller space*

**Abstract:**
Standard notions of nonpositive curvature do
not hold for the Teichmuller metric, so we introduce a
new notion which does capture some of the hyperbolic
behavior in that metric. Using this condition, we show
that the random action of the mapping class group is
well approximated by the geodesic flow.

Mikhail Belolipetsky (Durham)

*Finite groups and hyperbolic manifolds*

**Abstract:**
The isometry group of a compact n-dimensional hyperbolic
manifold is known to be finite. We show that for every n > 2, every
finite group is realized as the full isometry group of some compact
hyperbolic n-manifold. The cases n = 2 and n = 3 have been proven by
Greenberg and Kojima, respectively. Our proof is non constructive: it
uses counting results from subgroup growth theory and the strong
approximation theorem to show that such manifolds exist. This is joint
work with Alex Lubotzky.

Caroline Series (Warwick)

*Lines of minima and Teichmüller geodesics I*

Caroline Series (Warwick)

*Lines of minima and Teichmüller geodesics II*

James Giblin (Warwick)

*Classification of continuously transitive Circle Groups*

Brian Bowditch (Southampton)

*The Ending Lamination Conjecture I*

Brian Bowditch (Southampton)

*The Ending Lamination Conjecture II*

Brian Bowditch (Southampton)

*The Ending Lamination Conjecture III*

Pascal Hubert

* Veech groups and the dynamics of flat surfaces*

Emmanuel Royer

*Quasimodular forms and applications*

Jack Button (Cambridge)

*3-manifold groups, the BNS invariant and the Alexander
polynomial*

Anthony O'Farrell (NUI, Maynooth)

*Reversibility and its Connections*

Etienne Ghys (ENS Lyon)

*Rotation numbers for surface automorphisms*

Vladimir Bozin (Warwick)

*Quasiconformally homogenous surfaces *

Greg McShane (Toulouse)

*The Hitchin component, higher Teichmueller space and shear
coordinates*

Susan Hermiller (University of Nebraska - Lincoln)

*Almost convexities and tame combings for groups*

Martin Bridson (Imperial College)

*Snowflake groups, Perron-Frobenius exponents and isoperimetric
spectra*

Hugo Parlier (University of Geneva)

*Gaps left by simple closed geodesics on surfaces*

Robert Silhol (Montpellier)

*Hyperbolic Lego, translation surfaces and uniformization
*

Frank Herrlich (Karlsruhe)

* Equations for Teichmüller curves: some examples*

Anthony O'Farrell (Maynooth)

* Reversibility and its Connections*

Akshay Venkatesh (Currently visiting the Institute for Advanced Study, Princeton)

*TBA*