Geometry/Topology Seminar
Fall 2006
Thursdays (and sometimes Tuesdays) 2-3pm, in
Eckhart 206
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- Thursday October 5 at 2pm in E312
- Ben Schmidt, University of Chicago
- Weakly hyperbolic group actions
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Abstract: I'll define the terms and try to make a
case for the following analogy: weak hyperbolicity is to
group actions as Anosov is to diffeomorphisms.
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- Please note the room change
- Thursday October 12 at 2pm in E206
- Jayadev Athreya, Yale University
- Counting problems for Right-angled billiards
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Abstract: In joint work with A. Eskin and A. Zorich,
we prove results for aysmptotics of the number of closed
(and other special) trajectories for billiards in
right-angled Euclidean (not neccesarily convex) polygons.
This leads one to calculations of volumes of moduli spaces
of meromorphic quadratic differentials on spheres.
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- Thursday October 19 at 2pm in E206
- Anna Wienhard, University of Chicago
- Representations of surface groups and rotation numbers
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Abstract: In recent years several people have
studied special connected components of the space of
representations of the fundamental group of a surface of
genus g>1 into a noncompact semisimple Lie
group of higher rank. These components are of interest
because they generalize Teichmüller space (viewed here
as space of discrete faithful representations into
PSL(2,R)). In my talk I plan to give first
some background and then focus on the special case when the
Lie group is of Hermitian type to discuss some recent joint
work with Marc Burger and Alessandra Iozzi, which involves
rotation numbers to get a better understanding of these
generalized Teichmüller components for surfaces with
boundary.
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- Thursday October 26 at 2pm in E206
- Moon Duchin, UC Davis
- Intermediate divergence of geodesics
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Abstract: If you follow a pair of geodesic rays a
distance t from a common basepoint, how long does
it take to get from one to the other without backtracking
towards the basepoint? This rate is called geodesic
divergence. Hyperbolicity turns out to be equivalent to the
property that the divergence grows exponentially, while flat
spaces are characterized by linear divergence. I'll discuss
joint work with Kasra Rafi, showing that two well-loved
spaces, Teichmüller space and the mapping class group,
have intermediate divergence in an appropriate sense.
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- Thursday November 2 at 2pm in E206
- Ruth Charney, Brandeis University
- Outer Space for right-angled Artin groups
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Abstract: Associated to any finite simplicial graph
\Gamma is a right-angled Artin group
A\Gamma whose generators are the
vertices of \Gamma and whose relators are
commutators of adjacent pairs of vertices. This class of
groups may be thought of as interpolating between free
groups (\Gamma has no edges) and free abelian
groups (\Gamma is a complete graph). Thus,
automorphism groups of right-angled Artin groups
“interpolate" between Out(Fn)
and GL(n,Z). We study the (outer) automorphism
group of a right-angled Artin group
A\Gamma in the case where the defining
graph \Gamma is connected and triangle-free. We
construct Outer Space for A\Gamma, a
finite dimensional, contractible space with a proper action
of Out(A\Gamma). We show that Outer
Space retracts onto a lower dimensional spine, and we obtain
upper and lower bounds on the virtual cohomological
dimension of Out(A\Gamma). (Joint work
with John Crisp and Karen Vogtmann)
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- Thursday November 9 at 2pm in E206
- Miklos Abert, University of Chicago
- The growth of rank in residually finite groups and the rank vs
Heegaard genus problem
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Abstract: We relate the growth of rank in residually
finite groups to the cost of measure equivalence relations
using actions on infinite rooted trees. Amenability,
property (tau) and L2 Betti numbers
are part of the picture, too. As a byproduct, we show that
for hyperbolic 3-manifolds, the ratio of the Heegard genus
and the rank can get arbitrarily large.
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- Tuesday November 14 at 2pm in E206
- Igor Rivin, Temple University
- Aspects of ideal polyhedra
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Abstract: We discuss various aspects of geometry,
analysis, and computation related to ideal polyhedra in
H3.
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- Tuesday November 21 at 2pm in E206
- Bruno Klinger, University of Chicago
- About the Andre-Oort Conjecture
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- Tuesday November 28 at 2pm in E206
- Matthew Bainbridge, University of Chicago
- Teichmüller curves, Hilbert modular surfaces,
and billiards in L-shaped tables with barriers.
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Abstract: A Teichmüller curve is a curve in the
moduli space of Riemann surfaces which is isometrically
immersed with respect to the Teichmüller metric.
Recently, McMullen and Calta discovered an interesting
family of Teichmüller curves in the moduli space of
genus two Riemann surfaces. I'll discuss a formula for the
Euler characteristics of these curves in terms of the
volumes of Hilbert modular surfaces, and I'll give
applications to counting closed billiards paths on L-shaped
polygons with barriers (pictured below).
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- Thursday November 30 at 2pm in E206
- Christopher Connell, Indiana University
- Smooth volume rigidity for manifolds with
negatively curved targets
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Abstract: The basic question of when a map between
manifolds is homotopic to a homeomorphism turns out to be
especially delicate in the smooth category, even in the
category of negatively curved manifolds. I will present
volume and entropy conditions for a degree one map between
any closed orientable smooth manifold of dimension
n>4 and a negatively curved manifold to be
homotopic to a diffeomorphism. I'll discuss some
applications including the distribution of critical points
of C1 maps and related finiteness
theorems.
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- Tuesday December 5 at 2pm in E206
- Barak Weiss, Ben Gurion University
- Finiteness results for Veech groups
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Abstract: Veech groups are stabilizers of flat
surfaces which arise in the study of billiards on rational
polygons and Teichmüller theory. These are discrete
subgroups of SL(2,R) which could be
lattices but might not be (they are sometimes infinitely
generated!), and are the topic of lots of recent research. I
will describe work with John Smillie in which we
characterize the Veech groups which are lattices from 3
points of view: the dynamics of the straight flow on the
flat surface (Veech dichotomy and its variants); the
dynamics of the associated flows on the moduli space of flat
surfaces; the geometry of the flat surface. We also provide
new restrictions in case the groups are not lattices.
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- Thursday December 7 at 2pm in E206
- Raul Quiroga-Barranco, CIMAT
- Actions of simple Lie groups preserving a pseudoRiemannian metric
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Abstract: Let G be a noncompact simple
Lie group acting ergodically on a manifold M and
preserving a finite volume. A fundamental family of examples
is given by actions for which M = K\H/ \Gamma,
where H is a semisimple Lie group containing
G, \Gamma is an irreducible lattice in
H and K is a compact subgroup of the
centralizer of G in H. R. Zimmer has
proposed the problem of determining to what extent an
arbitrary action as above can be obtained from these
algebraic examples. In this talk we will discuss some
results for the case when G preserves a finite
volume pseudoRiemannian metric on M. In
particular, we will discuss some instances where the
existence of an invariant pseudoRiemannian metric implies
that M is (up to a finite covering) a double
coset K\H/ \Gamma as above.
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- Thursday January 11 at 2pm in E206
- Rich Schwartz, Brown University
- Outer Billiards, the Penrose Kite, and the Neumann Problem
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Abstract: Outer billiards is a simple dynamical
system, based on an arbitrarily chosen planar convex shape.
It was introduced by B. Neumann in the 1950's. Some consider
outer billiards to serve as a toy model for celestial
mechanics. All along, one of the central questions about
outer billiards has been: Can there be an outer billiards
system with an unbounded orbit? I recently discovered that
outer billiards, defined relative to the Penrose kite, has
some unbounded orbits. The Penrose kite is the convex
quadrilateral that appears in the Penrose tiling. In my talk
I will present vivid computer evidence for the truth of this
assertion, and also sketch a proof. The proof (still in
progress) has to do with polygon exchange maps, dynamics in
a number ring, and self-similar tilings.
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- Thursday February 8 at 2pm
- Lizhen Ji, University of Michigan
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- Thursday March 15 at 2pm in E206
- Julien Paupert, Johns Hopkins University
- Discrete complex reflection groups in PU(2,1)
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Abstract: The group PU(n,1) of
holomorphic isometries of complex hyperbolic space is one of
the two occurrences (with PO(n,1))of a simple
real Lie group of rank 1 where Margulis superrigidity does
not hold. The only known examples of nonarithmetic lattices
in PU(2,1) were constructed by Mostow in the
1980's. We will recall the construction of these lattices,
which are generated by complex reflections, and we will show
how to find new examples of the same kind in a family of
configuration polygons. This is joint work with John Parker
(Durham).
For questions, contact