Back to WTC Summer 2007
Matt Bainbridge
Billiards in
L-shaped tables with barriers
I'll discuss the computation
of the volumes of some moduli spaces of holomorphic 1-forms on Riemann
surfaces, and I'll give applications to counting closed billiards
trajectories in certain L-shaped polygons with barriers.
Nathan Broaddus
Irreducible
Sp-representations and distortion in the mapping class group
In this report on joint work
with B. Farb and A. Putman I will show that the Torelli subgroup of the
mapping class group is at least
exponentially distorted, answering a question of Hamenst\"adt. I
will then outline a
generalization (for the surface with one boundary component) that shows that nontrivial finitely generated
normal subgroups of the mapping
class group contained in the Torelli group are at least exponentially distorted. To prove this kind of
general result on normal subgroups, without actually knowing what these
groups might be, we must rely on constraints from symplectic
representation theory. A preprint of these and other distortion results is available at
arXiv:0707.2262.
Kai-Uwe Bux
Finitenes
properties of arithmetic groups and some relatives (with Amir Mohammadi
and Kevin Wortman)
We showed that the group
SL_n( Z[t] ) is not of type FP_{n-1}. The proof uses the action on an
associated Euclidean building. Although SL_n( Z[t] ) is not an
arithmetic group, the method of proof draws from methods used to study
SL_n( F_q[t] ). I would like to give an overview of the
geometric ideas used to study finiteness properties of artihmetic
groups and variations like SL_n( Z[t] ).
Jim Conant
The topology of
sets of Boolean formulae
The k-SAT problem in computer
science asks whether a given formula in a specified canonical form (a
conjunction of disjunctions of k literals) can be satisfied by some
assignment of truth values to the variables. It is well known that the
2-SAT problem can be solved in polynomial time, whereas the 3-SAT
problem is NP-complete. The set of satisfiable formulae in k-SAT is a
poset under logical implication, and in this talk we will study the
topology of this poset. The goal of the project is to find topological
differences between 2-SAT and 3-SAT with an eye toward separating the
classes of P and NP problems. At this stage, the general topological
behavior of these posets remains mysterious.
W. Patrick Hooper
An infinite
surface with the lattice property and applications
We will describe a
construction which yields a translation surface, infinite in both genus
and area, which has the lattice property. The surface arises as a limit
of some of Veech's original lattice surfaces, the regular 2n-gons with
opposite sides identified.
A similar construction yields this infinite surface as a limit of
translation surfaces corresponding to billiards in some irrational
polygons. We will utilize this surface to study periodic billiard paths
in certain irrational polygons. For instance, we provide lower bounds
for the growth rate of periodic billiard trajectories in some
irrational polygons.
Dan Margalit
Cohomological
dimension of the Torelli group
We prove that the
cohomological
dimension of the Torelli group for a surface of genus g is 3g-5, and
the cohomological dimension of the Johnson kernel is 2g-3. We
also
show that the (3g-5)-th integral homology of the Torelli group is
infinitely generated. Finally, we give a topological proof of the
theorem of Mess which gives a precise description of the Torelli group
in genus 2. The main tool is a new contractible complex on which
the
Torelli group acts.
Jessica Purcell
Volume bounds
from knot and link diagrams
Given the diagram of a
hyperbolic knot or link, one ought to be able to determine geometric
information on the link complement, such as its volume. However,
it seems to be difficult to read geometry off a diagram. We recently
proved a theorem giving bounds on the change of volume of a hyperbolic
3-manifold under Dehn filling. We will discuss how this result applies
to give volume bounds on large classes knots and links. This work is
joint with David Futer and Effie Kalfagianni.
Anne Thomas
Lattices for
Fuchsian buildings and Platonic complexes
Let L be a graph. A (k,L)-complex
is a polygonal complex X such that each 2-cell is a regular k-gon, and
the link of each vertex is the graph L. Examples include Euclidean
buildings, products of trees and Fuchsian (hyperbolic) buildings.
A (k,L)-complex X is Platonic if Aut(X) acts transitively on the set of
adjacent triples (vertex, edge, face) in X. An example is when L is the
Petersen graph. We study lattices in Aut(X), for X a Fuchsian
building or a Platonic complex. We give new constructions of
uniform and nonuniform lattices, characterize the set of covolumes of
lattices, and consider towers (infinite ascending sequences) of lattices.