Abstract:  This talk is about the Thompson group F and T; T is the group 
of all PL dyadic homeomorphisms of the circle, and F is the subgroup which 
fixes some base point.  Both are finitely presented groups of type FP_\infty.  
I will discuss two new results.
 
The first result (joint work with Bieri and Kochloukova) is a complete
description of the Bieri-Neumann-Strebel-Renz invariants for F.

The second result (joint work with Marco Varisco) is our proof that the
Whitehead group of T is non-trivial.  As far as I know, this is the first
known non-trivial K-theory of a Thompson group.  It is an application of
general work of Lueck-Reich-Rognes-Varisco. We also get other K- theoretic
information about T.

Abstract: Through highly non-constructive methods, works by Bestvina, 
Culler, Feighn, Morgan, Paulin, Rips, Shalen, and Thurston show that if a 
finitely presented group does not split over a small subgroup, then the space 
of its discrete and faithful actions on H^n, modulo conjugation, is compact
for all dimensions.  We make this result effective for Coxeter groups.
We find that either the group splits over a small subgroup or there is a 
constant C and a point in H^n that is moved no more than C by any generator.

Abstract: Recently, Gromov defined an $m$-point curvature
condition.  A curvature condition is a way of specifying constraints
on distances between $m$-point subsets of a given metric space.
Particular instances of these curvature conditions, including Gromov's
$\delta$-hyperbolicity and the $CAT(\kappa)$ conditions, have had a
profound impact on metric geometry, topology, and geometric group
theory, among other areas.  In this talk I plan to discuss some work
in progress on the relationships between various curvature conditions,
and in particular their implications for bordifications of spaces.

Abstract: TBA 

 
The usual Topology Seminar time for this semester is (tentatively) Thursday at 11:00 am (Rm 118 BAC). 
Changes in time or day will be noted in the schedule below. 

Abstract:  By considering the set of surfaces that a knot bounds in the
four-dimensional ball, we can define an equivalence relation on knots called 
concordance.  The set of equivalence classes of concordant knots forms a 
group -- the concordance group -- which has been studied by topologists for 
over forty years.  Another equivalence relation on knots which gives a second 
group structure to the set of knots is double null concordance.    I will 
discuss the relationship between these two groups and the recent progress that
has been made toward understanding their structure.

Abstract: In this talk I will report on joint work with Wolfgang L\"uck, 
Holger Reich, and John Rognes. 
We use topological cyclic homology and the cyclotomic trace to detect elements 
in \mbox{$K_n({\Bbb Z}G)\otimes_{{\Bbb Z}}{\Bbb Q}$}, the rationalized higher 
algebraic $K$-theory groups of an integral group ring.  Modulo a deep conjecture 
in number theory, the so-called Schneider conjecture, and under mild homological 
finiteness conditions on the group~$G$, we prove that the Farrell-Jones assembly 
map in connective algebraic $K$-theory with respect to the family of virtually cyclic
subgroups is rationally injective.  This generalizes a result of B\"okstedt, Hsiang, 
and Madsen---and leads to a concrete description of a large direct summand inside 
\mbox{$K_n({\Bbb Z}G)\otimes_{{\Bbb Z}}{\Bbb Q}$}. Along the way we also prove 
integral splitting and isomorphism results for assembly maps in topological 
Hochschild homology and topological cyclic homology with arbitrary coefficients.

Abstract: Two natural and well studied properties of a topological space are 
the hereditary separability and the hereditary Lindelof property.  These properties 
are equivalent in the class of metric spaces and seem very closely related in general.  
While these properties are topological, the study of their relationship is essentially 
Ramsey theoretic in nature.  The S and L space problems --- which concern whether one 
property implies the other --- were focal problems in set theory in the 1970s and 1980s.  
In this talk, I will give an exposition and set theoretic analysis of these problems and 
their solutions.  The talk will finish with my recent construction of an L space --- a
non-separable, hereditarily Lindelof space.

 
The usual Topology Seminar time for this semester is Wednesday at 1:00 pm (Rm 112 BAC). Changes
in time or day will be noted in the schedule below. 

Abstract: We introduce the new notion of a regressive Kurepa-tree and 
show that it has various applications in set theory. One example is the 
construction of a Boolean algebra with an omega_2-closed dense subset but 
no omega_2-directed-closed dense subset. On the other hand, regressive 
Kurepa-trees can witness the failure of certain partition relations and 
we discuss results related to that.

Abstract: This talk will discuss three families of groups, $ZW_n$,
$PL(I^n)$, and $PL(S^n)$ (these last two families consist of groups of
homeomorphisms of $n$-cubes and $n$-spheres).  We will see that for every
positive index $n$, $ZW_n$ embeds in $PL(I^n)$ and $PL(S^n)$.  This easy
result is in counterpoint to the harder result that $ZW_2$ does not embed in
$PL(I^1)$ or in $PL(S^1)$.

Next, we will discuss the proof of the non-embedding results.  I hope the
discussion will indicate why corresponding non-embedding results may hold
true in higher dimensions; underscoring a potential relationship between the
structure of the groups $ZW_n$, and the dimension of the spaces upon which
they can act in a piecewise-linear fashion.  

Abstract:  Sequences which guess club sets of ordinals have been well studied over
the past thirty years. The club filter on ordinals is generalized to sets of ordinals 
by considering sets whose members are those sets closed under a given finitary 
function. It turns out that the generalized filter admits club-guessing principles 
much stronger than those possible for sets of ordinals. Time permitting, we will present 
an application of this fact to a well known question on stationary reflection. This is 
joint work with Bernhard Koenig and Yasuo Yoshinobu.

Abstract: In the 1960's, Richard Thompson invented a triple of groups 
$F \subset T \subset V$, which have since appeared throughout many different
branches of mathematics. For example, they have provided a technique for
constructing an elementary example of a finitely presented group which has
unsolvable word problem, the universal obstruction to a problem in homotopy
theory, the structure group for the associative law, and the first example
of a group which is torsion-free, infinite dimensional, and of type
$\mathcal{F_{\infty}}$.
    The group $F$ is conjectured to be an example of a finitely presented,
nonamenable group which has no free subgroup on two generators. The speaker
gives a criterion which states that $F$ is nonamenable if and only if there
exist finite subsets in $F$ which satisfy certain properties.

Abstract: What influence does the collision of light in a compact Riemannian manifold 
have on geometric features of the manifold?  I'll discuss two conjectures and supporting results 
motivated by this question.  This is based on joint work with Jean Lafont and independent
joint work by Keith Burns and Eugene Gutkin.

Abstract: Thompson's group F, named for Richard Thompson, who
first discovered it in the 1960s, is a certain group of piecewise
linear homeomorphisms of the real line.  

This talk will take up the argument I was making in the seminar last
week.  I will introduce the idea of the link of a vertex in a cubical
complex and cover two applications.  I will show:

i)  that Thompson's group F admits a classifying complex with finitely
many cells in each dimension, and

ii) that the classifying complex for F (introduced last time) admits
a metric of non-positive curvature.

Abstract: Thompson's group F, named for Richard Thompson, who
first discovered it in the 1960s, is a certain group of piecewise
linear homeomorphisms of the real line.  

This talk will take up the argument I was making in the seminar three
weeks ago.  I will introduce the idea of the link of a vertex in a cubical
complex and cover two applications.  I will show:

i)  that Thompson's group F admits a classifying complex with finitely
many cells in each dimension, and

ii) that the classifying complex for F (introduced last time) admits
a metric of non-positive curvature.

Abstract: Consider the following three topological statements:
NS: Every normal first-countable space is collectionwise Hausdorff 
   (i.e., every closed discrete subspace can be separated).
CPS: Every countably paracompact first-countable space is collectionwise Hausdorff.
CMGD: In every countably metacompact first-countable space all closed discrete 
   subspaces are $G_\delta$-sets.
   These statements are individually known to be true in some models of ZFC and not true in 
other models.  We discuss some of the history and reasons for considering these statements. 
In order to better understand the relationships we give purely combinatorial equivalences for 
each of them.

Abstract: A $D$-space was introduced by van Douwen in 70's. For example, every 
compact space is a $D$-space. He asked if every Lindel{\"o}f space is a $D$-space, and 
it is still open. I will present a related result and a space suggested by Gruenhage as a 
potential counterexample.

Abstract: Thompson's group $F$, named for Richard Thompson, who first discovered it 
in the 1960s, is a certain group of piecewise linear homeomorphisms of the real line.  
I will discuss some of the basic properties of $F$, including presentations for the group and
the method of representing elements of $F$ by tree pairs.
   The main goal of the talk will be to show that $F$ is of type $F$-infinity, 
that is, that $F$ admits an aspherical classifying complex with finitely many cells in each 
dimension.  In the process, I will introduce a cubical complex $X$ on which $F$ acts cellularly 
and freely.