For more information, contact Ken Bromberg at kbromber@math.lsa.umich.edu or Bryan Mosher at bmosher@math.lsa.umich.edu.

Schedule of Talks

Date	Speaker	Title (click on title to view abstract)
September 19	Max Karoubi, Paris	The role of noncommutative differential forms in algebraic topology
October 2	Peter Scott, UM	Regular Neighbourhoods in Group Theory
October 9	Gadde Swarup, Melbourne	Canonical decompositions for groups
October 16	Ken Bromberg, UM	Projective structures with degenerate holonomy
October 23	Michael Sullivan, UM	Legendrian submanifolds in $R^{2n+1}$
October 30	Heiner Dovermann, Michigan State	Algebraic models of smooth manifolds and vector bundles
December 4	Jeff Brock, Chicago	Rank and curvature of moduli space
February 5	Karsten Grove, Maryland	Exotic actions
February 19	Rita Gitik	Algorithmic problems in negatively curved groups
March 12	Chris Hruska, Cornell	CAT(0) groups with well-defined boundaries
April 16	Tibor Beke, UM

Abstracts

The relation between algebraic topology and usual differential forms on a manifold X is well known ; for instance, a closed form of degree n on X is exact iff its suitably defined class in H^n(X) is trivial. In this talk, we reverse the procedure and use differential forms (in a noncommutative setting) to build the algebraic topology of any space X (not necessarily a manifold). With this purpose in mind, we have to use a new calculus (more precisely "quantum calculus") in order to get a better understanding of the topological complexity of the space X.

October 2
Peter Scott, UM
Regular Neighbourhoods in Group Theory

This talk describes joint work with Gadde Swarup of the University of Melbourne.

If one considers two simple closed curves on a surface, there is a standard topological idea of a regular neighbourhood of their union. The algebraic analogue of a simple closed curve on a surface is a splitting of a group over a subgroup. Surprisingly, given two splittings of a group there is a natural algebraic idea of a regular neigbourhood of these two splittings which corresponds to the topological idea. This idea is due to Fujiwara and Papasoglu in some special cases. The talk will describe the ideas needed to make this work in general. These ideas generalise further to an analogue of a regular neighbourhood of non-simple closed curves on a surface, and I will give some indication of what is involved if time permits.

Swarup and I have used this work on regular neighbourhoods to re-cast the theory of JSJ decompositions of finitely presented groups, and obtain new results which parallel the results from 3-manifold theory far more closely than previous results.

October 9
Gadde Swarup, Melbourne
Canonical decompositions for groups

(This is joint work with Peter Scott)

This talk is about obtaining decompositions for finitely presented groups which specialize to JSJ-decompositions when restricted to fundamental groups of Haken 3-manifolds. There have been various such studies but so far none of them gives the usual JSJ when one specializes to 3-manifold groups. The new ingredient is the development of the analogue of regular neighbourhoods for families of codimension one immersions of groups. The analogue of a codimension one immersion is called an almost invariant set and these have been studied by group theorists since Stallings' work on groups with infinitely many ends. Recently Peter Scott and I showed (Geometry and Topology, Vol. 4 (2000), pp 179-218) that many properties of immersions in low dimensional topology carry over to almost invariant sets. Concepts and results from this paper will be used.

October 16
Ken Bromberg, UM
Projective structures with degenerate holonomy

By a theorem of Gallo, Kapovich and Marden any PSL2C representation of the fundamental group of a surface is the holonomy of a (possibly branched) projective structure. In earlier work, Goldman showed that all projective structures with quasifuchsian holonomy are obtained by grafting a quasifuchsian projective structure. In this talk we will discuss how Goldman's work can be extended to discrete and faithful degenerate representations.

October 23
Michael Sullivan, UM
Legendrian submanifolds in $R^{2n+1}$

This talk is on joint work in progress with Tobias Ekholm and John Etnyre. I first review the basic definitions of contact geometry, the odd-dimensional analogy of symplectic geometry. Then I present some ideas on how to classify, up to Legendrian isotopy, Legendrian submanifolds in the contact manifold $(R^{2n+1}, dz-ydx)$. The main concept is examining pseudo-holomorphic curves in the Lagrangian projection, $R^{2n}$. This theory is connected to contact homology, a special case of the symplectic field theory currently being developed by Eliashberg, Givental and Hofer.

October 30
Heiner Dovermann, Michigan State
Algebraic models of smooth manifolds and vector bundles

Let G be a compact Lie group and M a closed smooth G manifold. We like to impose an equivariant real algebraic structure on M. We will discuss assumptions under which this is possible, how it is done, and how many inequivalent structures there are.

March 12
Chris Hruska, Cornell
CAT(0) groups with well-defined boundaries

The visual boundary of a CAT(0) space $X$ is the space of all geodesic rays in $X$ emanating from a single basepoint. Suppose a group $G$ acts properly and cocompactly on two different CAT(0) spaces $X$ and $Y$. If $G$ is word hyperbolic, then $X$ and $Y$ have homeomorphic boundaries. Thus, in that case, the boundary is really a topological invariant of the group. However, in the broader CAT(0) setting, Croke and Kleiner have constructed examples of groups possessing multiple boundaries. I will describe my recent discovery of a large class of CAT(0) groups with well-defined boundaries. The groups in question are those which act properly and cocompactly on CAT(0) 2-complexes whose flat planes are isolated from each other. In particular, it follows that the fundamental group of the figure eight knot complement has a well-defined boundary.