Past Topology Seminar

10 October 2016
Chris Douglas

The dimension of a finite-dimensional vector space V can be computed as the trace of the identity endomorphism id_V.  This dimension is also the value F_V(S^1) of the circle in the 1-dimensional field theory F_V associated to the vector space.  The trace of any endomorphism f:V-->V can be interpreted as the value of that field theory on a circle with a defect point labeled by the endomorphism f.  This last invariant makes sense even when the vector space is infinite-dimensional, and gives the trace of a trace-class operator on Hilbert space.  We introduce a 2-dimensional analog of this invariant, the `2-trace'.  The 2-dimension of a finite-dimensional separable k-algebra A is the dimension of the center of the algebra.  This 2-dimension is also the value F_A(S^1 x S^1) of the torus in the 2-dimensional field theory F_A associated to the algebra. Given a 2-endomorphism p of the algebra (that is an element of the center), the 2-trace of p is the value of the field theory on a torus with a defect point labeled by p.  Generalizations of this invariant to other defect configurations make sense even when the algebra is not finite-dimensional or separable, and this leads to a general notion of 2-trace class and 2-trace in any 2-category.  This is joint work with Andre Henriques.

20 June 2016
Jacob Lurie

Let X be a complex algebraic variety containing a point x. One of the central ideas of deformation theory is that the local structure of X near the point x can be encoded by a differential graded Lie algebra. In this talk, Jacob Lurie will explain this idea and discuss some generalizations to more exotic contexts.

6 June 2016
​​​Yong Hou

In this talk I'll give a general presentation of my recent work that a purely loxodromic Kleinian group of Hausdorff dimension<1 is a classical Schottky group. This gives a complete classification of all Kleinian groups of dimension<1. The proof uses my earlier result on the classification of Kleinian groups of sufficiently small Hausdorff dimension. This result in conjunction to another work (joint with Anderson) provides a resolution to Bers uniformization conjecture. No prior knowledge on the subject is assumed.

6 June 2016
Thomas Schick

We want to discuss a collection of results around the following Question: Given a smooth compact manifold $M$ without boundary, does $M$ admit a Riemannian metric of positive scalar curvature?

We focus on the case of spin manifolds. The spin structure, together with a chosen Riemannian metric, allows to construct a specific geometric differential operator, called Dirac operator. If the metric has positive scalar curvature, then 0 is not in the spectrum of this operator; this in turn implies that a topological invariant, the index, vanishes.

We use a refined version, acting on sections of a bundle of modules over a $C^*$-algebra; and then the index takes values in the K-theory of this algebra. This index is the image under the Baum-Connes assembly map of a topological object, the K-theoretic fundamental class.

The talk will present results of the following type:
If $M$ has a submanifold $N$ of codimension $k$ whose Dirac operator has non-trivial index, what conditions imply that $M$ does not admit a metric of positive scalar curvature? How is this related to the Baum-Connes assembly map? 

We will present previous results of Zeidler ($k=1$), Hanke-Pape-S. ($k=2$), Engel and new generalizations. Moreover, we will show how these results fit in the context of the Baum-Connes assembly maps for the manifold and the submanifold. 

23 May 2016
Panos Papazoglou

It is known that if the boundary of a 1-ended
hyperbolic group G has a local cut point then G splits over a 2-ended group. We prove a similar theorem for CAT(0)
groups, namely that if a finite set of points separates the boundary of a 1-ended CAT(0) group G
then G splits over a 2-ended group. Along the way we prove two results of independent interest: we show that continua separated
by finite sets of points admit a tree-like decomposition and we show a splitting theorem for nesting actions on R-trees.
This is joint work with Eric Swenson.

16 May 2016
Yevgeni Liokumovich

We will prove an upper bound for the volume of a minimal
hypersurface in a closed Riemannian manifold conformally equivalent to
a manifold with $Ric > -(n-1)$.  In the second part of the talk we will
construct a sweepout of a closed 3-manifold with positive Ricci
curvature by 1-cycles of controlled length and prove an upper bound
for the length of a stationary geodesic net. These are joint works
with Parker Glynn-Adey (Toronto) and Xin Zhou (MIT).

9 May 2016
Karen Vogtmann

Motivated by work of Borel and Serre on arithmetic groups, Bestvina and Feighn defined a bordification of Outer space; this is an enlargement of Outer space which is highly-connected at infinity and on which the action of $Out(F_n)$ extends, with compact quotient. They conclude that $Out(F_n)$ satisfies a type of duality between homology and cohomology.  We show that Bestvina and Feighn’s  bordification can be realized as a deformation retract of Outer space instead of an extension, answering some questions left open by Bestvina and Feighn and considerably simplifying their proof that the bordification is highly connected at infinity.