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).

# Past Topology Seminar

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.

Finiteness properties of groups come in many flavours, I will discuss topological finiteness properties. These relate to the finiteness of skelata in a classifying space. Groups with interesting finiteness properties have been found in many ways, however all such examples contains free abelian subgroups of high rank. I will discuss some constructions of groups discussing the various ways we can reduce the rank of a free abelian subgroup.

Bordered Floer homology is a variant of Heegaard Floer homology adapted to manifolds with boundary. I will describe a class of three-manifolds with torus boundary for which these invariants may be recast in terms of immersed curves in a punctured torus. This makes it possible to recast the paring theorem in bordered Floer homology in terms of intersection between curves leading, in turn, to some new observations about Heegaard Floer homology. This is joint work with Jonathan Hanselman and Jake Rasmussen.

For the programme see

The Cartan model computes the equivariant cohomology of a smooth manifold X with

differentiable action of a compact Lie group G, from the invariant functions on

the Lie algebra with values in differential forms and a deformation of the de Rham

differential. Before extracting invariants, the Cartan differential does not square

to zero. Unrecognised was the fact that the full complex is a curved algebra,

computing the quotient by G of the algebra of differential forms on X. This

generates, for example, a gauged version of string topology. Another instance of

the construction, applied to deformation quantisation of symplectic manifolds,

gives the BRST construction of the symplectic quotient. Finally, the theory for a

X point with an additional quadratic curving computes the representation category

of the compact group G.

I will explain some recent work using minimal surfaces to address problems in 3-manifold topology. Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface. If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting. I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.