L3

: I will review various constructions and properties of complete constant scalar curvature metrics. I will emphasize the role played by the so called "Fowler's solutions" which give rise to metrics with cylindrical ends. I will also draw the parallel between these constructions and similar constructions which surprisingly (or not) appear in a different context : constant mean curvature surfaces and more recently the Allen-Cahn equation and some equation in the biological theory of pattern formation.

If $G$ is a group and $g$ an element of the derived subgroup $[G,G]$, the commutator length of $g$ is the least positive integer $n$ such that $g$ can be written as a product of $n$ commutators. The commutator width of $G$ is the maximum of the commutator lengths of elements of $[G,G]$. Until 1991, to my knowledge, it has not been known whether there exist simple groups of commutator width greater than $1$. The same question for finite simple groups still remains unsolved. In 1992, Jean Barge and Étienne Ghys showed that the commutator width of certain simple groups of diffeomorphisms is infinite. However, those groups are not finitely generated. Finitely generated infinite simple groups of infinite commutator width can be constructed using "small cancellations." Additionally, finitely generated infinite boundedly simple groups of arbitrarily large (but necessarily finite) commutator width can be constructed in a similar way.

The Borel conjecture asserts that aspherical manifolds are topologically rigid, i.e., every homotopy equivalence between such manifolds is homotopic to a homeomorphism. This conjecture is strongly related to the Farrell-Jones conjectures in algebraic K- and L-theory. We will give an introduction to these conjectures and discuss the proof of the Borel conjecture for high-dimensional aspherical manifolds with word-hyperbolic fundamental groups.

Taking the intersection form of a 4n-manifold defines a functor from a category of cobordisms to a symmetric monoidal category of quadratic forms. I will present the theory of the Maslov index and some higher-categorical constructions as variations on this theme.

I will describe a new family of groups exhibiting wild geometric and computational features in the context of their Conjugacy Problems. These features stem from manifestations of "Hercules versus the hydra battles."

This is joint work with Martin Bridson.

The arc complex is a combinatorial moduli space, very similar to the curve complex. Using the techniques of Masur and Minsky, as well as new ideas, I'll sketch the theorem of the title. (Joint work with Howard

Masur.) If time permits, I'll discuss an application to the cusp shapes of fibred hyperbolic three-manifolds. (Joint work with David Futer.)

We are planning to have dinner at Chiang Mai afterwards.

If anyone would like to join us, please can you let me know today, as I plan to make a booking this evening. (Chiang Mai can be very busy even on a Monday.)