Noel Brady - Snowflake subgroups of CAT(0) groups

We construct CAT(0) groups which contain snowflake subgroups. This expands the known range of isoperimetric behavior of subgroups of CAT(0) groups. This is joint with Max Forester.

Mark Hagen - Cubulating mapping tori of some free group endomorphisms

Let V be a finite connected graph and let f be a pi_1-injective map from V to V sending vertices to vertices and edges to combinatorial paths. Let X be the mapping torus of f. Under the assumption that no positive power of f sends any edge of V to itself, we describe a general procedure for constructing immersed walls W in X. Under the additional assumption that pi_1X is word-hyperbolic, we give technical conditions on X guaranteeing that pi_1X acts freely and cocompactly on a CAT(0) cube complex. We use this result to prove that pi_1X is cocompactly cubulated when f represents an atoroidal, fully irreducible automorphism of a finitely-generated free group.

Nathan Geer - The Turaev-Viro invariant and some of its relatives

Following V. Turaev and O. Viro, I will discuss a construction which leads to information about the topology of a 3-manifold from one of its triangulation. This construction is based on algebraic tools which are 6-parameter quantities called 6j-symbols. In the first part of the talk, I will recall the Turaev-Viro invariant of 3-manifolds arising from restricted quantum sl(2) at a root of unity. The underlying category of modules associated to this invariant is semi-simple and all the simple modules have non-vanishing quantum dimension. In the second part of the talk, I will explain how the Turaev-Viro invariant can be modify to ﬁt the context of non-restricted quantized sl(2) at a root of unity. Here the underlying category is not semi-simple and many of the simple modules have vanishing quantum dimensions. This modified Turaev-Viro invariant is closely related Kashaev's invariant defined in his foundational paper where he first stated the volume conjecture. This is joint work with B. Patureau and V. Turaev.

Yair Glasner - (Almost) all dense subgroups of SL_2(Q_p) are created equal, while these in SL_2(C) are not

I will survey two contrasting papers. The first one is mine: establishing the first half of the tile. The second, due to Yair Minsky, proves the second part.

Patrick Hooper - Cutting and Resewing Pillow Cases

I will discuss the dynamics of a fairly simple piecewise isometry of a square pillowcase. We cut the pillowcase along two horizontal edges we obtain a cylinder, which we can rotate and then sew back together. We can then do the same in the vertical direction. The composition of these two cutting and resewing operations yields a piecewise isometry of the pillowcase with interesting dynamics. We will describe how in some cases the collection of aperiodic points forms a fractal curve, and the dynamics on this curve is topologically conjugate to a rotation (modulo concerns related to discontinuities). Properties of this map such as the existence of this curve depend on the even continued fraction expansions of the parameters.

Kathyrn Mann - Components of representation spaces

Let G be a group of homeomorphisms of the circle, and Γ the fundamental group of a closed surface. The representation space Hom(Γ, G) is a basic example in geometry and topology: it parametrizes flat circle bundles over the surface with structure group G, or G-actions of the surface group on the circle. Goldman proved that connected components of Hom(Γ, PSL(2,R)) are completely determined by the Euler number, a classical invariant. By contrast, the space Hom(Γ, Homeo+(S^1)) is relatively unexplored -- for instance, it is an open question whether this space has finitely or infinitely many components (!)

We report on recent work and new tools to distinguish connected components of Hom(Γ, Homeo+(S^1)). In particular, we give a new lower bound on the number of components, show that there are multiple components on which the Euler number takes the same value -- in contrast to the PSL(2,R) case -- and we identify certain representations which exhibit surprising rigidity. A key technique is the study of rigidity phenomena in rotation numbers, using recent ideas of Calegari-Walker.

Ric Wade - Automorphisms of right-angled Artin groups

When looking at the groups SL_n(Z) and Out(F_n) there is a clear distinction between the case n=2, where both groups are virtually free, and their behaviour when n is greater than 2. Automorphism groups of right-angled Artin groups can behave in a similar way. We look at some examples where the outer automorphism group of a RAAG is virtually a RAAG (e.g virtually free or virtually free abelian), and give some partial results aiming to describe when this happens in general.

Alex Wright - GL(2,R) orbit closures of translation surfaces

A translation surface can be thought of as a polygon in R^2, with parallel side identifications. For example, a regular octagon with opposite sides identified gives a genus 2 translation surface, which is flat everywhere except at one "singularity". The standard linear action of GL(2,R) on the plane induces an action of GL(2,R) on the moduli space of all translation surfaces. Over the past three decades it has been discovered that understanding the closure of the orbit of a translation surface is necessary for understanding the geometry and dynamics present on the translation surface. We will survey recent progress on the problem of classifying orbit closures, as well as hopes for the future. Parts of the talk may include joint work with D. Aulicino and D.-M. Nguyen.