One of the main themes in geometric group theory is Gromov's program to classify finitely generated groups up to quasi-isometry. We show that under certain situations, a quasi-isometry preserves commensurator subgroups. We will focus on the case where a finitely generated group G contains a coarse PD_n subgroup H such that G=Comm(H). Such groups can be thought of as coarse fibrations whose fibres are cosets of H; quasi-isometries of G coarsely preserve these fibres. This  generalises work of Whyte and Mosher--Sageev--Whyte.

Over 50 years ago, Bondi, Sachs, Newman, Penrose and others laid down foundations for the theory of gravitational waves in full non-linear general relativity. In particular, numerical simulations of binary mergers used in the recent discovery of gravitational waves are based on this theory. However, over the last 2-3 decades, observations have also revealed that the universe is accelerating in a manner consistent with the presence of a positive cosmological constant $\Lambda$. Surprisingly, it turns out that even the basic notions of the prevailing theory of gravitational waves --the Bondi news, the radiation field, the Bondi-Sachs 4-momentum-- do not easily generalize to this context, {\it no matter how small $\Lambda$ is.} Even in the weak field limit, it took a hundred years to find an appropriate generalization of Einstein's celebrated quadrupole formula to accommodate a positive cosmological constant. I will summarize the main issues and then sketch the current state of the art.
 

In the late 1880's Goursat investigated what we now call rigid local systems, classically described as linear differential equations without accessory parameters. In this talk I will discuss some arithmetic and geometric aspects of certain particular cases of Goursat's in rank four. For example, I will discuss what are likely to be all cases where the monodromy group is finite. This is joint work with Danylo Radchenko.

3d N=2 Chern-Simons-matter theories have a large variety of boundary conditions that preserve 2d N=(0,2) supersymmetry, and support chiral algebras. I'll discuss some examples of how the chiral algebras transform across dualities. I'll then explain how to construct duality interfaces in 3d N=2 theories, and relate dualities *of* duality interfaces to "Pachner moves" in triangulations of 4-manifolds. Based on recent and upcoming work with K. Costello, D. Gaiotto, and N. Paquette.

In this talk, we shall provide a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space.
This variational principle has many applications in partial differential equations while unifies and generalizes several results in nonlinear Analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality.