The symmetric groups, the general linear groups, and the automorphism groups of free groups are examples of families of groups that arise as symmetry groups of algebraic structures but that are also dear to topologists. There are many other less obvious examples of interest. For instance, in joint work with Nathalie Wahl, this point of view has led to a computation of the homology of the Higman-Thompson groups. It is also key to progress on groups related to mapping class groups and braid groups that I would like to present in this talk.

I will begin by reviewing the foundations of equivariant and non-equivariant stable motivic homotopy theory.

In the non-equivariant stable motivic homotopy category, I have constructed a motivic spectrum $MGLO$ whose topological realization over $\mathbb C$ is unoriented cobordism $MO$. Similar to the story of $MO$, the spectrum $MGLO$ turns out to be a wedge sum of suspensions of motivic $H\mathbb Z/2$. This answers a question of Jack Morava.

In 2011, Hu, Kriz, and Ormsby constructed algebraic Hermitian cobordism $MGL\mathbb R$. The topological realization of $MGL\mathbb R$ over $\mathbb C$ is $MU_\mathbb R$. In view of the classical Landweber-Araki analysis of $MU_\mathbb R$, $MGLO$ should play an important role in computing the coefficients of $MGL\mathbb R$. I have made progress in this direction, and in this talk, I will give a full description of the coefficients of $MGL\mathbb R$ over $\mathbb C$ completed at the prime 2 after inverting two ``twists.??

We will show that, for "almost" all arithmetic hyperbolic manifolds with dimension >3, their fundamental groups are not LERF. The main ingredient in the proof is a study of certain graph of groups with hyperbolic 3-manifold groups being the vertex groups. We will also show that a compact irreducible 3-manifold with empty or tori boundary does not support a geometric structure if and only if its fundamental group is not LERF.

Given a Lie group G acting on a symplectic manifold preserving a pair of Lagrangians setwise, I will describe a construction of G-equivariant Lagrangian Floer homology. This does not require G-equivariant transversality, which allows the construction to be flexible. Time permitting, I will talk about applying this for the O(2)-action on Seidel-Smith's symplectic Khovanov homology. This is joint with Kristen Hendricks and Robert Lipshitz.

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)

I will discuss joint work with Ben Elias in which we introduce the notion of a diagonalizable functor and give a categorical analogue of the usual minimal polynomial condition for diagonalizability. As our main application we prove that the Rouquier complex associated to the full-twist braid acts diagonalizably on the category of Soergel bimodues. This has important consequences for the triply graded Khovanov-Rozansky link homology, which I will explain. I will conclude by discussing connections with some recent, very exciting work of Gorsky-Negut-Rasmussen, which suggests that categorical diagonalization is the key to understanding a deep (conjectural) connection between Khovanov-Rozansky homology and Hilbert schemes.