I will describe an extremely easy construction with formal group laws, and a

slightly more subtle argument to show that it can be done in a coordinate-free

way with formal groups. I will then describe connections with a range of other

phenomena in stable homotopy theory, although I still have many more

questions than answers about these. In particular, this should illuminate the

relationship between the Lambda algebra and the Dyer-Lashof algebra at the

prime 2, and possibly suggest better ways to think about related things at

odd primes. The Morava K-theory of symmetric groups is well-understood

if we quotient out by transfers, but somewhat mysterious if we do not pass

to that quotient; there are some suggestions that dilation will again be a key

ingredient in resolving this. The ring $MU_*(\Omega^2S^3)$ is another

object for which we have quite a lot of information but it seems likely that

important ideas are missing; dilation may also be relevant here.

# Topology Seminar

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: Out(F_n), n>1, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least 2, etc. Roughly speaking, a group G is acylindrically hyperbolic if there is a (possibly infinite) generating set X of G such that the Cayley graph \Gamma(G,X) is hyperbolic and the action of G on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups.

In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties.(The talk will be based on joint work with Denis Osin.)