The talk will explain what an operad is, give some examples, and illustrate how operads come up in various mathematical settings.

The calculus of homotopy functors provides a systematic way to approximate a given functor (say from based spaces to spectra) by so-called `polynomial' functors. Each functor F that preserves weak equivalences has a `Taylor tower' (analogous to the Taylor series of ordinary calculus) which in turn is built from homogeneous pieces that are classified by certain `derivatives' for F. I will review this material and consider the problem of how the Taylor tower of F can be reconstructed from its derivatives. We will discuss some important examples built from mapping spaces. Then. if time permits, I will us this approach to give a classification of analytic functors from based spaces to spectra and try to describe some connections to the Goodwillie-Weiss manifold calculus

: I will give an introduction to Morita theory in stable homotopy theory. This will include the context of differential graded algebra and of spectral algebra. I will also survey some recent related results.

Topological cyclic homology is a topological refinement of Connes' cyclic homology. It was introduced twenty-five years ago by Bökstedt-Hsiang-Madsen who used it to prove the K-theoretic Novikov conjecture for discrete groups all of whose integral homology groups are finitely generated. In this talk, I will give an introduction to topological cyclic homology and explain how results obtained in the intervening years lead to a short proof of this result in which the necessity of the finite generation hypothesis becomes transparent. In the end I will explain how one may hope to remove this restriction and discuss number theoretic consequences that would ensue.

I will give an overview of the rich structure in the stable homotopy groups of spheres, both giving a summary of what we do know, and what we don't know.

We will begin with an introduction to stable motivic homotopy theory. Then we will use the motivic version of the Adams spectral sequence to compute stable motivic homotopy groups. We will discuss a number of open questions concerning these computations. Along the way, we will encounter some new results about classical stable homotopy groups and equivariant stable homotopy groups.

This will be a survey talk on subjects related to the J-homomorphism. In addition to the original geometric construction which assigns to a vector bundle its associated spherical fibration, we will touch on more algebraic constructions derived form it, in particular via Picard and unit spectra. We will examine the map from a computational point of view, studying its p-localisation and interpretation in terms of K(1)-local homotopy theory. We will almost certainly fail to do justice to any of these ideas in the time available. At the end, we will briefly extend some of these results to a higher chromatic setting.

I will give an introduction to the theory of representation stability, through the lens of its applications in homological stability. I'll focus on three applications: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; and stability for twisted homology such as H_i( GL_n(R); R^n ), where the coefficients change along with the groups.

I will first discuss some fairly classical homological stability phenomena: spaces of 0-manifolds (e.g. configuration spaces), and spaces of 2-manifolds (e.g. Riemann's moduli space). I will explain the general method, introduced by Quillen, for proving these stability theorems, and then explain some recent work with Søren Galatius which proves such a stability theorem for moduli spaces of 2n-dimensional manifolds (for n > 2).

I will discuss recent joint work with Oscar Randal-Williams, aimed at calculating the cohomology of BDiff(W) and related spaces, where W is a smooth 2n-dimensional manifold, Diff(W) is the topological group of diffeomorphisms of W, and BDiff(W) is its classifying space. Surprisingly, the cohomology ring turns out to be partially independent of W through a range of degrees (homological stability). In this talk, I will discuss how infinite loop spaces can be used to describe the cohomology in this stable range.

In this survey talk, I will shall the role that loop groups have played in the construction of topological gauge theories in dimensions 2 and possibly 3.