It is hard to detect the exotic nature of an exotic n-sphere M 
in homotopical features of the diffeomorphism group Diff(M). The well 
known reason is that Diff(M) contains a big topological subgroup H which 
is identified with the group of diffeomorphisms rel boundary of the 
n-disk, with a small coset space Diff(M)/H which is invariably homotopy 
equivalent to O(n+1). Therefore it seems that our only chance to detect 
the exotic nature of M in homotopical features of Diff(M) is to see 
something in this extension.  (To make sense of "homotopical features of 
Diff(M)" one should think of Diff(M) as a space with a multiplication 
acting on an n-sphere.) I am planning to report on PhD work of O Sommer 
and calculations due to myself and Sommer which, if all goes well, would 
show that Diff(M) has some exotic homotopical properties in the case 
where M is the 7-dimensional exotic sphere of Kervaire-Milnor fame which 
bounds a compact smooth framed 8-manifold of signature 8. The 
theoretical work is based on classical smoothing theory and the 
calculations would be based on ever-ongoing (>30 years) joint work 
Weiss-Williams, and might give me and Williams another valuable 
incentive to finish it.

We construct the diffeomorphism-equivariant “scanning map” associated to the configuration spaces of manifolds with twisted summable labels. The scanning map is also functorial with respect to embeddings of manifolds. To adapt P. Salvatore's idea of non-commutative summation into twisted setting, we define a bundle of Fulton-MacPherson operads over a manifold M whose fibres are built within tangent spaces of M.

In 1956 Milnor published a paper proving that there are manifolds homeomorphic to the 7-sphere but not diffeomorphic to it. Seeking to generalise this example, he was led in around 1960 to introduce a construction for  killing homotopy groups of manifolds. When this was generalised to killing relative homotopy groups it became a general and powerful method of construction. An obstruction arises to killing the last group, and the analysis of this obstruction in general leads to a new theory.

For mapping class groups of surfaces it is well-understood that their homology stability is closely related to the fact that they give rise to an infinite loop space. Indeed, they define an operad whose algebras group complete to infinite loop spaces.

In recent work with Basterra, Bobkova, Ponto and Yaekel we define operads with homology stability (OHS) more generally and prove that they are infinite loop space operads in the above sense. The strong homology stability results of Galatius and Randal-Williams for moduli spaces of manifolds can be used to construct examples of OHSs. As a consequence the map to K-theory defined by the action of the diffeomorphisms on the middle dimensional homology can be shown to be a map of infinite loop spaces.

Localization and holography are powerful approaches to the computation of supersymmetric observables. The computations may, however, include divergences. Therefore, one needs renormalization schemes preserving supersymmetry. I will consider minimal gauged supergravity in five dimensions to demonstrate that the standard holographic renormalization scheme breaks supersymmetry, and propose a set of non-standard boundary counterterms that restore supersymmetry. I will then show that for a certain class of solutions the improved on-shell action correctly reproduces an intrinsic observable of four-dimensional SCFTs, the supersymmetric Casimir energy.

We study D3 brane theories that are described as deformations of N=2 SCFTs. They arise at the self-intersection of a 7-brane in F-Theory. As we shall explain, the associated string junctions and their monodromies can be studied via torus knots or links. The monodromy reduces (potentially different) flavor algebras of dual deformations of N=2 theories and projects out charged states, leading to N=1 SCFTs. We propose an explanation for these effects in terms of an electron-monopole-dyon condensate.

 Anyone who has worked in $\beta $N will not be surprised to learn that some of the algebraically defined subsets of $\beta N$ are not topologically simple, even though their algebraic definition may be very simple.  I shall show that the following subsets of $\beta N$ are not Borel: $N^*+N^*$; the smallest ideal of $\beta N$; the set of idempotents in $\beta N$; any semiprincipal right ideal in $\beta N$; the set of idempotents in any left ideal in $\beta N$.

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.