L3

It is known for long that the set of possible compactifications of a topological space (up to homeomorphism) is in order-preserving bijection to "strong inclusion" relations on the lattice of open sets. Since these relations do not refer to points explicitly, this bijection has been generalised to point-free topology (a.k.a. locales). The strong inclusion relations involved are typically "witnessed" relations. For example, the Stone-Cech compactification has a strong inclusion witnessed by real-valued functions. This makes it natural to think of compactification in terms of d-frames, a category invented by Jung and Moshier for bitopological Stone duality. Here, a witnessed strong inclusion is inherent to every object and plays a central role.

We present natural analogues of the topological concepts regularity, normality, complete regularity and compactness in d-frames. Compactification is then a coreflection into the category of d-frames dually equivalent to compact Hausdorff spaces. The category of d-frames has a few surprising features. Among them are:

• The real line with the bitopology of upper and lower semicontinuity admits precisely one compactification, the extended reals.
• Unlike in the category of topological spaces (or locales), there is a coreflection into the subcategory of normal d-frames, and every compactification can be factored as "normalisation" followed by Stone-Cech compactification.

In field theory simple forms of certain scattering amplitudes in four dimensional theories with massless particles are known. This has been shown to be closely related to underlying (super)symmetries and has been a source of inspiration for much development in the last years. Away from four dimensions much less is known with some concrete development only in six dimensions. I will show how to construct promising on-shell superspaces in eight and ten dimensions which permit suggestively simple forms of supersymmetric four point scattering amplitudes with massless particles. Supersymmetric on-shell recursion relations which allow one to compute in principle any amplitude are constructed, as well as the three point `seed' amplitudes to make these work. In the three point case I will also present some classes of supersymmetric amplitudes with a massive particle for the type IIB superstring in a flat background.

Distortion is an asymptotic invariant of the embeddings

of finitely generated algebras. For group embeddings,

it has been introduced by M.Gromov. The main part of

the talk will be based on a recent work with Yu.Bahturin,

where we consider the behavior of distortion functions

for subalgebras of associative and Lie algebras.

In the 1990's Haagerup discovered a new subfactor, and hence a new topological quantum field theory, that has so far proved inaccessible by the methods of quantum groups and conformal field theory. It was the subfactor of smallest index beyond 4. This led to a classification project-classify all subfactors to as large an index as possible. So far we have gone as far as index 5. It is known that at index 6 wildness phenomena occur which preclude a simple listing of all subfactors of that index. It is possible that wildness occurs at a smaller index value, the main candidate being approximately 5.236.