Joint with Sarah Whitehouse

Tall-Wraith monoids were introduced in The Hunting of the Hopf Ring to describe the algebraic structure on the set of unstable operations of a suitable generalised cohomology theory. In this paper we begin the study of Tall-Wraith monoids in an algebraic and categorical setting. We show that for V a variety of algebras, applying the free V-algebra functor to a monoid in Set produces a Tall-Wraith monoid. We also study the example of the Tall-Wraith monoid defined by the self set-maps of a finite ring, an example closely related to the original motivation for Tall-Wraith monoids.

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We define the notion of a co-Riemannian structure and show how it can be used to define the Dirac operator on an appropriate infinite dimensional manifold. In particular, this approach works for the smooth loop space of a so-called string manifold.

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We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.

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We consider the problem of defining the structure of a smooth manifold on the various spaces of piecewise-smooth loops in a smooth finite dimensional manifold. We succeed for a particular type of piecewise-smooth loops.

We also examine the action of the diffeomorphism group of the circle. It is not a useful action on the manifold that we define. We consider how one might fix this problem and conclude that it can only be done by completing to the space of loops of bounded variation.

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with Sarah Whitehouse

We describe the families of degree zero operations in the first Morava K-theory, and consequently in mod p K-theory, in terms of Adams' operations.

When devising problems for a linear algebra course it is desirable to have an extensive source of integral matrices of determinant one. In this short paper we give a straightforward method of generating some such matrices from binomial coefficients.

This paper has now been superseded by The Co-Riemannian Structure of a Smooth Loop Space and How to Construct a Dirac Operator in Infinite Dimensions.

We construct a natural inner product on the cotangent spaces of the free loop space of a Riemannian manifold. As an application of, and motivation for, this construction we show how to construct a Dirac operator on the loop space of a string manifold.

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Welcome to the Enchanted Forest. The pictures displayed here were originally designed for the papers by Nils Baas:

• On Many Body System Interactions (arXiv:1012.2705)
• New States of Matter Suggested by New Topological Structures (arXiv:1012.2698)

TheEnchantedForest