David Barnes

Rational SO(2)-equivariant spectra

The category of rational O(2)-equivariant cohomology theories has an algebraic model A(O(2)), as established by work of Greenlees. That is, there is an equivalence of categories between the homotopy category of rational O(2)-equivariant spectra and the derived category of the abelian model DA(O(2)). In this paper we lift this equivalence of homotopy categories to the level of Quillen equivalences of model categories. This Quillen equivalence is also compatible with the Adams short exact sequence of the algebraic model.

Splitting monoidal stable model categories

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S,S] C . An idempotent e of this ring will split the homotopy category: [X,Y ] C ∼ e[X,Y ] C ⊕(1−e)[X,Y] C . We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that is, C is Quillen equivalent to LeSC × L(1 e)SC and [X,Y ] L eSC ∼ = e[X,Y ] C . This Quillen equivalence is strong monoidal and is symmetric when the monoidal product of C is.

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories.

Model categories for orthogonal calculus

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O.n/‐equivariance clearer. Thus we develop model structures for the category of n‐polynomial and n‐homogeneous functors, along with Quillen pairs relating them. We then classify n‐homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O.n/‐ action. This improves upon the classification theorem of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra. 55P42, 55P91, 55U35

Rational equivariant rigidity

The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m<n. This complex is defined purely algebraically, in terms of characteristic structures of E_n-operads. Our construction involves: the free complete algebra in one variable associated to any E_n-operad; and an element in this free complete algebra, which is associated to a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. We deduce our main theorem from: a connection between the cohomology of iterated loop spaces and the cohomology of algebras over E_n-operads; and a Koszul duality result for E_n-operads.We analyze the stable isomorphism type of polynomial rings on degree 1 generators as modules over the sub-algebra A(1) = of the mod 2 Steenrod algebra. Since their augmentation ideals are Q_1-local, we do this by studying the Q_i-local subcategories and the associated Margolis localizations. The periodicity exhibited by such modules reduces the calculation to one that is finite. We show that these are the only localizations which preserve tensor products, by first computing the Picard groups of these subcategories and using them to determine all idempotents in the stable category of bounded-below A(1)-modules. We show that the Picard groups of the whole category are detected in the local Picard groups, and show that every bounded-below A(1) -module is uniquely expressible as an extension of a Q_0-local module by a Q_1-local module, up to stable equivalence. Applications include correct, complete proofs of Ossas theorem, applications to Powells work describing connective K-theory of classifying spaces of elementary abelian groups in functorial terms, and Aults work on the hit problem.

DEM Modelling of a New ‘Sphere Filling’ Approach for Optimising Motion Control of Rotational Moulding Processes

Rotational moulding is a polymer forming process used to create hollow, stress-free products using both heat and rotation. The basic principles behind the machines which execute the process of rotational moulding have not changed significantly over the last 60 years. A factor restricting the growth of the rotational moulding industry is the limited wall thickness uniformity that can be achieved using the current machines which have limited motion control. Improved flexibility of motion control over the mould is now available and will be investigated with the aim of providing a more efficient process and higher quality products. Using a mathematical ‘sphere filling’ curve approach, a rotational path can be designed which allows every area of a spherical mould to spend a more uniform time period in contact with the powder pool (Wall thickness uniformity is affected by the powder-wall contact time). This paper proposes a new approach to mould motion control optimisation and provides validation using Discrete Element Method (DEM) simulations. This method has been found to increase the uniformity of powder-wall contact time by up to 19%.

Rational

We find a simple algebraic model for rational Zp -equivariant spectra, via a series of Quillen equivalences. This model, along with an Adams short exact sequence, will allow us to easily perform constructions and calculations.