Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.where ρ := ρ r,s : Spin r,s → GL(Σ r,s ) is a complex spinor representation, Θ g : Spin g M → SO g M is a metric spin structure and Spin r,s ⊂ GL + m is the spin group. Main Theorem 1 (universal spinor bundle, cf. Theorem 2.25). There exists a finite dimensional vector bundleπ Σ SM :ΣM → J 1 π r,s such that for each metric g ∈ S r,s (M ), there existsῑ g such thatcommutes. Here, J 1 π r,s denotes the first jet bundle of π r,s . Moreover,π Σ SM carries a connection, a metric and a Clifford multiplication such thatῑ g is a morphism of (generalized) Dirac bundles (see Definition 2.27). In addition,π Σ SM is natural with respect to spin diffeomorphisms. ♦The claims of the last assertions mean that not only the vector bundle structure of any metric spinor bundle π g M can be recovered fromπ Σ SM , but also its spinorial connection, its metric and its Clifford multiplication, see Theorem 2.25 for the precise meaning and a proof. We would like to emphasize that the morphism j 1 (g) in (1.2) is the 1-jet of the metric g and that the naturality assertion is formulated with respect to diffeomorphisms and not just isometries, see Section 2.6 for details. arXiv:1504.01034v3 [math.DG] 6 Oct 2015 ∼ = 9 9 We setκ V := κ V • θ. ♦ Diagram (2.1) easily extends to spin manifolds as follows: Let {ρ r,s : Spin r,s → GL(Σ r,s )} r,s∈N be a fixed choice of Spin r,s -representations. The construction of κ V andκ V induces bundle maps κ M : GL + M → S r,s M,κ M : GL + M → S r,s M, (2.2) by setting κ M | GL + x M := κ TxM andκ M | GL + x M :=κ TxM for any x ∈ M . Definition 2.3 (universal spinor bundle). The map π Σ SM : ΣM → S r,s M [b, σ] →κ M (b), where ΣM := GL + M × ρr,s Σ r,s is called universal spinor (vector) bundle. We obtain the following diagram ΣM π Σ SM / / π Σ M 9 9 S r,s M π r,s / / M, (2.3)where π Σ M := π r,s • π Σ SM . The map π Σ M is called universal spinor (fiber) bundle of M . Its sections are called universal spinor fields.♦

Abstract. It is a well-known fact that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function Z → R and endow the space of all spectra with an arsinh-uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all Riemannian metrics, which represents the entire Dirac spectrum at any metric. We also show that in general these functions do not descend to the space of Riemannian metrics modulo spin diffeomorphisms due to spectral flow.

In this article, we prove that on any compact spin manifold of dimension m ≡ 0, 6, 7 mod 8, there exists a metric, for which the associated Dirac operator has at least one eigenvalue of multiplicity at least two. We prove this by "catching" the desired metric in a subspace of Riemannian metrics with a loop that is not homotopically trivial. We show how this can be done on the sphere with a loop of metrics induced by a family of rotations. Finally, we transport this loop to an arbitrary manifold (of suitable dimension) by extending some known results about surgery theory on spin manifolds.

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CrowdEmotion produce software to measure a person's emotions based on analysis of microfacial expressions using a machine learning algorithm to recognize which features correspond with which emotions. The features are derived by applying a bank of Gabor filters to a set of frames. CrowdEmotion needed to improve the accuracy, processing speed and cost-efficiency of the tool. In particular they wanted to know if a subset of the bank of Gabor filters was sufficient, and whether the image filtering stage could be implemented on a GPU. A framework for choosing the optimum set of Gabor filters was established and ways of reducing the dimensionality of this were interrogated. Taking a subset of Local Binary Patterns was found to be fully justified. Meanwhile choosing a gridding pattern is open to interpretation; some suggestions were made about how this choice might be improved.