L2 harmonic forms for a class of complete Kahler metrics

The Hodge theorem for compact manifolds states that every real cohomology class of a compact manifold M is represented by a unique harmonic form. That is, the space of solutions to the differential equation .d Cd / D 0 on L2 forms over M; a space that depends on the metric on M; is canonically isomorphic to the purely topological real cohomology space of M: This isomorphism is enormously useful because it provides a way to transform theorems from geometry into theorems in topology and vice versa. No such result holds in general for complete noncompact manifolds, but in many specific cases there are Hodge-type theorems. One of the oldest is the description, due to Atiyah, Patodi, and Singer [1], of the space of L2 harmonic forms on a manifold with complete cylindrical ends. By calculating the solutions to the equation for harmonic forms on the cylindrical ends, they showed that the space of L2 harmonic forms is isomorphic to the image of the relative cohomology of the manifold in the absolute cohomology. Another Hodge-type result was found by Zucker [14] for a natural class of metrics called Poincaré metrics. These metrics, first constructed by Cornalba and Griffiths [4], are complete Kähler metrics with hyperbolic cusp-type singularities at isolated points on a Riemann surface. Zucker showed that the space of L2 forms on a Riemann surface that are harmonic with respect one of these metrics is isomorphic to the standard cohomology of the surface. This result was extended by Cattani, Kaplan, and Schmid [3] to analogous metrics on bundles over projective varieties with singularities along a divisor. These metrics can be thought of as complete Kähler metrics on the noncompact manifold given by removing the divisor

Hodge and signature theorems for a family of manifolds with fibre bundle boundary

Over the past fifty years, Hodge and signature theorems have been proved for various classes of noncompact and incomplete Riemannian manifolds. Two of these classes are manifolds with incomplete cylindrical ends and manifolds with cone bundle ends, that is, whose ends have the structure of a fibre bundle over a compact oriented manifold, where the fibres are cones on a second fixed compact oriented manifold. In this paper, we prove Hodge and signature theorems for a family of metrics on a manifold M with fibre bundle boundary that interpolates between the incomplete cylindrical metric and the cone bundle metric on M . We show that the Hodge and signature theorems for this family of metrics interpolate naturally between the known Hodge and signature theorems for the extremal metrics. The Hodge theorem involves intersection cohomology groups of varying perversities on the conical pseudomanifold X that completes the cone bundle metric on M . The signature theorem involves the summands τ i of Dai’s τ invariant [J Amer Math Soc 4 (1991) 265–321] that are defined as signatures on the pages of the Leray–Serre spectral sequence of the boundary fibre bundle of M . The two theorems together allow us to interpret the τ i in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X

Weighted Hodge cohomology of iterated fibred cusp metrics

On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted L2 cohomology for iterated bred cusp metrics on the smooth stratum

Hodge theory for intersection space cohomology

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincare duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted L2 harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well

Additivity and non-additivity for perverse signatures

A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary. Wall showed that this property is not true of signatures on manifolds with boundary and that the difference from additivity could be described as a certain Maslov triple index. Perverse signatures are signatures defined for any oriented stratified pseudomanifold, using the intersection homology groups of Goresky and MacPherson. In the case of Witt spaces, the middle perverse signature is the same as the Witt signature. This paper proves a generalization to perverse signatures of Wall's non-additivity theorem for signatures of manifolds with boundary. Under certain topological conditions on the dividing hypersurface, Novikov additivity for perverse signatures may be deduced as a corollary. In particular, Siegel's version of Novikov additivity for Witt signatures is a special case of this corollary

Isospectrality for orbifold lens spaces

We answer Mark Kac’s famous question [K], “can one hear the shape of a drum?” in the positive for orbifolds that are 3-dimensional and 4-dimensional lens spaces; we thus complete the answer to this question for orbifold lens spaces in all dimensions. We also show that the coefficients of the asymptotic expansion of the trace of the heat kernel are not sufficient to determine the above results

Building efficient deep Hebbian networks for image classification tasks

Multi-layer models of sparse coding (deep dictionary learning) and dimensionality reduction (PCANet) have shown promise as unsupervised learning models for image classification tasks. However, the pure implementations of these models have limited generalisation capabilities and high computational cost. This work introduces the Deep Hebbian Network (DHN), which combines the advantages of sparse coding, dimensionality reduction, and convolutional neural networks for learning features from images. Unlike in other deep neural networks, in this model, both the learning rules and neural architectures are derived from cost-function minimizations. Moreover, the DHN model can be trained online due to its Hebbian components. Different configurations of the DHN have been tested on scene and image classification tasks. Experiments show that the DHN model can automatically discover highly discriminative features directly from image pixels without using any data augmentation or semi-labeling

Neural networks for efficient nonlinear online clustering

Unsupervised learning techniques, such as clustering and sparse coding, have been adapted for use with data sets exhibiting nonlinear relationships through the use of kernel machines. These techniques often require an explicit computation of the kernel matrix, which becomes expensive as the number of inputs grows, making it unsuitable for efficient online learning. This paper proposes an algorithm and a neural architecture for online approximated nonlinear kernel clustering using any shift-invariant kernel. The novel model outperforms traditional low-rank kernel approximation based clustering methods, it also requires significantly lower memory requirements than those of popular kernel k-means while showing competitive performance on large data sets

TeStED Project- Transitioning without A2 level mathematics

The research presented is the first stage of a project to support students entering STEM degrees. The study aims to investigate and address the mathematical difficulties that many students present transitioning to undergraduate Engineering courses. To this end data were collected to identify how age, gender, mathematical background, and preferred learning styles relate to outcomes on a mathematics diagnostic test. Both quantitative and qualitative methods of analysis were used to analyse the data. Our findings complement findings from previous research. We compared students with BTEC, GCSE and A/AS level qualifications, and related qualifications to study habits

Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions.

Let V be a real valued potential that is smooth everywhere on R 3 , except at a periodic, discrete set S of points, where it has singularities of the Coulomb-type Z/r . We assume that the potential V is periodic with period lattice L . We study the spectrum of the Schrödinger operator H=−Δ+V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k . Let T≔R 3 /L . Let u be an eigenfunction of H with eigenvalueλ and let ϵ>0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u∊H 5/2−ϵ (T) in the usual Sobolev spaces, and u∊K m 3/2−ϵ (T\S) in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k , we also show that H has compact resolvent and hence a complete eigenfunction expansion. The case of the hydrogen atom suggests that our regularity results are optimal. We present two applications to the numerical approximation of eigenvalues: using wave functions and using piecewise polynomials