Eugenie Hunsicker

Hodge cohomology of gravitational instantons

This article was published in the Duke Mathematical Journal [© Duke University Press] and is also available at: http://projecteuclid.org/euclid.dmj/1082665286

Harmonic forms on manifolds with edges

Let (X, g) be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of X, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai [5]. Because the metric g is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of L2 harmonic forms for any complete edge metric on the regular part of X.

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

Let V be a real valued potential that is smooth everywhere on R3, 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 Schrodinger operator H=−Δ+V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let T≔R3/L. Let u be an eigenfunction of H with eigenvalue λ and let ϵ>0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u∊H5/2−ϵ(T) in the usual Sobolev spaces, and u∊K3/2−ϵm(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 usi...

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 i of Dai’s invariant [10] 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 i in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X .

Additivity and non-additivity for perverse signatures

Abstract 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 Walls 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, Siegels version of Novikov additivity for Witt signatures is a special case of this corollary.

Scattering theory of the p-form Laplacian on manifolds with generalized cusps

In this paper we consider scattering theory on manifolds with special cusp-like metric singularities of warped product type g=dx^2 + x^(-2a)h, where a>0. These metrics form a natural subset in the class of metrics with warped product singularities and they can be thought of as interpolating between hyperbolic and cylindrical metrics. We prove that the resolvent of the Laplace operator acting on p-forms on such a manifold extends to a meromorphic function defined on the logarithmic cover of the complex plane with values in the bounded operators between weighted L^2-spaces. This allows for a construction of generalized eigenforms for the Laplace operator as well as for a meromorphic continuation of the scattering matrix. We give a precise description of the asymptotic expansion of generalized eigenforms on the cusp and find that the scattering matrix satisfies a functional equation.

A Parametrix Construction for the Laplacian on ℚ-rank 1 Locally Symmetric Spaces

This paper presents the construction of parametrices for the Gauss–Bonnet and Hodge Laplace operators on noncompact manifolds modelled on ℚ-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, φ-differential operators; that is, they live in the generalised φ-calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the φ-calculus. We construct parametrices for them in this paper using a combination of the b-pseudodifferential operator calculus of R. Melrose and the φ-pseudodifferential operator calculus. The construction simplifies and generalizes the construction of a parametrix for the Dirac operator done by Vaillant in his thesis. In addition, we study the mapping properties of these operators and determine the appropriate Hilbert spaces between which the Gauss–Bonnet and Hodge Laplace operators are Fredholm. Finally, we establish regularity results for elements of the kernels of these operators.

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.

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.

Extended Hodge theory for fibred cusp manifolds

For a particular class of pseudo manifolds, we show that the intersection cohomology groups for any perversity may be naturally represented by extended weighted