Hukuhara's Topological Degree for non Compact Valued Multifunctions

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Interpolation in Wavelet Spaces and the HRT-Conjecture

We investigate the wavelet spaces $\mathcal{W}_{g}(\mathcal{H}_{\pi})\subset L^{2}(G)$ arising from square integrable representations $\pi:G \to \mathcal{U}(\mathcal{H}_{\pi})$ of a locally compact group $G$. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong conditions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time-frequency analysis, this problem turns out to be equivalent to the HRT-Conjecture. Finally, we consider the problem of whether all the wavelet spaces $\mathcal{W}_{g}(\mathcal{H}_{\pi})$ of a locally compact group $G$ collectively exhaust the ambient space $L^{2}(G)$. We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.Comment: Added a relevant citation and made minor modifications to the expositio

Fukaya Categories as Categorical Morse Homology

The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplectic analogue of the well-known algebraic and topological theories of (micro)localization

Spacelike Singularities and Hidden Symmetries of Gravity

We review the intimate connection between (super-)gravity close to a spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody algebras. We show that in this limit the gravitational theory can be reformulated in terms of billiard motion in a region of hyperbolic space, revealing that the dynamics is completely determined by a (possibly infinite) sequence of reflections, which are elements of a Lorentzian Coxeter group. Such Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras, suggesting that these algebras yield symmetries of gravitational theories. Our presentation is aimed to be a self-contained and comprehensive treatment of the subject, with all the relevant mathematical background material introduced and explained in detail. We also review attempts at making the infinite-dimensional symmetries manifest, through the construction of a geodesic sigma model based on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case of the hyperbolic algebra E10, which is conjectured to be an underlying symmetry of M-theory. Illustrations of this conjecture are also discussed in the context of cosmological solutions to eleven-dimensional supergravity.Comment: 228 pages. Typos corrected. References added. Subject index added. Published versio