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A four-day conference, "Functional Analysis on the Eve of the Twenty­ First Century," was held at Rutgers University, New Brunswick, New Jersey, from October 24 to 27, 1993, in honor of the eightieth birthday of Professor Israel Moiseyevich Gelfand. He was born in Krasnye Okna, near Odessa, on September 2, 1913. Israel Gelfand has played a crucial role in the development of functional analysis during the last half-century. His work and his philosophy have in fact helped to shape our understanding of the term "functional analysis" itself, as has the celebrated journal Functional Analysis and Its Applications, which he edited for many years. Functional analysis appeared at the beginning of the century in the classic papers of Hilbert on integral operators. Its crucial aspect was the geometric interpretation of families of functions as infinite-dimensional spaces, and of op­ erators (particularly differential and integral operators) as infinite-dimensional analogues of matrices, directly leading to the geometrization of spectral theory. This view of functional analysis as infinite-dimensional geometry organically included many facets of nineteenth-century classical analysis, such as power series, Fourier series and integrals, and other integral transforms.



Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures

Our journey starts with a macroscopic view of Riemannian manifolds with positive scalar curvature and terminates with a glimpse of the proof of the homotopy invariance of some Novikov higher signatures of non-simply connected manifolds. Our approach focuses on the spectra of geometric differential operators on compact and non-compact manifolds V where the link with the macroscopic geometry and topology is established with suitable index theorems for our operators twisted with almost flat bundles over V. Our perspective mainly comes from the asymptotic geometry of infinite groups and foliations.
M. Gromov

Geometric Construction of Polylogarithms, II

The purpose of this paper is to construct Grassmannian p-cocycles (and in particular Grassmannian p-logarithms) for every positive integer p. A Grassmannian p-cocycle is a collection of holomorphic differential forms, each one on a Grassmannian variety, satisfying a cocyle condition. (See §9 for the precise definition. We refer to [HM-M] and the references there for background on Grassmannian p-cocycles, and for a description of their utility.)
Masaki Hanamura, Robert MacPherson

A Note on Localization and the Riemann-Roch Formula

Let M be a compact symplectic manifold of (real) dimension 2m, equipped with the Hamiltonian action of a compact connected Lie group K with maximal torus T; we denote the moment map for this action by μ : M → k*. In this note, we shall treat some properties of the symplectic quotient M red = μ−1(0)/K, whose symplectic structure ω0 descends from the symplectic structure on M. (We assume that 0 is a regular value of μ, so that M red has at worst finite quotient singularities.) We shall describe some applications of the main result of [16] (Theorem 8.1, the residue formula): this formula specifies the evaluation on the fundamental class of M red of η0 e ω0, for any class η0H *(M red). The residue formula relates cohomology classes on M red to the equivariant cohomology H K * (M) M, via the natural ring homomorphism κ0 : H K * (M) → H *(M red) whose surjectivity was proved in [20].
Lisa C. Jeffrey, Frances C. Kirwan

A Note on ODEs from Mirror Symmetry

We give close formulas for the counting functions of rational curves on complete intesection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the Prepotential.
A. Klemm, B. H. Lian, S. S. Roan, S. T. Yau


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