Functional Analysis on the Eve of the 21st Century: Volume I: In Honor of the Eightieth Birthday of I. M. Gelfand (Progress in Mathematics, 131/132)

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. A four-day conference, "Functional Analysis on the Eve of the TwentyƯ 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 opƯ 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 These two volumes contain eighteen invited papers by distinguished mathematicians in honor of the eightieth birthday of Israel M. Gelfand, one of the most remarkable mathematicians of our time. 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 shape our understanding of the term 'functional analysis'. The papers in these volumes largely concern areas in which Gelfand has a very strong interest today, including geometric quantum field theory, representation theory, combinatorial structures underlying various 'continuous' constructions, quantum groups and geometry. The second of the two volumes contains the somewhat more 'geometric' papers, although such a designation is to a certain extent arbitrary, because of the breadth of the papers. Front Matter....Pages i-xxiii Connection formulas in the q -analog de Rham cohomology....Pages 1-12 Lagrangian Models of Minimal Representations of E 6 , E 7 and E 8 ....Pages 13-63 Trigonometric Solutions of the Yang-Baxter Equation, Nets, and Hypergeometric Functions....Pages 65-118 Analogies between the Langlands Correspondence and Topological Quantum Field Theory....Pages 119-151 “Forms” of the Principal Series for GL n ....Pages 153-171 Geometry of determinants of elliptic operators....Pages 173-197 Quantum groups at ν = ∞....Pages 199-221 The Symplectic Operad....Pages 223-243 Quadratic unipotent representations of p-adic groups....Pages 245-262 On the Master Field in Two Dimensions....Pages 263-281 Physical Methods Applied to Donaldson Theory....Pages 283-292 Back Matter....Pages 293-296 Edited By Simon Gindikin, James Lepowsky, Robert L. Wilson.