James Lepowsky - Böcker

This work is motivated by and develops connections between several branches of mathematics and physics--the theories of Lie algebras, finite groups and modular functions in mathematics, and string theory in physics. The first part of the book presents a new mathematical theory of vertex operator algebras, the algebraic counterpart of two-dimensional holomorphic conformal quantum field theory. The remaining part constructs the Monster finite simple group as the automorphism group of a very special vertex operator algebra, called the "moonshine module" because of its relevance to "monstrous moonshine."

The theory of 'Monstrous Moonshine' has been a major development in mathematics since 1979. Beginning with remarkable conjectures relating finite group theory and number theory that stimulated an outpouring of new ideas, 'Monstrous Moonshine' deeply involves many different areas of mathematics, as well as string theory and conformal field theory in physics. It has changed the landscape of many fields. This volume consists of 17 papers by research leaders, based on talks presented at a workshop held to mark the anniversary of the 'Monstrous Moonshine' conjectures. The papers, which include surveys of many important developments, will give the reader a view of a wide range of current activity and promising directions for future research.

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

In the past few years, vertex operator algebra theory has been growing both in intrinsic interest and in the scope of its interconnections with areas of mathematics and physics. The structure and representation theory of vertex operator algebras is deeply related to such subjects as monstrous moonshine, conformal field theory and braid group theory. Vertex operator algebras are the mathematical counterpart of chiral algebras in conformal field theory. In the Introduction which follows, we sketch some of the main themes in the historical development and mathematical and physical motivations of these ideas, and some of the current issues. Given a vertex operator algebra, it is important to consider not only its modules (representations) but also intertwining operators among the mod­ ules. Matrix coefficients of compositions of these operators, corresponding to certain kinds of correlation functions in conformal field theory, lead natu­ rally to braid group representations. In the specialbut important case when these braid group representations are one-dimensional, one can combine the modules and intertwining operators with the algebra to form a structure satisfying axioms fairly close to those for a vertex operator algebra. These are the structures which form the main theme of this monograph. Another treatment of similar structures has been given by Feingold, Frenkel and Ries (see the reference [FFR] in the Bibliography), and in fact the material de­ veloped in the present work has close connections with much work of other people, as we explain in the Introduction and throughout the text.

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, directlyleading 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.

The Seminar has taken place at Rutgers University in New Brunswick, New Jersey, since 1990 and it has become a tradition, starting in 1992, that the Seminar be held during July at IHES in Bures-sur-Yvette, France. This is the second Gelfand Seminar volume published by Birkhauser, the first having covered the years 1990-1992. Most of the papers in this volume result from Seminar talks at Rutgers, and some from talks at IHES. In the case of a few of the papers the authors did not attend, but the papers are in the spirit of the Seminar. This is true in particular of V. Arnold's paper. He has been connected with the Seminar for so many years that his paper is very natural in this volume, and we are happy to have it included here. We hope that many people will find something of interest to them in the special diversity of topics and the uniqueness of spirit represented here. The publication of this volume would be impossible without the devoted attention of Ann Kostant. We are extremely grateful to her. I. Gelfand J. Lepowsky M. Smirnov Questions and Answers About Geometric Evolution Processes and Crystal Growth Fred Almgren We discuss evolutions of solids driven by boundary curvatures and crystal growth with Gibbs-Thomson curvature effects. Geometric measure theo­ retic techniques apply both to smooth elliptic surface energies and to non­ differentiable crystalline surface energies.

This is Volume 2 of a two-volume set which contains 18 invited papers in honour of the 80th birthday of mathematician Israel M. Gelfand, who is considered to have played an important role in the development of functional analysis in the latter half of the 20th century. The papers cover areas in which Gelfand has a strong interest, including geometric quantum field theory, representation theory, combinatorial structures underlying various "continuous" constructions, quantum groups and geometry. This second volume contains the more "geometric" papers.

These two volumes contain 18 invited papers in honour of the 80th birthday of mathematician Israel M. Gelfand, who is considered to have played an important role in the development of functional analysis in the latter half of the 20th century. The papers cover areas in which Gelfand has a strong interest, including geometric quantum field theory, representation theory, combinatorial structures underlying various "continuous" constructions, quantum groups and geometry. The second volume contains the more "geometric" papers.

* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.

In the past few years, vertex operator algebra theory has been growing both in intrinsic interest and in the scope of its interconnections with areas of mathematics and physics. The structure and representation theory of vertex operator algebras is deeply related to such subjects as monstrous moonshine, conformal field theory and braid group theory. Vertex operator algebras are the mathematical counterpart of chiral algebras in conformal field theory. In the Introduction which follows, we sketch some of the main themes in the historical development and mathematical and physical motivations of these ideas, and some of the current issues. Given a vertex operator algebra, it is important to consider not only its modules (representations) but also intertwining operators among the mod­ ules. Matrix coefficients of compositions of these operators, corresponding to certain kinds of correlation functions in conformal field theory, lead natu­ rally to braid group representations. In the specialbut important case when these braid group representations are one-dimensional, one can combine the modules and intertwining operators with the algebra to form a structure satisfying axioms fairly close to those for a vertex operator algebra. These are the structures which form the main theme of this monograph. Another treatment of similar structures has been given by Feingold, Frenkel and Ries (see the reference [FFR] in the Bibliography), and in fact the material de­ veloped in the present work has close connections with much work of other people, as we explain in the Introduction and throughout the text.

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.

The Seminar has taken place at Rutgers University in New Brunswick, New Jersey, since 1990 and it has become a tradition, starting in 1992, that the Seminar be held during July at IHES in Bures-sur-Yvette, France. This is the second Gelfand Seminar volume published by Birkhauser, the first having covered the years 1990-1992. Most of the papers in this volume result from Seminar talks at Rutgers, and some from talks at IHES. In the case of a few of the papers the authors did not attend, but the papers are in the spirit of the Seminar. This is true in particular of V. Arnold's paper. He has been connected with the Seminar for so many years that his paper is very natural in this volume, and we are happy to have it included here. We hope that many people will find something of interest to them in the special diversity of topics and the uniqueness of spirit represented here. The publication of this volume would be impossible without the devoted attention of Ann Kostant. We are extremely grateful to her. I. Gelfand J. Lepowsky M. Smirnov Questions and Answers About Geometric Evolution Processes and Crystal Growth Fred Almgren We discuss evolutions of solids driven by boundary curvatures and crystal growth with Gibbs-Thomson curvature effects. Geometric measure theo­ retic techniques apply both to smooth elliptic surface energies and to non­ differentiable crystalline surface energies.

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.