V. S. Varadarajan

Harmonic Analysis of Spherical Functions on Real Reductive Groups

Contents: The Concept of a Spherical Function Structure of Semisimple Lie Groups and Differential Operators on Them The Elementary Spherical Functions The Harish-Chandra Series for and the c-Function Asymptotic Behaviour of Elementary Spherical Functions The L2-Theory. The Harish-Chandra Transform on the Schwartz Space of G//K LP-Theory of Harish-Chandra Transform. Fourier Analysis on the Spaces CP(G//K) Bibliography Subject Index.

Supersymmetry for Mathematicians: An Introduction

Introduction The concept of a supermanifold Super linear algebra Elementary theory of supermanifolds Clifford algebras, spin groups, and spin representations Fine structure of spin modules Superspacetimes and super Poincare groups.

Local moduli for meromorphic differential equations

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Path Integrals for a Class of P-Adic Schrödinger Equations

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. If the group is a vector space of finite dimension over a non-Archimedean locally compact division ring, it is of interest to examine the structure of dynamical systems defined by Hamiltonians analogous to those encountered over the field of real numbers. In this Letter, a path integral formula for the imaginary time propagators of these Hamiltonians is derived.

Linear meromorphic differential equations: A modern point of view

A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substantial links of this theory to the theory of algebraic groups, commutative algebra, resurgent functions, and Galois differential methods have been discovered. This survey attempts a general introduction to some of these aspects, with emphasis on reduction theory, asymptotic analysis, Stokes phenomena, and certain moduli problems.

The Minkowski and conformal superspaces

We define complex Minkowski superspace in four dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this superflag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, based on the real conformal supergroup and its Lie superalgebra.

Euler and his work on infinite series

Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously. It was also said (by Laplace) that all mathematicians were his students. It is appropriate in this, the tercentennial year of his birth, to revisit him and survey his work, its offshoots, and the remarkable vitality of his themes which are still flourishing, and to immerse ourselves once again in the universe of ideas that he has created. This is not a task for a single individual, and appropriately enough, a number of mathematicians are attempting to do this and present a picture of his work and its modern resonances to the general mathematical community. To be honest, such a project is Himalayan in its scope, and it is impossible to do full justice to it. In the following pages I shall try to make a very small contribution to this project, discussing in a sketchy manner Euler’s work on infinite series and its modern outgrowths. My aim is to acquaint the generic mathematician with

On the concept of Einstein–Podolsky–Rosen states and their structure

In this paper the notion of an EPR state for the composite S of two quantum systems S1,S2, relative to S2 and a set O of bounded observables of S2, is introduced in the spirit of the classical examples of Einstein–Podolsky–Rosen and Bohm. We restrict ourselves mostly to EPR states of finite norm. The main results are contained in Theorems 3–6 and imply that if EPR states of finite norm relative to (S2,O) exist, then the elements of O have discrete probability distributions and the Von Neumann algebra generated by O is essentially imbeddable inside S1 by an antiunitary map. The EPR states then correspond to the different imbeddings and certain additional parameters, and are explicitly given by formulas which generalize the famous example of Bohm. If O generates all bounded observables, S2 must be of finite dimension and can be imbedded inside S1 by an antiunitary map, and the EPR states relative to S2 are then in canonical bijection with the different imbeddings of S2 inside S1; moreover they are then give...

Su (2) Poisson–Lie T Duality

Poisson–Lie target space duality is a framework where duality transformations are properly defined. In this Letter, we investigate the dual pair of σ-models defined by the double SO(3,1) in the Iwasawa decomposition.

Variations on a theme of Schwinger and Weyl

The theme of doing quantum mechanics on all Abelian groups goes back to Schwinger and Weyl. This theme was studied earlier from the point of view of approximating quantum systems in infinite-dimensional spaces by those associated to finite Abelian groups. This Letter links this theme to deformation quantization, and explores the set of noncommutative associative algebra structures on the Schwartz-Weil algebra of any locally compact separable Abelian group. If the group is a vector space of even dimension over a non-Archimedean local fieldK, there exists a family of noncommutative (Moyal) structures parametrized by the local field and containing membersarbitrarily close to the classical one, although the classical algebra is rigid in the sense of deformation theory. The-products are defined by Fourier integral operators. The problem of constructing sucharithmetic Moyal structures on the algebra of Schwartz-Bruhat functions on manifolds that are locally likeK2n is raised.