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Introduction
Anthony Knapp is retired from the Department of Mathematics, Stony Brook University. He does research in Algebra, Geometry and Topology, and Analysis. His most recent book is `Stokes's Theorem and Whitney Manifolds,' original digital edition.
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September 1997 - present
July 1975 - June 1990
July 1970 - June 1975
Publications
Publications (160)
"Stokes's Theorem and Whitney Manifolds" is mathematically the third volume in a five-volume series that systematically develops concepts and tools in algebra and real analysis that are vital to any mathematician, whether pure or applied, aspiring or established. Four of the volumes were written earlier, namely the pair "Basic Algebra" and "Advance...
This is a corrected version of the electronic second edition of the book Advanced Real Analysis published by Birkhauser in 2005. It may be freely downloaded and freely transmitted, but there are restrictions on printing copies. See the terms and conditions inside the book.
This is an electronic version of the second edition of the book Advanced Algebra published by Birkhauser in 2007. It may be freely downloaded and freely transmitted, but there are restrictions on printing copies. See the terms and conditions inside the book.
This is an electronic version of the second edition of the book Advanced Real Analysis published by Birkhauser in 2005. It may be freely downloaded and freely transmitted, but there are restrictions on printing copies. See the terms and conditions inside the book. This version is now obsolete; the current version is the Corrected Version 2017.
This is the final list of corrections to the first edition, which was published in 2005.
This is an electronic version of the second edition of the book Basic Real Analysis published by Birkhauser in 2005. It may be freely downloaded and freely transmitted, but there are restrictions on printing copies. See the terms and conditions inside the book.
This is an electronic version of the second edition of the book Basic Algebra published by Birkhauser in 2006. It may be freely downloaded and freely transmitted, but there are restrictions on printing copies. See the terms and conditions inside the book.
This is the final list of corrections to the first edition of Basic Algebra. Many of the corrections were kindly sent to the author by Skip Garibaldi of Emory University and Ario Contact of Shiraz, Iran. The long correction for pages 596-598 resulted from a discussion with Qiu Ruyue.
This is the final list of corrections to the first edition.
This is the final list of corrections to the first edition. Many of these corrections were kindly sent to the author either by S. H. Kim of South Korea or by Jacques Larochelle of Canada. The correction on pages 39 and 40 was kindly sent by Glenn Jia of China. The long correction for pages 596-598 resulted from a discussion with Qiu Ruyue.
The following people have kindly pointed out short corrections and other remarks concerning Elliptic Curves: Bryan J. Birch, Robin Chapman, Pierre-Yves Gaillard, Yves Hellegouarch, Chan Heng Huat, Günter Köhler, Robert P. Langlands, Kim Shun Enoch Lee, Dino Lorenzini, Jean-Pierre Serre, Allan Trojan, Yu Zhao, and Yuecheng Zhu. Two corrections found...
“This article describes ways in which to get an overview of some of the goals and methods of the Langlands program, and it points to treatments of some examples that are worth keeping in mind.” This article should be read and used together with the preceding overview by Knapp (see Zbl 1236.11096).
“For learning about the Langlands program, knowledge of Lie-group structure theory, algebraic number theory, algebraic geometry, modular forms, and infinite-dimensional representation theory is appropriate. This article describes how one can get a glimpse of the program with less background than this.” The article contains recommendations of books...
"For a connected linear semisimple Lie group G, this paper considers those nonzero limits of discrete series representations having infinitesimal character~0, calling them ""totally degenerate"". Such representations exist if and only if G has a compact Cartan subgroup, is quasisplit, and is acceptable in the sense of Harish-Chandra. Totally degene...
Basic Real Analysis and Advanced Real Analysis (available separately or together as a Set) systematically develop those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. These works present a comprehensive treatment with a global view of the subject, emphasizing the connecti...
Basic Real Analysis and Advanced Real Analysis (available separately or together as a Set) systematically develop those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. These works present a comprehensive treatment with a global view of the subject, emphasizing the connecti...
Reprint of editor's note that occurs in the memorial article "Armand Borel (1923-2003)," Notices of the American Mathematical Society, 51 (2004), 498-524.
Memorial article with Knapp as editor and with the other nine people as authors
For 2⩽m⩽l/2, let G be a simply connected Lie group with g0=so(2m,2l−2m) as Lie algebra, let g=k⊕p be the complexification of the usual Cartan decomposition, let K be the analytic subgroup with Lie algebra k∩g0, and let U(g) be the universal enveloping algebra of g. This work examines the unitarity and K spectrum of representations in the “analytic...
The question of unitarity of representations in the analytic continuation of discrete series from a Borel-de Siebenthal chamber is considered for those linear equal-rank classical simple Lie groups G that have not been treated fully before. Groups treated earlier by other authors include those for which G has real rank one or has a symmetric space...
D.E. Littlewood proved two branching theorems for decomposing the restriction of an irreducible finite-dimensional representation of a unitary group to a symmetric subgroup. One is for restriction of a representation of U(n) to the rotation group SO(n) when the given representation τλ of U(n) has nonnegative highest weight λ of depth ⩽n/2. It says...
Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie Algebras Universal Enveloping Algebra Compact Lie Groups Finite-Dimensional Representations Structure Theory of Semisimple Groups Advanced Structu...
A compact symmetric space, for purposes of this article, is a quotient G/K, where G is a compact connected Lie group and K is the identity component of the subgroup of fixed points of an involution. A branching theorem describes how an irreducible representation decomposes upon restriction to a subgroup. The article deals with branching theorems fo...
Elie Cartan's classification of the simple Lie algebras over R is derived quickly from some structure theory over Rand the classification over C . ' Elie Cartan classified the simple Lie algebras over R for the first time in 1914. There have been a number of simplifications in the proof since then, and these are described in [3, p. 537]. All proofs...
This is front matter of the paperback version of the book. The hardrback version was published in 1986. The only differences in what the two versions contain is that the paperback version contains an additional preface. The lists of corrections for the two books are the same.
The Selberg-Arthur trace formula is one of the tools available for approaching the conjecture of global functoriality in the Langlands program. Global functoriality is described within this volume in [Kn2]. We start with reductive groups G and H, say over the rationals Q for simplicity. We assume that G is quasisplit, and we suppose that we are giv...
This paper is an introduction to some ways that the trace formula can be applied to proving global functoriality. We rely heavily on the ideas and techniques in (Kn1), (Kn2), and (Ro4) in this volume. To address functoriality with the help of the trace formula, one compares the trace formulas for two different groups. In particular the trace formul...
This article is an introduction to automorphic forms on the adeles of a linear reductive group over a number fleld. The flrst half is a summary of aspects of local and global class fleld theory, with emphasis on the local Weil group, the L functions of Artin and Hecke, and the role of Artin reciprocity in relating the two kinds of L functions. The...
This article provides a review of the elementary theory of semisimple Lie al-gebras and Lie groups. It is essentially a summary of much of [K3]. The four sections treat complex semisimple Lie algebras, finite-dimensional representations of complex semisimple Lie algebras, compact Lie groups and real forms of complex Lie algebras, and structure theo...
This file contains the front matter of the proceedings of the Edinburgh conference "Representation Theory and Automorphic Forms" of 1996.
Let G be a noncompact simple Lie group with finite center, let K be a maximal compact subgroup, and suppose that rank G = rank K. If G=K is not Hermitian symmetric, then a theorem of Borel and de Siebenthal gives the existence of a system of positive roots relative to a compact Cartan subalgebra so that there is just one noncompact simple root and...
The essence of harmonic analysis is to decompose complicated expressions into pieces that reflect the structure of a group action when there is one. The goal is to make some difficult analysis manageable. The tool for taking advantage of an action by a nonabelian group is the associated group representation.
What are group representations, why are they so pervasive in mathematics, and where is their theory headed?
Élie Cartan's classification of the simple Lie algebras over ℝ is derived quickly from some structure theory over ℝ and the classification over C.
This second Part deals with the development of harmonic analysis during the nineteenth and twentieth centuries. [Part I in ibid. 43, 410–415 (1996; Zbl 1044.43501)].
Let be the quotient of a semisimple Lie group G by the centralizer L of a torus. The space of Dolbeault cohomology sections of a holomorphic line bundle over is a natural place to realize interesting irreducible unitary representations of G and was first studied for this purpose by Bott and Schmid. Zuckerman and Vogan later introduced derived funct...
See the linked data set "Notes of 1988 Zagier Course" for notes that formed the starting point for this book.
For a linear semisimple Lie group with a compact Cartan subgroup, the authors obtain formulas for the action of intertwining operators on certain subspaces of standard induced representations. These formulas provide explicit limitations on the pool of candidates for irreducible unitary representations, since the only possible invariant inner produc...
Building on joint work [8] with B. Speh, D. A. vogan [9,10] has obtained an algorithm for computimg composition series of the standard induced representations of a semisimple Lie group G. The algorithm takes a particularly simple form in the case that the representations are induced from a maximal parabolic subgroup. On the one hand, this algorithm...
This is an authorized Russian translation of the second edition of "Denumerable Markov Chains."
It is known that the problem of classifying the irreducible unitary representations of a linear connected semisimple Lie group G comes down to deciding which Langlands quotients J(MAN, σ, v) are infinitesimally unitary. Here MAN is any cuspidal parabolic subgroup, σ is any discrete series or nondegenerate limit of discrete series representation of...
This set of notes is a preliminary version of the first three chapters in the book of Knapp and Vogan entitled "Cohomological Induction and Unitary Representations." The notes are preliminary in two ways. One is that the notes work with K finite functions on K, while the book works with K finite distributions on K. Although the two approaches come...
This is front matter of the hardback version of the book. The paperback version was published in 2001. The only differences in what the two versions contain is that the paperback version contains an additional preface. The lists of corrections for the two books are the same.
For a wide class of linear connected semisimple Lie groups, one obtains formulas limiting the Langlands parameters of irreducible unitary representations obtained from maximal parabolic subgroups. The formulas relate unitarity to the number of roots satisfying certain conditions. Some evidence is presented that the formulas are sharp. The results c...
Guide to proof of minimal K type formula:
Let G be a linear connected reductive group. If K is a maximal compact subgroup of G, the published paper "Minimal K-type formula" gives a formula for the minimal K types of all standard induced representations of G. A proof of the published formula appears in the two linked data sets of accompanying handwr...
