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Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings

Author: Wolfgang Bertram

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 218

ISBN-13: 0821840916

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The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

Book Synopsis Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings by : Wolfgang Bertram

Download or read book Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings written by Wolfgang Bertram and published by American Mathematical Soc.. This book was released on 2008 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed, without any restriction on the dimension or on the characteristic. Two basic features distinguish the author's approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent- and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.

Differential Geometry, Lie Groups, and Symmetric Spaces

Author: Sigurdur Helgason

Publisher: Academic Press

Published: 1979-02-09

Total Pages: 628

ISBN-13: 9780080873961

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The present book is intended as a textbook and reference work on three topics in the title. Together with a volume in progress on "Groups and Geometric Analysis" it supersedes my "Differential Geometry and Symmetric Spaces," published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful.

Book Synopsis Differential Geometry, Lie Groups, and Symmetric Spaces by : Sigurdur Helgason

Download or read book Differential Geometry, Lie Groups, and Symmetric Spaces written by Sigurdur Helgason and published by Academic Press. This book was released on 1979-02-09 with total page 628 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present book is intended as a textbook and reference work on three topics in the title. Together with a volume in progress on "Groups and Geometric Analysis" it supersedes my "Differential Geometry and Symmetric Spaces," published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful.

Invariant Differential Operators for Quantum Symmetric Spaces

Author: Gail Letzter

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 104

ISBN-13: 0821841319

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This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.

Book Synopsis Invariant Differential Operators for Quantum Symmetric Spaces by : Gail Letzter

Download or read book Invariant Differential Operators for Quantum Symmetric Spaces written by Gail Letzter and published by American Mathematical Soc.. This book was released on 2008 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of Harish-Chandra and Helgason: There is a Harish-Chandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials.

The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra

Author: Michael Kapovich

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 98

ISBN-13: 0821840541

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In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over $\mathbb{Q}$ and its complex Langlands' dual. The authors give a new proof of the Saturation Conjecture for $GL(\ell)$ as a consequence of their solution of the corresponding saturation problem for the Hecke structure constants for all split reductive algebraic groups over $\mathbb{Q}$.

Book Synopsis The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra by : Michael Kapovich

Download or read book The Generalized Triangle Inequalities in Symmetric Spaces and Buildings with Applications to Algebra written by Michael Kapovich and published by American Mathematical Soc.. This book was released on 2008 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper the authors apply their results on the geometry of polygons in infinitesimal symmetric spaces and symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the nonvanishing of the structure constants of the (spherical) Hecke and representation rings associated with a split reductive algebraic group over $\mathbb{Q}$ and its complex Langlands' dual. The authors give a new proof of the Saturation Conjecture for $GL(\ell)$ as a consequence of their solution of the corresponding saturation problem for the Hecke structure constants for all split reductive algebraic groups over $\mathbb{Q}$.

Sum Formula for SL$_2$ over a Totally Real Number Field

Author: Roelof W. Bruggeman

Publisher: American Mathematical Soc.

Published: 2009-01-21

Total Pages: 96

ISBN-13: 0821842021

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The authors prove a general form of the sum formula $\mathrm{SL}_2$ over a totally real number field. This formula relates sums of Kloosterman sums to products of Fourier coefficients of automorphic representations. The authors give two versions: the spectral sum formula (in short: sum formula) and the Kloosterman sum formula. They have the independent test function in the spectral term, in the sum of Kloosterman sums, respectively.

Book Synopsis Sum Formula for SL$_2$ over a Totally Real Number Field by : Roelof W. Bruggeman

Download or read book Sum Formula for SL$_2$ over a Totally Real Number Field written by Roelof W. Bruggeman and published by American Mathematical Soc.. This book was released on 2009-01-21 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors prove a general form of the sum formula $\mathrm{SL}_2$ over a totally real number field. This formula relates sums of Kloosterman sums to products of Fourier coefficients of automorphic representations. The authors give two versions: the spectral sum formula (in short: sum formula) and the Kloosterman sum formula. They have the independent test function in the spectral term, in the sum of Kloosterman sums, respectively.

The Recognition Theorem for Graded Lie Algebras in Prime Characteristic

Author: Georgia Benkart

Publisher: American Mathematical Soc.

Published: 2009

Total Pages: 164

ISBN-13: 0821842269

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"Volume 197, number 920 (second of 5 numbers)."

Book Synopsis The Recognition Theorem for Graded Lie Algebras in Prime Characteristic by : Georgia Benkart

Download or read book The Recognition Theorem for Graded Lie Algebras in Prime Characteristic written by Georgia Benkart and published by American Mathematical Soc.. This book was released on 2009 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Volume 197, number 920 (second of 5 numbers)."

Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups

Author: John Rognes

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 154

ISBN-13: 0821840762

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The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

Book Synopsis Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups by : John Rognes

Download or read book Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups written by John Rognes and published by American Mathematical Soc.. This book was released on 2008 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

Developments and Trends in Infinite-Dimensional Lie Theory

Author: Karl-Hermann Neeb

Publisher: Springer Science & Business Media

Published: 2010-10-17

Total Pages: 492

ISBN-13: 0817647414

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This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups. Contributors: B. Allison, D. Beltiţă, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

Book Synopsis Developments and Trends in Infinite-Dimensional Lie Theory by : Karl-Hermann Neeb

Download or read book Developments and Trends in Infinite-Dimensional Lie Theory written by Karl-Hermann Neeb and published by Springer Science & Business Media. This book was released on 2010-10-17 with total page 492 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie (super-)algebras, geometry of infinite-dimensional Lie (transformation) groups, and representation theory of infinite-dimensional Lie groups. Contributors: B. Allison, D. Beltiţă, W. Bertram, J. Faulkner, Ph. Gille, H. Glöckner, K.-H. Neeb, E. Neher, I. Penkov, A. Pianzola, D. Pickrell, T.S. Ratiu, N.R. Scheithauer, C. Schweigert, V. Serganova, K. Styrkas, K. Waldorf, and J.A. Wolf.

The History of Continua

Author: Stewart Shapiro

Publisher: Oxford University Press, USA

Published: 2021

Total Pages: 593

ISBN-13: 0198809646

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Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.

Book Synopsis The History of Continua by : Stewart Shapiro

Download or read book The History of Continua written by Stewart Shapiro and published by Oxford University Press, USA. This book was released on 2021 with total page 593 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.

Toroidal Dehn Fillings on Hyperbolic 3-Manifolds

Author: Cameron Gordon

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 154

ISBN-13: 082184167X

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The authors determine all hyperbolic $3$-manifolds $M$ admitting two toroidal Dehn fillings at distance $4$ or $5$. They show that if $M$ is a hyperbolic $3$-manifold with a torus boundary component $T 0$, and $r,s$ are two slopes on $T 0$ with $\Delta(r,s) = 4$ or $5$ such that $M(r)$ and $M(s)$ both contain an essential torus, then $M$ is either one of $14$ specific manifolds $M i$, or obtained from $M 1, M 2, M 3$ or $M {14}$ by attaching a solid torus to $\partial M i - T 0$.All the manifolds $M i$ are hyperbolic, and the authors show that only the first three can be embedded into $S3$. As a consequence, this leads to a complete classification of all hyperbolic knots in $S3$ admitting two toroidal surgeries with distance at least $4$.

Book Synopsis Toroidal Dehn Fillings on Hyperbolic 3-Manifolds by : Cameron Gordon

Download or read book Toroidal Dehn Fillings on Hyperbolic 3-Manifolds written by Cameron Gordon and published by American Mathematical Soc.. This book was released on 2008 with total page 154 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors determine all hyperbolic $3$-manifolds $M$ admitting two toroidal Dehn fillings at distance $4$ or $5$. They show that if $M$ is a hyperbolic $3$-manifold with a torus boundary component $T 0$, and $r,s$ are two slopes on $T 0$ with $\Delta(r,s) = 4$ or $5$ such that $M(r)$ and $M(s)$ both contain an essential torus, then $M$ is either one of $14$ specific manifolds $M i$, or obtained from $M 1, M 2, M 3$ or $M {14}$ by attaching a solid torus to $\partial M i - T 0$.All the manifolds $M i$ are hyperbolic, and the authors show that only the first three can be embedded into $S3$. As a consequence, this leads to a complete classification of all hyperbolic knots in $S3$ admitting two toroidal surgeries with distance at least $4$.