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Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

Mark Green, Phillip A. Griffiths, and Matt Kerr

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691154244
eISBN:
9781400842735
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691154244.001.0001
Subject:
Mathematics, Analysis

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive ... More


RADON TRANSFORM ON GROUPS

Leon Ehrenpreis

in The Universality of the Radon Transform

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198509783
eISBN:
9780191709166
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509783.003.0007
Subject:
Mathematics, Mathematical Physics

This chapter applies the book's methods to various Lie groups. In particular, to the horocyclic and geodesic transforms on G/K.


Higgs Bundles and Geometric Structures on Surfaces

William M. Goldman

in The Many Facets of Geometry: A Tribute to Nigel Hitchin

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199534920
eISBN:
9780191716010
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199534920.003.0008
Subject:
Mathematics, Geometry / Topology

The theory of Higgs bundles, pioneered by Hitchin and Simpson, provides an analytic approach to studying surface group representations and their deformation space. This chapter describes the basic ... More


Introduction

Mark Green, Phillip Griffiths, and Matt Kerr

in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691154244
eISBN:
9781400842735
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691154244.003.0001
Subject:
Mathematics, Analysis

This book deals with Mumford-Tate groups, the fundamental symmetry groups in Hodge theory. Much, if not most, of the use of Mumford-Tate groups has been in the study of polarized Hodge structures of ... More


Hodge Representations and Hodge Domains

Mark Green, Phillip Griffiths, and Matt Kerr

in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691154244
eISBN:
9781400842735
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691154244.003.0005
Subject:
Mathematics, Analysis

This chapter deals with Hodge representations and Hodge domains. For general polarized Hodge structures, it considers which semi-simple ℚ-algebraic groups M can be Mumford-Tate groups of polarized ... More


Arithmetic Aspects of Mumford-Tate Domains

Mark Green, Phillip Griffiths, and Matt Kerr

in Mumford-Tate Groups and Domains: Their Geometry and Arithmetic (AM-183)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691154244
eISBN:
9781400842735
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691154244.003.0007
Subject:
Mathematics, Analysis

This chapter describes the arithmetic aspects of Mumford-Tate domains and Noether-Lefschetz loci. It first clarifies a few points concerning the structure and construction of Mumford-Tate domains ... More


Lie Groups and Lie Algebras: A Physicist's Perspective

Adam M. Bincer

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199662920
eISBN:
9780191745492
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199662920.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This book is based on lectures given to graduate students in physics at the University of Wisconsin-Madison. Group theory has been around for many years and the only thing new in this book is my ... More


Derrida's Secret: Perjury, Testimony, Oath

Charles Barbour

Published in print:
2017
Published Online:
January 2018
ISBN:
9781474424998
eISBN:
9781474434911
Item type:
book
Publisher:
Edinburgh University Press
DOI:
10.3366/edinburgh/9781474424998.001.0001
Subject:
Philosophy, Political Philosophy

Against the backdrop of a series of recent political events (the Snowden Affair, the Manning Affair, Wikileaks, Clinton’s private email account, and so forth), this book argues that secrecy is a ... More


Breaking Symmetries

Gian Francesco Giudice

in A Zeptospace Odyssey: A Journey into the Physics of the LHC

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199581917
eISBN:
9780191723001
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199581917.003.0008
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

Symmetries play a fundamental role in modern physics. This chapter introduces this concept by discussing the meaning of symmetries in mathematics and in physics, explaining the significance of Lie ... More


Symmetries And Renormalization

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0013
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter deals with linear continuous symmetries corresponding to compact Lie groups because they imply interesting formal properties; consequences of discrete symmetries can also be studied but ... More


Non-Abelian Gauge Theories: Introduction

JEAN ZINN-JUSTIN

in Quantum Field Theory and Critical Phenomena

Published in print:
2002
Published Online:
January 2010
ISBN:
9780198509233
eISBN:
9780191708732
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509233.003.0019
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter constructs a field theory invariant under local, that is, space-dependent, transformations of a general compact Lie group G. Inspired by the abelian example of Chapter 18, it introduces ... More


An Alternative Approach to Lie Groups and Geometric Structures

Ercüment H. Ortaçgil

Published in print:
2018
Published Online:
September 2018
ISBN:
9780198821656
eISBN:
9780191860959
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821656.001.0001
Subject:
Mathematics, Pure Mathematics

This book is about the foundations of geometric symmetry, namely, Lie groups and differential geometry. Although this is a classical subject about which hundreds of books have been written, this book ... More


Lie groups and Lie algebras

Adam M. Bincer

in Lie Groups and Lie Algebras: A Physicist's Perspective

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199662920
eISBN:
9780191745492
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199662920.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Lie groups and Lie algebras are defined. SU(2), the group whose elements are 2x2 unitary unimodular matrices is described providing an example of a 3-dimensional Lie group. Infinitesimal generators ... More


The Linear Spencer Sequence

Ercüment H. Ortaçgil

in An Alternative Approach to Lie Groups and Geometric Structures

Published in print:
2018
Published Online:
September 2018
ISBN:
9780198821656
eISBN:
9780191860959
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821656.003.0012
Subject:
Mathematics, Pure Mathematics

The three-term nonlinear Spencer sequence constructed in Chapter 8 extends to a linear sequence. This construction explains the “deformation cohomology” of Chapter 9 and raises some intriguing ... More


Introduction

Ercüment H. Ortaçgil

in An Alternative Approach to Lie Groups and Geometric Structures

Published in print:
2018
Published Online:
September 2018
ISBN:
9780198821656
eISBN:
9780191860959
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821656.003.0001
Subject:
Mathematics, Pure Mathematics

This introductory chapter explain the unique feature of this book that makes it different from the classical books on the subject.


Geometric Integrators

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0015
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric ... More


The Old Lie

Glenn Watkins

in Proof through the Night: Music and the Great War

Published in print:
2002
Published Online:
May 2012
ISBN:
9780520231580
eISBN:
9780520927896
Item type:
chapter
Publisher:
University of California Press
DOI:
10.1525/california/9780520231580.003.0004
Subject:
Music, History, American

Just as the call to arms in 1914 triggered a search for national identity and increased the pull of patriotism, so it soon became clear that not all the voices of caution and dissent could be ... More


Matrices and Lie groups

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0002
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This chapter discusses the following: the general linear group; Lie groups; examples of Lie groups; some complex Lie groups; the three most commonly seen subgroups of Gl(n; ℂ): Sl(n; ℂ), U(n) and ... More


Maps and vector bundles

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0005
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This discussion of maps and vector bundles covers the pull-back construction; pull-backs and Grassmannians; pull-back of differential forms and push-forward of vector fields; invariant forms and ... More


The covariant derivative and the curvature

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0063
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter first considers the tangent spaces of a non-connected manifold, in which the tangent t at the set of points p in the manifold is an element of the tangent space at p. Afterward, the ... More


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