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The Fourth Janko Group

Alexander A. Ivanov

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198527596
eISBN:
9780191713163
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198527596.001.0001
Subject:
Mathematics, Pure Mathematics

This book illustrates how different methods of finite group theory including representation theory, cohomology theory, combinatorial group theory, and local analysis, are combined to construct one of ... More


Derived Category and Canonical Bundle --II

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199296866
eISBN:
9780191711329
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296866.003.0006
Subject:
Mathematics, Geometry / Topology

Based on the work of Orlov, Kawamata, and others, this chapter shows that the (numerical) Kodaira dimension and the canonical ring are preserved under derived equivalence. The same techniques can be ... More


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


Foliations

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

Published in print:
2007
Published Online:
January 2008
ISBN:
9780198564959
eISBN:
9780191713712
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198564959.003.0003
Subject:
Mathematics, Geometry / Topology

As Sasakian manifolds are all examples of Riemannian foliations with one-dimensional leaves, this chapter goes into the world of foliations with a particular focus on the Riemannian case. Haeiger ... More


Fundamentals of Orbifolds

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

Published in print:
2007
Published Online:
January 2008
ISBN:
9780198564959
eISBN:
9780191713712
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198564959.003.0005
Subject:
Mathematics, Geometry / Topology

A key tool that allows for connecting Sasakian structures to other geometric structures is the theory of Kähler orbifolds and their orbifold bundles or ‘orbibundles’. Orbifolds, just as manifolds, ... More


K-Contact and Sasakian Structures

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

Published in print:
2007
Published Online:
January 2008
ISBN:
9780198564959
eISBN:
9780191713712
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198564959.003.0008
Subject:
Mathematics, Geometry / Topology

This chapter begins the study of Sasakian geometry, where all important structure theorems are presented. It then gathers all the results concerning the geometry, topology, and curvature properties ... More


PENTAD GROUP 23+12 · (L 3(2) × Sym5)

A. A. Ivanov

in The Fourth Janko Group

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198527596
eISBN:
9780191713163
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198527596.003.0004
Subject:
Mathematics, Pure Mathematics

This chapter discusses the geometric subgroup at level 2 of the modified amalgam G, known as the pentad group, which has a rather complicated structure. The structure can be described in terms of the ... More


COHOMOLOGICAL METHODS IN INFINITE SOLUBLE GROUP THEORY

John C. Lennox and Derek J. S. Robinson

in The Theory of Infinite Soluble Groups

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198507284
eISBN:
9780191709326
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507284.003.0010
Subject:
Mathematics, Pure Mathematics

This chapter talks about cohomological methods in soluble group theory, soluble groups with finite (co)homological dimension, vanishing theorems, applications to near splitting and conjugacy, and ... More


HOMOTOPY AND HOMOLOGY

Andrew Ranicki

in Algebraic and Geometric Surgery

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198509240
eISBN:
9780191708725
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509240.003.0003
Subject:
Mathematics, Geometry / Topology

This chapter serves as a recollection of basic algebraic topology, including mention of homology and cohomology, and homotopy groups. It details intersections and cup product pairing. Attaching cells ... More


An Introduction to Bundle Gerbes

Michael K. Murray

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/9780199534920.003.0012
Subject:
Mathematics, Geometry / Topology

This chapter focuses on bundle gerbes, which provide a geometric realisation of the three dimensional cohomology of a manifold. It shows that the pattern of a p-gerbe being defined as a (p - 1)-gerbe ... More


The Hodge star

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/9780199605880.003.0019
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This chapter first provides a definition of the Hodge star. It then discusses representatives of De Rham cohomology; presents the Hodge theorem; and explains the notion of self-duality.


Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

Claire Voisin

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160504
eISBN:
9781400850532
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160504.001.0001
Subject:
Mathematics, Geometry / Topology

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. ... More


Hodge Theory (MN-49)

Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161341
eISBN:
9781400851478
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161341.001.0001
Subject:
Mathematics, Geometry / Topology

This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The ... More


The Hodge Theory of Maps

Mark Andrea de Cataldo, Luca Migliorini Lectures 4–5, and Mark Andrea de Cataldo

Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)

in Hodge Theory (MN-49)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161341
eISBN:
9781400851478
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161341.003.0006
Subject:
Mathematics, Geometry / Topology

This chapter showcases two further lectures on the Hodge theory of maps, and they are mostly composed of exercises. The first lecture details a minimalist approach to sheaf cohomology, and then turns ... More


Differential geometry, topology and fibre bundles

Reinhold A. Bertlmann

in Anomalies in Quantum Field Theory

Published in print:
2000
Published Online:
January 2010
ISBN:
9780198507628
eISBN:
9780191706400
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507628.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces all necessary mathematical concepts. Section 2.1 briefly summarizes some topological definitions. Section 2.2 explains the homotopy of maps and the homotopy of groups. Section ... More


The de Rham Cohomology of an LLG

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/9780198821656.003.0011
Subject:
Mathematics, Pure Mathematics

This chapter is a preparation for Chapter 11, which is a continuation of Chapter 9. It reformulates some known facts in the new setup.


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


From Sheaf Cohomology to the Algebraic de Rham Theorem

Fouad El Zein and Loring W. Tu

Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng (eds)

in Hodge Theory (MN-49)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161341
eISBN:
9781400851478
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161341.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter proves Grothendieck's algebraic de Rham theorem. It first proves Grothendieck's algebraic de Rham theorem more or less from scratch for a smooth complex projective variety X, namely, ... More


Cohomology of Higgs isocrystals

Takeshi Tsuji

in The p-adic Simpson Correspondence (AM-193)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691170282
eISBN:
9781400881239
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691170282.003.0004
Subject:
Mathematics, Algebra

This chapter describes the cohomology of Higgs isocrystals, which are introduced to replace the notion of Higgs bundles. The link between these two notions uses Higgs envelopes and calls to mind the ... More


Almost étale coverings

Ahmed Abbes, Michel Gros, and Takeshi Tsuji

in The p-adic Simpson Correspondence (AM-193)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691170282
eISBN:
9781400881239
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691170282.003.0005
Subject:
Mathematics, Algebra

This chapter explains Faltings' theory of almost étale extensions, a tool that has become essential in many questions in arithmetic geometry, even beyond p-adic Hodge theory. It begins with a brief ... More


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