## 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