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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


The Many Facets of Geometry: A Tribute to Nigel Hitchin

Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon (eds)

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199534920
eISBN:
9780191716010
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199534920.001.0001
Subject:
Mathematics, Geometry / Topology

Few people have proved more influential in the field of differential and algebraic geometry, and in showing how this links with mathematical physics, than Nigel Hitchin. Oxford University's Savilian ... More


Various Generalizations and Deformations of PSL(2,ℝ) Surface Group Representations and their Higgs Bundles

Brian Collier

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0019
Subject:
Mathematics, Geometry / Topology

The goal of this chapter is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus g into PSL(2, R) and their associated Higgs bundles ... More


S‐Duality in HyperkäHler Hodge Theory

TamáS Hausel

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/9780199534920.003.0016
Subject:
Mathematics, Geometry / Topology

This chapter surveys the motivations, related results, and progress made towards the following problem, raised by Hitchin in 1995: What is the space of L2 harmonic forms on the moduli space of Higgs ... 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


Parabolic Higgs Bundles for Real Reductive Lie Groups: A Very Basic Introduction

Ignasi Mundet i Riera

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0027
Subject:
Mathematics, Geometry / Topology

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter ... More


Involutions of Rank 2 Higgs Bundle Moduli Spaces

Oscar García-Prada and S. Ramanan

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0022
Subject:
Mathematics, Geometry / Topology

This chapter considers the moduli space of rank 2 Higgs bundles with fixed determinant over a smooth projective curve X of genus 2 over ℂ, and studies involutions defined by tensoring the vector ... More


Mirror Symmetry with Branes by Equivariant Verlinde Formulas

Tamás Hausel, Anton Mellit, and Du Pei

in Geometry and Physics: Volume I: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802013
eISBN:
9780191840500
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802013.003.0009
Subject:
Mathematics, Geometry / Topology

This chapter finds an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL2 Higgs moduli space on a Riemann surface. On one side of the agreement, ... More


Facsimile : A p-adic Simpson correspondence

Gerd Faltings

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/9780691170282.003.0007
Subject:
Mathematics, Algebra

This chapter presents the facsimile of Gerd Faltings' article entitled “A p-adic Simpson Correspondence,” reprinted from Advances in Mathematics 198(2), 2005. In this article, an equivalence between ... More


Mathematical Work of Nigel Hitchin

Sir Michael Atiyah

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/9780199534920.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter reviews selected topics covered in Nigel Hitchin's papers. These include his introduction of the spectral curve of a magnetic monopole, the introduction by Nigel of ‘Higgs bundles’ in ... More


Irreducibility of Moduli of Semi-Stable Chains and Applications to U(p, q)-Higgs Bundles

Steven Bradlow, Oscar García-Prada, Peter Gothen, and Jochen Heinloth

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0018
Subject:
Mathematics, Geometry / Topology

This chapter gives necessary and sufficient conditions for moduli spaces of semi-stable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the ... More


Representations of the fundamental group and the torsor of deformations. An overview

Ahmed Abbes and Michel Gros

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/9780691170282.003.0001
Subject:
Mathematics, Algebra

This chapter provides an overview of a new approach to the p-adic Simpson correspondence, focusing on representations of the fundamental group and the torsor of deformations. The discussion covers ... More


SL(∞,ℝ), Higgs Bundles and Quantization

Olivier Biquard

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0016
Subject:
Mathematics, Geometry / Topology

Nigel Hitchin recently proposed a theory of SL(∞, R)-Higgs bundles which should parametrize a Hitchin component of representations of surface groups into SL(∞, R). This chapter discusses some ... More


Brauer Group of Moduli of Higgs Bundles and Connections

David Baraglia, Indranil Biswas, and Laura P. Schaposnik

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0014
Subject:
Mathematics, Geometry / Topology

Given a compact Riemann surface X and a semi-simple affine algebraic group G defined over C, there are moduli spaces of Higgs bundles and of connections associated to (X, G). The chapter computes the ... More


The p-adic Simpson Correspondence (AM-193)

Ahmed Abbes, Michel Gros, and Takeshi Tsuji

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691170282
eISBN:
9781400881239
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691170282.001.0001
Subject:
Mathematics, Algebra

The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms ... More


Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada (eds)

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.001.0001
Subject:
Mathematics, Geometry / Topology

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and ... More


Geometry and Physics: Volume I: A Festschrift in honour of Nigel Hitchin

Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada (eds)

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802013
eISBN:
9780191840500
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802013.001.0001
Subject:
Mathematics, Geometry / Topology

These volumes contain the proceedings of the conference held at Aarhus, Oxford and Madrid in September 2016 to mark the seventieth birthday of Nigel Hitchin, one of the world’s foremost geometers and ... More


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