*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number ...
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This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.Less

This chapter deals with topological invariants and differential geometry. It first considers a topological space *X* for which singular homology and cohomology are defined, along with the Euler number *e*(*X*). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space *X*. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold *X*.

*Graham Ellis*

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0001
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology

This chapter introduces some of the basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: regular CW-complex, ...
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This chapter introduces some of the basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: regular CW-complex, non-regular CW-complex, simplicial complex, cubical complex, permutahedral complex, simple homotopy, set of path-components, fundamental group, van Kampen’s theorem, knot quandle, Alexander polynomial of a knot, covering space. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.Less

This chapter introduces some of the basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: regular CW-complex, non-regular CW-complex, simplicial complex, cubical complex, permutahedral complex, simple homotopy, set of path-components, fundamental group, van Kampen’s theorem, knot quandle, Alexander polynomial of a knot, covering space. These are illustrated using computer examples involving digital images, protein backbones, high-dimensional point cloud data, knot complements, discrete groups, and random simplicial complexes.

*Gerd Faltings*

- 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 ...
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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 the category of Higgs bundles and that of “generalized representations” of the étale fundamental group is constructed for curves over a p-adic field. The definition of “generalized representations” uses p-adic Hodge theory and almost étale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group. The method used in the proofs is the theory of almost étale extensions. A nonabelian Hodge–Tate theory is also developed.Less

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 the category of Higgs bundles and that of “generalized representations” of the étale fundamental group is constructed for curves over a *p*-adic field. The definition of “generalized representations” uses *p*-adic Hodge theory and almost étale coverings, and it includes usual representations which form a full subcategory. The equivalence depends on the choice of an exponential function for the multiplicative group. The method used in the proofs is the theory of almost étale extensions. A nonabelian Hodge–Tate theory is also developed.

*Takeshi Tsuji*

- 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 ...
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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 link between classical crystals and modules with integrable connections. After discussing Higgs isocrystals and Higgs crystals, cohomology of Higgs isocrystals, and representations of the fundamental group, the chapter presents the main result: the construction of a fully faithful functor from the category of Higgs (iso)crystals satisfying an overconvergence condition to that of small generalized representations. It also proves the compatibility of this functor with the natural cohomologies and concludes by comparing the cohomology of Higgs isocrystals with Faltings cohomology.Less

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 link between classical crystals and modules with integrable connections. After discussing Higgs isocrystals and Higgs crystals, cohomology of Higgs isocrystals, and representations of the fundamental group, the chapter presents the main result: the construction of a fully faithful functor from the category of Higgs (iso)crystals satisfying an overconvergence condition to that of small generalized representations. It also proves the compatibility of this functor with the natural cohomologies and concludes by comparing the cohomology of Higgs isocrystals with Faltings cohomology.