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Area-preserving diffeomorphisms

Dusa McDuff and Dietmar Salamon.

in Introduction to Symplectic Topology

March 2017; p ublished online June 2017 .

Chapter. Subjects: Mathematics; Geometry. 6582 words.

This chapter is devoted to two-dimensional symplectomorphisms, which are just area- and orientation-preserving diffeomorphisms. The chapter includes an exposition of Birkhoff’s proof of...

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Arithmetic Compactifications of PEL-Type Shimura Varieties

Kai-Wen Lan.

March 2013; p ublished online October 2017 .

Book. Subjects: Geometry. 584 pages.

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the...

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An Arithmetical Analogue

Umberto Zannier and David Masser.

in Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

March 2012; p ublished online October 2017 .

Chapter. Subjects: Geometry. 14707 words.

This chapter slightly moves in another direction to describe a purely arithmetical analogue of Theorem 1.3. This material was not mentioned in the lectures, for lack of time and because of...

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The arnold conjecture

Dusa McDuff and Dietmar Salamon.

in Introduction to Symplectic Topology

March 2017; p ublished online June 2017 .

Chapter. Subjects: Mathematics; Geometry. 21002 words.

This chapter contains a proof of the Arnold conjecture for the standard torus, which is based on the discrete symplectic action. The symplectic part of this proof is very easy. However, for...

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Asymptotic Dimension

Greg Bell.

in Office Hours with a Geometric Group Theorist

July 2017; p ublished online May 2018 .

Chapter. Subjects: Geometry. 7789 words.

This chapter considers the notion of asymptotic dimension. It first provides an overview of asymptotic dimension before explaining the asymptotic dimensions of free abelian and nonabelian...

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Automorphisms of Free Groups

Matt Clay.

in Office Hours with a Geometric Group Theorist

July 2017; p ublished online May 2018 .

Chapter. Subjects: Geometry. 7820 words.

This chapter discusses the automorphisms of free groups. Every group is the collection of symmetries of some object, namely, its Cayley graph. A symmetry of a group is called an...

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Auxiliary Results

Isroil A. Ikromov and Detlef Müller.

in Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (AM-194)

May 2016; p ublished online October 2017 .

Chapter. Subjects: Geometry. 13006 words.

This chapter compiles various auxiliary results; including variants of van der Corput-type estimates for one-dimensional oscillatory integrals and related sublevel estimates through...

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Basic definitions

Simon Donaldson.

in Riemann Surfaces

March 2011; p ublished online December 2013 .

Chapter. Subjects: Geometry. 6116 words.

This chapter presents definitions of Riemann surfaces and holomorphic maps, and provides examples, including algebraic curves and quotients.

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Boundary Value Problems for the Lorentzian Dirac Operator

Christian Bär and Sebastian Hannes.

in Geometry and Physics: Volume I

October 2018; p ublished online December 2018 .

Chapter. Subjects: Geometry. 8759 words.

On a compact globally hyperbolic Lorentzian spin manifold with smooth space-like Cauchy boundary, the (hyperbolic) Dirac operator is known to be Fredholm when Atiyah–Patodi–Singer boundary...

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Boundary Value Problems in Dimensions 7, 4 and 3 Related to Exceptional Holonomy

Simon Donaldson.

in Geometry and Physics: Volume I

October 2018; p ublished online December 2018 .

Chapter. Subjects: Geometry. 10030 words.

The variational point of view on exceptional structures in dimensions 6, 7 and 8 is one of Nigel Hitchin’s seminal contributions. One feature of this point of view is that it motivates the...

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