The Resource Mostly surfaces, Richard Evan Schwartz

Mostly surfaces, Richard Evan Schwartz

Label
Mostly surfaces
Title
Mostly surfaces
Statement of responsibility
Richard Evan Schwartz
Creator
Subject
Language
eng
Summary
The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description
Member of
Cataloging source
DLC
http://library.link/vocab/creatorName
Schwartz, Richard Evan
Illustrations
illustrations
Index
index present
LC call number
QA571
LC item number
.S385 2011
Literary form
non fiction
Nature of contents
bibliography
Series statement
Student mathematical library
Series volume
v. 60
http://library.link/vocab/subjectName
  • Hypersurfaces
  • Riemann surfaces
  • Surfaces, Algebraic
  • Hypersurfaces
  • Riemann surfaces
  • Surfaces, Algebraic
  • Fläche
  • Hyperfläche
  • Riemannsche Fläche
  • Oberfläche
  • Algebraic geometry
  • Functions of a complex variable
  • Several complex variables and analytic spaces
  • Dynamical systems and ergodic theory
  • Differential geometry
  • Geometry
  • Several complex variables and analytic spaces
Label
Mostly surfaces, Richard Evan Schwartz
Instantiates
Publication
Bibliography note
Includes bibliographical references (p. 309-310) and index
Carrier category
volume
Carrier category code
nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
  • Cone surfaces and translation surfaces
  • The modular group and the veech group
  • Moduli space
  • Dessert
  • Part 1.
  • Surfaces and topology
  • Chapter 2.
  • Definition of a surface
  • A word about sets
  • Metric spaces
  • Chapter 1.
  • Open and closed sets
  • Continuous maps
  • Homeomorphisms
  • Compactness
  • Surfaces
  • Manifolds
  • Chapter 3.
  • The gluing construction
  • Gluing spaces together
  • The gluing construction in action
  • Book overview
  • The classification of surfaces
  • The Euler characteristic
  • Chapter 4.
  • The fundamental group
  • A primer on groups
  • Homotopy equivalence
  • The fundamental group
  • Changing the basepoint
  • Functoriality
  • Some first steps
  • Behold, the torus!
  • Chapter 5.
  • Examples of fundamental groups
  • The Winding number
  • The Circle
  • The Fundamental theorem of algebra
  • The Torus
  • The 2-sphere
  • The projective plane
  • A lens space
  • The poincaré homology sphere
  • Gluing polygons
  • Chapter 6.
  • Covering spaces and the deck group
  • Covering spaces
  • The deck group
  • A flat torus
  • Drawing on a surface
  • Covering spaces
  • Hyperbolic geometry and the octagon
  • Complex analysis and Riemann surfaces
  • The main result
  • The covering property
  • Simple connectivity
  • Part 2.
  • Surfaces and geometry
  • Chapter 8.
  • Euclidean geometry
  • Euclidean space
  • The Pythagorean theorem
  • The X theorem
  • More examples
  • Pick's theorem
  • The polygon dissection theorem
  • Line integrals
  • Green's theorem for polygons
  • Chapter 9.
  • Spherical geometry
  • Metrics, tangent planes, and isometries
  • Geodesics
  • Geodesic triangles
  • Convexity
  • Simply connected spaces
  • Stereographic projection
  • The hairy ball theorem
  • Chapter 10.
  • Hyperbolic geometry
  • Linear fractional transformations
  • Circle preserving property
  • The upper half-plane model
  • Another point of view
  • Symmetries
  • Geodesics
  • The isomorphism theorem
  • The disk model
  • Geodesic polygons
  • Classification of isometries
  • Chapter 11.
  • Riemannian metrics on surfaces
  • Curves in the plane
  • Riemannian metrics on the plane
  • Diffeomorphisms and isometries
  • Atlases and smooth surfaces
  • Smooth curves and the tangent plane
  • The Bolzano-Weierstrass theorem
  • Riemannian surfaces
  • Chapter 12.
  • Hyperbolic surfaces
  • Definition
  • Gluing recipes
  • Gluing recipes lead to surfaces
  • Some examples
  • Geodesic triangulations
  • Riemannian covers
  • Hadamard's theorem
  • The lifting property
  • The hyperbolic cover
  • Part 3.
  • Surfaces and complex analysis
  • Chapter 13.
  • A primer on complex analysis
  • Basic definitions
  • Cauchy's theorem
  • The Cauchy integral formula
  • Differentiability
  • Proof of the isomorphism theorem
  • Chapter 7.
  • Existence of universal covers
  • Stereographic projection revisited
  • Chapter 15.
  • The Schwarz-Christoffel transformation
  • The basic construction
  • The inverse function theorem
  • Proof of theorem 15.1
  • The range of possibilities
  • Invariance of domain
  • The existence proof
  • Chapter 16.
  • The maximum principle
  • Riemann surface and uniformization
  • Riemann surfaces
  • Maps between Riemann surfaces
  • The Riemann mapping theorem
  • The uniformization theorem
  • The small Picard theorem
  • Implications for compact surfaces
  • Part 4.
  • Flat cone surfaces
  • Chapter 17.
  • Removable singularities
  • Flat cone surfaces
  • Sectors and Euclidean cones
  • Euclidean cone surfaces
  • The gauss-bonnet theorem
  • Translation surfaces
  • Billiards and translation surfaces
  • Special maps on a translation surface
  • Existence of periodic billiard paths
  • Chapter 18.
  • Translation surfaces and the Veech group
  • Power series
  • Affine automorphisms
  • The diffential representation
  • Hyperbolic group actions
  • Proof of theorem 18.1
  • Triangle groups
  • Linear and hyperbolic reflections
  • Behold, the double octagon!
  • Part 5.
  • The totality of surfaces
  • Chapter 19.
  • Taylor series
  • Continued fractions
  • The Gauss map
  • Continued fractions
  • The farey graph
  • Structure of the modular group
  • Continued fractions and the farey graph
  • The Irrational case
  • Chapter 20.
  • Teichmüller space and Moduli space
  • Parallelograms
  • Chapter 14.
  • Flat tori
  • The modular group again
  • Moduli space
  • Teichmüller group again
  • The mapping class group
  • 21.
  • Topology of teichmüller space
  • Pairs of pants
  • Pants decompositions
  • Special maps and triples
  • Disk and plane rigidity
  • Disk rigidity
  • Liouville's theorem
  • Depleted balls
  • The depleted ball theorem
  • The injective homomorphism
  • Chapter 23.
  • Dehn's dissection theorem
  • The result
  • Dihedral angles
  • Irrationality proof
  • Rational vector spaces
  • Dehn's invariant
  • The end of the proof
  • Clean dissections
  • The proof
  • Chapter 24.
  • The Cauchy rigidity theorem
  • The main result
  • The dual graph
  • Outline of the proof
  • Proof of Lemma 24.3
  • Proof of Lemma 24.2
  • Euclidean intuition does not work
  • Part 6.
  • Proof of Cauchy's arm Lemma
  • Dessert.
  • Chapter 22.
  • The Banach-Tarski theorem
  • The result
  • The Schroeder-Bernstein theorem
  • The doubling theorem
Dimensions
22 cm.
Extent
xiii, 314 pages
Isbn
9780821853689
Isbn Type
(pbk. : alk. paper)
Lccn
2011005544
Media category
unmediated
Media MARC source
rdamedia
Media type code
n
Other physical details
illustrations
System control number
  • (OCoLC)706677482
  • (OCoLC)ocn706677482
Label
Mostly surfaces, Richard Evan Schwartz
Publication
Bibliography note
Includes bibliographical references (p. 309-310) and index
Carrier category
volume
Carrier category code
nc
Carrier MARC source
rdacarrier
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
  • Cone surfaces and translation surfaces
  • The modular group and the veech group
  • Moduli space
  • Dessert
  • Part 1.
  • Surfaces and topology
  • Chapter 2.
  • Definition of a surface
  • A word about sets
  • Metric spaces
  • Chapter 1.
  • Open and closed sets
  • Continuous maps
  • Homeomorphisms
  • Compactness
  • Surfaces
  • Manifolds
  • Chapter 3.
  • The gluing construction
  • Gluing spaces together
  • The gluing construction in action
  • Book overview
  • The classification of surfaces
  • The Euler characteristic
  • Chapter 4.
  • The fundamental group
  • A primer on groups
  • Homotopy equivalence
  • The fundamental group
  • Changing the basepoint
  • Functoriality
  • Some first steps
  • Behold, the torus!
  • Chapter 5.
  • Examples of fundamental groups
  • The Winding number
  • The Circle
  • The Fundamental theorem of algebra
  • The Torus
  • The 2-sphere
  • The projective plane
  • A lens space
  • The poincaré homology sphere
  • Gluing polygons
  • Chapter 6.
  • Covering spaces and the deck group
  • Covering spaces
  • The deck group
  • A flat torus
  • Drawing on a surface
  • Covering spaces
  • Hyperbolic geometry and the octagon
  • Complex analysis and Riemann surfaces
  • The main result
  • The covering property
  • Simple connectivity
  • Part 2.
  • Surfaces and geometry
  • Chapter 8.
  • Euclidean geometry
  • Euclidean space
  • The Pythagorean theorem
  • The X theorem
  • More examples
  • Pick's theorem
  • The polygon dissection theorem
  • Line integrals
  • Green's theorem for polygons
  • Chapter 9.
  • Spherical geometry
  • Metrics, tangent planes, and isometries
  • Geodesics
  • Geodesic triangles
  • Convexity
  • Simply connected spaces
  • Stereographic projection
  • The hairy ball theorem
  • Chapter 10.
  • Hyperbolic geometry
  • Linear fractional transformations
  • Circle preserving property
  • The upper half-plane model
  • Another point of view
  • Symmetries
  • Geodesics
  • The isomorphism theorem
  • The disk model
  • Geodesic polygons
  • Classification of isometries
  • Chapter 11.
  • Riemannian metrics on surfaces
  • Curves in the plane
  • Riemannian metrics on the plane
  • Diffeomorphisms and isometries
  • Atlases and smooth surfaces
  • Smooth curves and the tangent plane
  • The Bolzano-Weierstrass theorem
  • Riemannian surfaces
  • Chapter 12.
  • Hyperbolic surfaces
  • Definition
  • Gluing recipes
  • Gluing recipes lead to surfaces
  • Some examples
  • Geodesic triangulations
  • Riemannian covers
  • Hadamard's theorem
  • The lifting property
  • The hyperbolic cover
  • Part 3.
  • Surfaces and complex analysis
  • Chapter 13.
  • A primer on complex analysis
  • Basic definitions
  • Cauchy's theorem
  • The Cauchy integral formula
  • Differentiability
  • Proof of the isomorphism theorem
  • Chapter 7.
  • Existence of universal covers
  • Stereographic projection revisited
  • Chapter 15.
  • The Schwarz-Christoffel transformation
  • The basic construction
  • The inverse function theorem
  • Proof of theorem 15.1
  • The range of possibilities
  • Invariance of domain
  • The existence proof
  • Chapter 16.
  • The maximum principle
  • Riemann surface and uniformization
  • Riemann surfaces
  • Maps between Riemann surfaces
  • The Riemann mapping theorem
  • The uniformization theorem
  • The small Picard theorem
  • Implications for compact surfaces
  • Part 4.
  • Flat cone surfaces
  • Chapter 17.
  • Removable singularities
  • Flat cone surfaces
  • Sectors and Euclidean cones
  • Euclidean cone surfaces
  • The gauss-bonnet theorem
  • Translation surfaces
  • Billiards and translation surfaces
  • Special maps on a translation surface
  • Existence of periodic billiard paths
  • Chapter 18.
  • Translation surfaces and the Veech group
  • Power series
  • Affine automorphisms
  • The diffential representation
  • Hyperbolic group actions
  • Proof of theorem 18.1
  • Triangle groups
  • Linear and hyperbolic reflections
  • Behold, the double octagon!
  • Part 5.
  • The totality of surfaces
  • Chapter 19.
  • Taylor series
  • Continued fractions
  • The Gauss map
  • Continued fractions
  • The farey graph
  • Structure of the modular group
  • Continued fractions and the farey graph
  • The Irrational case
  • Chapter 20.
  • Teichmüller space and Moduli space
  • Parallelograms
  • Chapter 14.
  • Flat tori
  • The modular group again
  • Moduli space
  • Teichmüller group again
  • The mapping class group
  • 21.
  • Topology of teichmüller space
  • Pairs of pants
  • Pants decompositions
  • Special maps and triples
  • Disk and plane rigidity
  • Disk rigidity
  • Liouville's theorem
  • Depleted balls
  • The depleted ball theorem
  • The injective homomorphism
  • Chapter 23.
  • Dehn's dissection theorem
  • The result
  • Dihedral angles
  • Irrationality proof
  • Rational vector spaces
  • Dehn's invariant
  • The end of the proof
  • Clean dissections
  • The proof
  • Chapter 24.
  • The Cauchy rigidity theorem
  • The main result
  • The dual graph
  • Outline of the proof
  • Proof of Lemma 24.3
  • Proof of Lemma 24.2
  • Euclidean intuition does not work
  • Part 6.
  • Proof of Cauchy's arm Lemma
  • Dessert.
  • Chapter 22.
  • The Banach-Tarski theorem
  • The result
  • The Schroeder-Bernstein theorem
  • The doubling theorem
Dimensions
22 cm.
Extent
xiii, 314 pages
Isbn
9780821853689
Isbn Type
(pbk. : alk. paper)
Lccn
2011005544
Media category
unmediated
Media MARC source
rdamedia
Media type code
n
Other physical details
illustrations
System control number
  • (OCoLC)706677482
  • (OCoLC)ocn706677482

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