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Conformal Dimension: Theory and Application

John M. Mackay University of Illinois at Urbana-Champaign, Urbana, IL
Jeremy T. Tyson University of Illinois at Urbana-Champaign, Urbana, IL
Available Formats:
Softcover ISBN: 978-0-8218-5229-3
Product Code: ULECT/54
List Price: $47.00 MAA Member Price:$42.30
AMS Member Price: $37.60 Electronic ISBN: 978-1-4704-1649-2 Product Code: ULECT/54.E List Price:$44.00
MAA Member Price: $39.60 AMS Member Price:$35.20
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List Price: $70.50 MAA Member Price:$63.45
AMS Member Price: $56.40 Click above image for expanded view Conformal Dimension: Theory and Application John M. Mackay University of Illinois at Urbana-Champaign, Urbana, IL Jeremy T. Tyson University of Illinois at Urbana-Champaign, Urbana, IL Available Formats:  Softcover ISBN: 978-0-8218-5229-3 Product Code: ULECT/54  List Price:$47.00 MAA Member Price: $42.30 AMS Member Price:$37.60
 Electronic ISBN: 978-1-4704-1649-2 Product Code: ULECT/54.E
 List Price: $44.00 MAA Member Price:$39.60 AMS Member Price: $35.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$70.50 MAA Member Price: $63.45 AMS Member Price:$56.40
• Book Details

University Lecture Series
Volume: 542010; 143 pp
MSC: Primary 30; 28;

Conformal dimension measures the extent to which the Hausdorff dimension of a metric space can be lowered by quasisymmetric deformations. Introduced by Pansu in 1989, this concept has proved extremely fruitful in a diverse range of areas, including geometric function theory, conformal dynamics, and geometric group theory.

This survey leads the reader from the definitions and basic theory through to active research applications in geometric function theory, Gromov hyperbolic geometry, and the dynamics of rational maps, amongst other areas. It reviews the theory of dimension in metric spaces and of deformations of metric spaces. It summarizes the basic tools for estimating conformal dimension and illustrates their application to concrete problems of independent interest. Numerous examples and proofs are provided.

Working from basic definitions through to current research areas, this book can be used as a guide for graduate students interested in this field, or as a helpful survey for experts. Background needed for a potential reader of the book consists of a working knowledge of real and complex analysis on the level of first- and second-year graduate courses.

Graduate students and research mathematicians interested in geometric function theory.

• Chapters
• Chapter 1. Background material
• Chapter 2. Conformal gauges and conformal dimension
• Chapter 3. Gromov hyperbolic groups and spaces and their boundaries
• Chapter 4. Lower bounds for conformal dimension
• Chapter 5. Sets and spaces of conformal dimension zero
• Chapter 6. Gromov–Hausdorff tangent spaces and conformal dimension
• Chapter 7. Ahlfors regular conformal dimension
• Chapter 8. Global quasiconformal dimension

• Requests

Review Copy – for reviewers who would like to review an AMS book
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Accessibility – to request an alternate format of an AMS title
Volume: 542010; 143 pp
MSC: Primary 30; 28;

Conformal dimension measures the extent to which the Hausdorff dimension of a metric space can be lowered by quasisymmetric deformations. Introduced by Pansu in 1989, this concept has proved extremely fruitful in a diverse range of areas, including geometric function theory, conformal dynamics, and geometric group theory.

This survey leads the reader from the definitions and basic theory through to active research applications in geometric function theory, Gromov hyperbolic geometry, and the dynamics of rational maps, amongst other areas. It reviews the theory of dimension in metric spaces and of deformations of metric spaces. It summarizes the basic tools for estimating conformal dimension and illustrates their application to concrete problems of independent interest. Numerous examples and proofs are provided.

Working from basic definitions through to current research areas, this book can be used as a guide for graduate students interested in this field, or as a helpful survey for experts. Background needed for a potential reader of the book consists of a working knowledge of real and complex analysis on the level of first- and second-year graduate courses.

Graduate students and research mathematicians interested in geometric function theory.

• Chapters
• Chapter 1. Background material
• Chapter 2. Conformal gauges and conformal dimension
• Chapter 3. Gromov hyperbolic groups and spaces and their boundaries
• Chapter 4. Lower bounds for conformal dimension
• Chapter 5. Sets and spaces of conformal dimension zero
• Chapter 6. Gromov–Hausdorff tangent spaces and conformal dimension
• Chapter 7. Ahlfors regular conformal dimension
• Chapter 8. Global quasiconformal dimension
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.