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The study of geometry is at least 2500 years old, and it is within
this field that the concept of mathematical proof - deductive
reasoning from a set of axioms - first arose. To this day geometry
remains a very active area of research in mathematics. This Very
Short Introduction covers the areas of mathematics falling under
geometry, starting with topics such as Euclidean and non-Euclidean
geometries, and ranging to curved spaces, projective geometry in
Renaissance art, and geometry of space-time inside a black hole.
Starting from the basics, Maciej Dunajski proceeds from concrete
examples (of mathematical objects like Platonic solids, or theorems
like the Pythagorean theorem) to general principles. Throughout, he
outlines the role geometry plays in the broader context of science
and art. Very Short Introductions: Brilliant, Sharp, Inspiring
ABOUT THE SERIES: The Very Short Introductions series from Oxford
University Press contains hundreds of titles in almost every
subject area. These pocket-sized books are the perfect way to get
ahead in a new subject quickly. Our expert authors combine facts,
analysis, perspective, new ideas, and enthusiasm to make
interesting and challenging topics highly readable.
Most nonlinear differential equations arising in natural sciences
admit chaotic behavior and cannot be solved analytically.
Integrable systems lie on the other extreme. They possess regular,
stable, and well behaved solutions known as solitons and
instantons. These solutions play important roles in pure and
applied mathematics as well as in theoretical physics where they
describe configurations topologically different from vacuum. While
integrable equations in lower space-time dimensions can be solved
using the inverse scattering transform, the higher-dimensional
examples of anti-self-dual Yang-Mills and Einstein equations
require twistor theory. Both techniques rely on an ability to
represent nonlinear equations as compatibility conditions for
overdetermined systems of linear differential equations.
The book provides a self-contained and accessible introduction to
the subject. It starts with an introduction to integrability of
ordinary and partial differential equations. Subsequent chapters
explore symmetry analysis, gauge theory, gravitational instantons,
twistor transforms, and anti-self-duality equations. The three
appendices cover basic differential geometry, complex manifold
theory, and the exterior differential system.
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