6. Other geometries

‘Other geometries’ offers an overview of some modern geometries and their links with other areas of mathematics, such as Gauss lemma in number theory and Gaussian distributions in statistics. The Atiyah–Singer index theorem, proved in 1963 by Michael Atiyah and Isadore Singer, is a celebrated result of geometry in the last 60 years. Felix Klein had the idea of defining geometry as a pair consisting of a space a two-dimensional plane and a group of all affine transformations of the plane. Grigori Perelman’s proof showed the Poincare conjecture, which was formulated in the early 20th century by French mathematician Henri Poincaré.