#### Jean-Michel Bismut & Jeff Cheeger

for their remarkable insights that have transformed, and continue to transform, modern geometry.

#### Contribution

The Shaw Prize in Mathematical Sciences 2021 is awarded in equal shares to Jean-Michel Bismut, Emeritus Professor, Mathematics Department, Université Paris-Saclay, France and Jeff Cheeger, Professor of Mathematics at the Courant Institute of Mathematical Sciences, New York University, USA for their remarkable insights that have transformed, and continue to transform, modern geometry.

Geometry is one of the oldest branches of mathematics, going back to the Greeks and beyond. A famous problem left open by the Greeks and not resolved until the 19th century was whether the parallel postulate, which states that, for two dimensions, given a line and a point not on that line, there is exactly one line through the point that does not meet the first line, could be deduced from Euclid’s other axioms. It was shown by Gauss, Bolyai and Lobachevsky that the answer was no, and that there are different geometries that are mathematically consistent in which the other axioms hold but the parallel postulate does not. Moreover, these non-Euclidean geometries, far from being mere curiosities, are fundamental to modern mathematics.

#### An Essay on the Prize

Geometry is one of the oldest branches of mathematics, going back to the Greeks and beyond. A famous problem left open by the Greeks and not resolved until the 19th century was whether the parallel postulate, which states that given a line in the plane and a point not on that line, there is exactly one line through the point that does not meet the first line, could be deduced from Euclid’s other axioms. It was shown by Gauss, Bolyai and Lobachevsky that the answer was no, and that there are different, mathematically consistent geometries in which the notions of Euclidean geometry such as points and lines have natural interpretations, and in these geometries the other axioms hold but the parallel postulate does not. This demonstrates that the parallel postulate cannot be a consequence of the other axioms. Moreover, these non-Euclidean geometries, far from being mere curiosities, are fundamental to modern mathematics.