Abstract
We review the remarkably fruitful interactions between mathematics and quantum physics in the past decades, pointing out some general trends and highlighting several examples, such as the counting of curves in algebraic geometry, invariants of knots and four-dimensional topology.
Highlights
The relation between mathematics and physics is one with a long tradition going back thousands of years and originating, to a great extent, in the great mystery of the cosmos as seen by shepherds on starry nights
Astronomy in the hands of Galileo ushered in the modern scientific era, and it was Galileo who said that the book of nature is written in the language of mathematics (Drake 1957, p. 237): Philosophy is written in this grand book, the universe, which stands continually open to our gaze
The book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth
Summary
The relation between mathematics and physics is one with a long tradition going back thousands of years and originating, to a great extent, in the great mystery of the cosmos as seen by shepherds on starry nights. At this stage both gravitation and electromagnetism were formulated as field theories in four-dimensional space– time, and this fusion of geometry and classical physics provided a strong stimulus to mathematicians in the field of differential geometry. The picture began to change around 1955, ironically the year of Einstein’s death, with the advent of the Yang–Mills equations, which showed that particle physics could be treated by the same kind of geometry as Maxwell’s theory, but with quantum mechanics playing a dominant role. It is a remarkable achievement that all the building blocks of this theory can be formulated in terms of geometrical concepts such as vector bundles, connections, curvatures, covariant derivatives and spinors This combination of geometrical field theory with quantum mechanics worked well for the structure of matter but seemed to face a brick wall when confronted with general relativity and gravitation. The most promising candidate for a solution to this problem is string theory, and the mathematical development of this theory, in its various forms, has been pursued with remarkable vigour and some success
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
