Eugene Wigner to wonder what accounts for the unreasonable effectiveness of mathematics in physics.

A more recent version of the same basic pattern is the Yang-Mills theory. Here again the physicists were struggling to develop a mathematical framework to handle the physical concepts they were developing, when in fact the mathematical framework, which in mathematics is known as connections on principal bundles and curvature, had already been introduced for mathematical reasons. Much of the recent history of quantum field theory has turned this model of interaction on its head. When quantum field theory was introduced in the 1940s and 1950s there was no appropriate mathematical context. Nevertheless, physicists were able to develop the art of dealing with these objects, at least in special cases. This line of reasoning, using as a central feature the Yang-Mills theory, led to the standard model of theoretical physics, which makes predictions that have been checked by experiment to enormous precision. Nevertheless, there was not then and still is not today a rigorous mathematical context for these computations. The situation became even worse with the advent of string theory, where the appropriate mathematical formulation seems even more remote. But the fact that the mathematical context for these theories did not exist and has not yet been developed is only part of the way that the current interactions between mathematics and physics differ from previous ones. As physicists develop and explore these theories, for which no rigorous mathematical formulation is known, they have increasingly used ever more sophisticated geometric and topological structures in their theories. As physicists explore these theories they come across mathematical questions and statements about the underlying geometric and topological objects in terms of which the theories are defined. Some of these statements are well-known mathematical results, but many turn out to be completely new types of mathematical statements.

These statements, conjectures, and questions have been one of the main forces driving geometry and topology for the last 20 to 25 years. Some of them have been successfully verified mathematically; some have not been proved but the mathematical evidence for them is overwhelming; and some are completely mysterious mathematically. One of the first results along these lines gave a formula for the number of lines of each degree in a general hypersurface in complex projective four-dimensional space given by a homogeneous degree-5 polynomial. Physics arguments produced a general formula for the number of such lines where the formula came from a completely different area of mathematics (power series solutions to certain ordinary differential equations). Before the input from physics, mathematicians had computed the answer for degrees 1 through 5 but had no conjecture for the general answer. The physics arguments provided the general formula, and this was later verified by direct mathematical argument.



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