“It’s one of the most well-known stories in the history of science,” says Reinhard Siegmund-Schultze, a math historian at the University of Agder in Norway. “Göttingen was so dominant in mathematics internationally.”
In 1933, that dominance came crashing down. On April 7, two months after Hitler became chancellor, Germany passed a law making it illegal for Jews — or rather those considered Jewish by the Nazis — and Communists to hold civil service jobs, with a few exceptions including for people who had served Germany in World War I. That immediately forced several Göttingen mathematicians from their jobs. The crisis snowballed, and over the course of the year, a total of 18 left or were driven out.
By the time of Hilbert’s legendary dinner with Rust, Germany had lost its status as the world’s foremost country for mathematical research. America took its place — and today, though globalization has spread the wealth, the U.S. has retained its eminence. From Princeton and Columbia to Berkeley and Stanford, it’s hard to find a great math department in the United States that was not shaped in part by European mathematicians who came to or stayed in the U.S. because of the Nazis.
There is a deep relationship between math and physics. Sometimes physicists point to new math, sometimes they mine odd and curious places for new approaches. Something is currently stirring - here's a nice non-technical description.
Feynman diagrams were devised by Richard Feynman in the 1940s. They feature lines representing elementary particles that converge at a vertex (which represents a collision) and then diverge from there to represent the pieces that emerge from the crash. Those lines either shoot off alone or converge again. The chain of collisions can be as long as a physicist dares to consider.
To that schematic physicists then add numbers, for the mass, momentum and direction of the particles involved. Then they begin a laborious accounting procedure — integrate these, add that, square this. The final result is a single number, called a Feynman probability, which quantifies the chance that the particle collision will play out as sketched.
“In some sense Feynman invented this diagram to encode complicated math as a bookkeeping device,” said Sergei Gukov, a theoretical physicist and mathematician at the California Institute of Technology.
Feynman diagrams have served physics well over the years, but they have limitations. One is strictly procedural. Physicists are pursuing increasingly high-energy particle collisions that require greater precision of measurement — and as the precision goes up, so does the intricacy of the Feynman diagrams that need to be calculated to generate a prediction.
The second limitation is of a more fundamental nature. Feynman diagrams are based on the assumption that the more potential collisions and sub-collisions physicists account for, the more accurate their numerical predictions will be. This process of calculation, known as perturbative expansion, works very well for particle collisions of electrons, where the weak and electromagnetic forces dominate. It works less well for high-energy collisions, like collisions between protons, where the strong nuclear force prevails. In these cases, accounting for a wider range of collisions — by drawing ever more elaborate Feynman diagrams — can actually lead physicists astray.
“We know for a fact that at some point it begins to diverge” from real-world physics, said Francis Brown, a mathematician at the University of Oxford. “What’s not known is how to estimate at what point one should stop calculating diagrams.”
Yet there is reason for optimism. Over the last decade physicists and mathematicians have been exploring a surprising correspondence that has the potential to breathe new life into the venerable Feynman diagram and generate far-reaching insights in both fields. It has to do with the strange fact that the values calculated from Feynman diagrams seem to exactly match some of the most important numbers that crop up in a branch of mathematics known as algebraic geometry. These values are called “periods of motives,” and there’s no obvious reason why the same numbers should appear in both settings. Indeed, it’s as strange as it would be if every time you measured a cup of rice, you observed that the number of grains was prime.
It may or may not pan out, but people continue to work.