An eminent mathematician reveals that his advances in the study of millennia-old mathematical questions owe to concepts derived from physics.
Quanta Magazine said:Mathematics is full of weird number systems that most people have never heard of and would have trouble even conceptualizing. But rational numbers are familiar. They’re the counting numbers and the fractions — all the numbers you’ve known since elementary school. But in mathematics, the simplest things are often the hardest to understand. They’re simple like a sheer wall, without crannies or ledges or obvious properties you can grab ahold of.
Minhyong Kim, a mathematician at the University of Oxford, is especially interested in figuring out which rational numbers solve particular kinds of equations. It’s a problem that has provoked number theorists for millennia. They’ve made minimal progress toward solving it. When a question has been studied for that long without resolution, it’s fair to conclude that the only way forward is for someone to come up with a dramatically new idea. Which is what Kim has done.
“There are not many techniques, even though we’ve been working on this for 3,000 years. So whenever anyone comes up with an authentically new way to do things it’s a big deal, and Minhyong did that,” said Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison.
Over the past decade Kim has described a very new way of looking for patterns in the seemingly patternless world of rational numbers. He’s described this method in papers and conference talks and passed it along to students who now carry on the work themselves. Yet he has always held something back. He has a vision that animates his ideas, one based not in the pure world of numbers, but in concepts borrowed from physics. To Kim, rational solutions are somehow like the trajectory of light.
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Spaces of Spaces
If you’re looking for a larger kind of space, along with clues about how to use symmetry to navigate it, physics is a good place to turn.
Generally speaking, a “space,” in the mathematical sense, is any set of points that has geometric or topological structure. One thousand points scattered willy-nilly won’t form a space — there’s no structure that ties them together. But a sphere, which is just a particularly coherent arrangement of points, is a space. So is a torus, or the two-dimensional plane, or the four-dimensional space-time in which we live.
In addition to these spaces, there exist even more exotic spaces, which you can think of as “spaces of spaces.” To take a very simple example, imagine that you have a triangle — that’s a space. Now imagine the space of all possible triangles. Each point in this larger space represents a particular triangle, with the coordinates of the point given by the angles of the triangles it represents.
That sort of idea is often useful in physics. In the framework of general relativity, space and time are constantly evolving, and physicists think of each space-time configuration as a point in a space of all space-time configurations. Spaces of spaces also come up in an area of physics called gauge theory, which has to do with fields that physicists layer on top of physical space. These fields describe how forces like electromagnetism and gravity change as you move through space. You can imagine that there’s a slightly different configuration of these fields at every point in space — and that all those different configurations together form points in a higher-dimensional “space of all fields.”
This space of fields from physics is a close analogue to what Kim is proposing in number theory. To understand why, consider a beam of light. Physicists imagine the light moving through the higher-dimensional space of fields. In this space, light will follow the path that adheres to the “principle of least action” — that is, the path that minimizes the amount of time required to go from A to B. The principle explains why light bends when it moves from one material to another — the bent path is the one that minimizes the time taken.
These larger spaces of spaces that come up in physics feature additional symmetries that are not present in any of the spaces they represent. These symmetries draw attention to specific points, emphasizing, for example, the time-minimizing path. Constructed in another way in another context, these same kinds of symmetries might emphasize other kinds of points — like the points corresponding to rational solutions to equations.
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