Physics and number theory - the math of Minhyong Kim

I met Minhyong in 1987 when I was a postdoc at Yale and he was a grad student there. He came to my course on nonlinear wave equations. He and his roommates started a tradition of "ice cream math socials". We had a huge amount of fun talking about math, physics and everything else. We've been doing it ever since, whenever we can. Now he's at Oxford, and I last saw him in his room at Merton College, which is about the closest thing to Hogwarts Academy that really exists. (They've got secret passages, and a library with medieval books in Latin chained to the desks.)

As a grad student he started out working with Greg Zuckerman on math connected to string theory, but then - much to my shock - he switched to working with Serge Lang on number theory. Ever since, he's done number theory influenced by physics.

If the connection sounds fantastical it’s because it is, even to mathematicians. And for that reason, Kim long kept it to himself. “I was hiding it because for many years I was somewhat embarrassed by the physics connection,” he said. “Number theorists are a pretty tough-minded group of people, and influences from physics sometimes make them more skeptical of the mathematics.”

But now Kim says he’s ready to make his vision known. “The change is, I suppose, simply a symptom of growing old!”

This is a quote from the Quanta article below. Read the whole thing!

The link is not as fantastical as the article makes it sound. For over a century, the most exciting parts of number theory are those that connect it to geometry. If you're looking for integer solutions to an equation like

x² + y² = z²

you might as well divide by z² and look for points with rational coordinates on the unit circle, so you're studying a curve "defined over the rationals". This idea is the tip of an iceberg called arithmetic geometry, where you accent the "e" in "arithmetic" so it becomes an adjective.

Also, for over a century the most exciting parts of fundamental physics are the parts that connect physics to geometry. Einstein realized that gravity is just the curvature of spacetime. Now we know the other forces are also various kinds of curvature, more abstract but still geometrical. By now, almost all the most exciting parts of geometry are inspired by physics. In particular, while string theory has been a dud when it comes to making experimental predictions, it's been an enormous boon for geometers.

So it makes some basic sense to connect number theory to the math coming from physics. But it's not so easy to make this work in practice! Especially if you're trying to impress number theorists.

The article says:

So far, Kim has made no mention of physics in his papers. Instead, he’s written about objects called Selmer varieties, and he’s considered relationships between Selmer varieties in the space of all Selmer varieties. These are recognizable terms to number theorists. But to Kim, they’ve always been another name for certain kinds of objects in physics.

Unfortunately this means I can't read his papers on Selmer varieties for information on how they're connected to physics. I'll have to ask him sometime.

But in some other papers, Minhyong is very explicit about the link between prime numbers, knots and a physical theory called Chern-Simons theory. It turns out that prime numbers are a lot like knots. They can get linked with each other... and you can use Chern-Simons theory to measure how linked they are!

This stuff is enormously fun. Especially because I remember Zuckerman shushing Minhyong and me when we were whispering at a talk by Stasheff on Chern-Simons theory!

If you're into math, check out this one:

• Minhyong Kim, Arithmetic Chern-Simons Theory I,

Abstract. In this paper, we apply ideas of Dijkgraaf and Witten on 2+1 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chern-Simons functionals on spaces of Galois representations. In the highly speculative section 5, we consider the far-fetched possibility of using Chern-Simons theory to construct L-functions.

You can see his self-deprecating sense of humor in that last sentence. I guess he wants to keep the skeptical number theorists at bay.

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