Sunday, April 2, 2017

Quantum Questions Inspire New Math

Mathematics has the wonderful ability to connect different worlds. The most overlooked symbol in any equation is the humble equal sign. Ideas flow through it, as if the equal sign conducts the electric current that illuminates the “Aha!” lightbulb in our mind. And the double lines indicate that ideas can flow in both directions. Albert Einstein was an absolute master of finding equations that exemplify this property. Take E = mc2, without a doubt the most famous equation in history. In all its understated elegance, it connects the physical concepts of mass and energy that were seen as totally distinct before the advent of relativity. Through Einstein’s equation we learn that mass can be transformed into energy, and vice versa. The equation of Einstein’s general theory of relativity, although less catchy and well-known, links the worlds of geometry and matter in an equally surprising and beautiful manner. A succinct way to summarize that theory is that mass tells space how to curve, and space tells mass how to move.

Mirror symmetry is another perfect example of the power of the equal sign. It is capable of connecting two different mathematical worlds. One is the realm of symplectic geometry, the branch of mathematics that underlies much of mechanics. On the other side is the realm of algebraic geometry, the world of complex numbers. Quantum physics allows ideas to flow freely from one field to the other and provides an unexpected “grand unification” of these two mathematical disciplines.

It is comforting to see how mathematics has been able to absorb so much of the intuitive, often imprecise reasoning of quantum physics and string theory, and to transform many of these ideas into rigorous statements and proofs. Mathematicians are close to applying this exactitude to homological mirror symmetry, a program that vastly extends string theory’s original idea of mirror symmetry. In a sense, they’re writing a full dictionary of the objects that appear in the two separate mathematical worlds, including all the relations they satisfy. Remarkably, these proofs often do not follow the path that physical arguments had suggested. It is apparently not the role of mathematicians to clean up after physicists! On the contrary, in many cases completely new lines of thought had to be developed in order to find the proofs. This is further evidence of the deep and as yet undiscovered logic that underlies quantum theory and, ultimately, reality.

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