Volume 2, Issue 12
Mathematics of Everything
Edward Frenkel is on a search for symmetry. Most simply, an object is symmetrical if it can be reflected, rotated, or "slid" from one area to another seemingly without changing it. Snowflakes are symmetrical. So are butterfly wings. And the human body. Indeed, the beauty of symmetry surrounds us. But the symmetries sought by Frenkel, a UC Berkeley mathematician, are hidden within esoteric branches of math and physics. In his eyes though, those symmetries are just as stunning.
"In mathematics, symmetries play a very important role in such diverse subjects as geometry, number theory, and algebra," he says. "And in physics like quantum field theory and string theory, symmetry is a focal point of research. I look at the fundamental properties of symmetry and how they play out in different domains so we can understand the underlying concepts and patterns."
As a high school student in Moscow, Edward Frenkel wanted to be a physicist. Then a mathematician friend of the family introduced Frenkel to the beauty of numbers. Frenkel's research enables him to pursue both passions.
Mathematical symmetry means that an expression remains unchanged even when certain operations are applied to it. For example, the formula a2c + 3ab + b2c is symmetrical because even if the "a" and "b" are swapped, the expression stays the same. Mathematicians study those kinds of symmetries by grouping together all of the operations that don't change a particular expression. In geometry, objects can be symmetrical. If an equilateral triangle is rotated 120 degrees or reflected along a vertical axis down the center, the shape stays the same.
Of course, much of the symmetry in geometry, algebra, and mathematics is far more difficult to spot. Frenkel's stomping ground is the Langlands Program, something of a "unifying theory" of mathematics based on symmetries. First proposed in 1967 by Robert Langlands of the Institute for Advanced Study, the conjectures boldly linked together seemingly unrelated objects in branches of mathematics like number theory and algebraic geometry. (British mathematicians Andrew Wiles and Richard Taylor built upon the Langlands Program to famously solve Fermat's Last Theorem in 1994, three hundred years after Fermat scribbled it in the margin of a book.)
Frenkel is co-managing a multi-university research project sponsored by the Defense Advanced Research Projects Agency (DARPA) to investigate the Geometric Langlands Program. As part of the effort, the researchers are applying Langlands' lessons to two of the most cutting-edge research thrusts in physics today--quantum field theory and superstring theory.
Quantum field theory is a framework to study elementary particles and their interactions. Superstring theory attempts to unite Einstein's general theory of relativity and quantum mechanics under one umbrella, or "theory of everything." According to superstring theory, all elementary particles are tiny vibrating strands of energy. Every matter particle also has a "superpartner," a particle that carries a fundamental force of nature such as gravity. The theory posits that superpartners are paired by supersymmetry.
"The question we're asking is, can we see the Langlands program in string theory like we do in number theory or geometry?" Frenkel says. "For me, the excitement of mathematics is recognizing when a phenomenon appears in many different contexts unexpectedly. First you admire that. Then you ask why."
Of course, string theory has not been experimentally confirmed. But postulating theories about theories is all part of the game, Frenkel adds.
"We don't know if string theory describes the universe, so this research may touch on something fundamental about reality or it may not," he says. "But at the very least, it produces very beautiful mathematics."