Larry Guth is a 2014 Simons Investigator in Mathematics and professor of mathematics at the Massachusetts Institute of Technology (MIT), where he also received his Ph.D. His research focuses on combinatorial geometry, a branch of mathematics that studies the combinations and organizations of geometric objects, as well as harmonic analysis and metric analysis. He received his undergraduate degree from Yale University and, before returning to MIT to teach, he worked at the University of Toronto and the Courant Institute of Mathematical Sciences at New York University.
Guth learned of combinatorial geometry as a postdoctoral researcher at Stanford University. A colleague told him about some important theorems in the field from the 1980s, and he was impressed to see interesting recent mathematics about lines in the plane. “I felt like these questions really could have been asked in the 1800s, except that nobody had this angle. I thought this was very engaging.”
Guth is also interested in mathematics that can be understood by a younger audience. “There are open problems that are similar to these ones that you can even explain to middle school students.” A friend who runs a summer program for middle school students in New York City invited Guth to teach there about problems related to the intersection pattern you can make with lines in the plane. He taught there for one summer and has visited since.
Recently, Guth began a collaboration with Nets Katz of the California Institute of Technology; the two were able to prove that a complex configuration of lines can be modeled with a low-degree polynomial. “It was counterintuitive to me,” says Guth. “The problem only involves lines, but the structure that appears is algebraic geometry — just polynomials.”
The combinatorial problems related to lines that Guth works with now are ‘overdetermined,’ that is, there are more equations than there are unknowns in those equations. “Your first instinct in seeing an overdetermined situation is that it’s just impossible,” says Guth. “That’s a valid first guess, but there are many situations that come up in math — and perhaps also in other parts of science — where that first guess is wrong. I would say that what we’re trying to find is the right second guess: if you have only ten parameters to play with and you’re trying to make one hundred things happen, what are the situations where that could work, and what should you be looking for?”