A mathematical problem that has vexed researchers for more than 100 years was finally solved on April 5, 2018, when Hertz Fellow Eric Larson (’13) submitted his doctoral thesis to the mathematics department at the Massachusetts Institute of Technology.
Larson earned the 2019 Hertz Thesis Prize from the Fannie and John Hertz Foundation for his work, which provided a proof for the problem. The Hertz Thesis Prize, which includes a $5,000 stipend, is given to Hertz Fellows for dissertations of overall excellence and for pertinence to high-impact applications in the physical sciences.
The topic of Larson’s thesis, “The Maximal Rank Conjecture,” addresses what mathematicians regard as one of the major unsolved problems of algebraic geometry. The conjecture was an unproved statement regarding the relationships between the two different ways of mathematically describing a given curve. Most previous work on the conjecture, and Larson’s proof, used an inductive argument that relates the truth of the conjecture for any given curve to the truth of the conjecture for simpler curves.
“This requires an enormously complicated induction,” said Harvard Professor of Mathematics Joe Harris, who chaired Larson’s thesis committee.
Hertz Fellow Thomas Weaver (’71), who recommended Larson’s thesis to the Hertz Foundation’s Thesis Prize reviewers, predicted that Larson’s methods would significantly impact pure mathematics.
“Surprisingly often, that turns out eventually to be applied to the physical world,” Weaver said. One of the most famous examples, he noted, was the application of group theory—the study of symmetry—to understand the orderly arrangement of atoms and molecules in solid materials. Eugene Wigner shared the 1963 Nobel Prize in physics for this insight.
Larson’s thesis dealt with the relationship between two different types of formulas that represent the shape of a curve. A simple example is a curve drawn on a piece of paper. One type of formula can generate all the points of the curve as it’s drawn on the paper. The other type of formula can determine whether or not a given point on the paper sits on the curve.
“You could say, ‘Let me tell you how to trace out the curve’” said Larson, now the Szego Assistant Professor of Mathematics at Stanford University. “Start here, move this way, do this, and then you get this curve. Or, I could give you a rule that says, ‘That point’s not on the curve, but that point is.’”
The two types of formulas both yield accurate results, but they are built on different principles. They were like two separate languages that could not be translated into each other. Larson has now connected the two types of formulas. He found that he could randomly generate a curve using the first type of formula, then predict the shape of the corresponding formula of the second type for the same curve.
The Maximal Rank Conjecture now represents a bridge between two worlds, said Harris. “Being able to pass back and forth between the two is a powerful tool.”
Larson began thinking about the Maximal Rank Conjecture during the summer after his junior year at Harvard University. He soon decided that working the problem of interpolation—how to pass a curve through a set of points in multi-dimensional space—might help him solve the conjecture.
Larson spent his first two years of graduate school working on the interpolation problem and generated some useful results. “I still haven’t solved it in the full generality that I would like to solve it,” he said. Eventually, however, he realized that what he knew about interpolation was enough to solve the Maximal Rank Conjecture. So, he began working once again on the conjecture and managed to solve it.
The Hertz Foundation typically awards its Thesis Prize, established in 1981, for work in fields such as electrical engineering, chemistry, applied physics, and biomedical engineering, said Hertz Fellow Carol Burns (’83), who chaired the Thesis Prize Committee.
“Eric’s work is certainly more pure math than is common for Thesis Prize winners, but his accomplishment in proving the Maximal Rank Conjecture was so seminal and his contributions so singular, that it rose to the top,” said Burns, Executive Officer to the Deputy Director at Los Alamos National Laboratory.
Weaver, a veteran Hertz Fellow interviewer, agreed. “To have a mathematics thesis win a Hertz Thesis Prize is extremely special. It’s never happened before,” Weaver said. “In my opinion, this is the most remarkable thesis in pure math that I’ve ever seen a Hertz Fellow produce.”
Larson’s thesis formally acknowledged the support he received from 2013 to 2018 as a Hertz Fellow, and as a National Defense Science and Engineering Graduate Fellow of the U.S. Department of Defense.
The Hertz Fellowship, in particular, allowed him to devote as much time as he needed to his research and enabled him to consult frequently and share insights with other Hertz Fellows.
“It certainly made a difference in my graduate career,” Larson said.
Larson continues to work on the interpolation problem with his wife, Isabel Vogt, who has completed her Ph.D. in mathematics at MIT. They will both complete postdoctoral appointments at Stanford next year, then begin tenure-track faculty positions at the University of Washington in 2020.