Mathematician Eric Larson Receives Prestigious Hertz Foundation Dissertation Award

Mathematician Eric Larson Receives Prestigious Hertz Foundation Dissertation Award

Livermore, Calif. – A mathematical problem that has vexed researchers for over 100 years was finally resolved on April 5, 2018, when Eric Larson (Hertz Fellow) presented his doctoral dissertation to the Massachusetts Institute of Technology's Department of Mathematics.

Larson received the Hertz 2019 Thesis Award from the Fannie and John Hertz Foundation for his work, which provided proof of the problem. The Hertz Dissertation Prize, which includes a $ 5,000 scholarship, is awarded to Hertz Fellows for their theses of overall excellence and relevance for high impact applications in the physical sciences.

The subject of Larson's thesis, "The Conjecture of Maximum Rank," addresses what mathematicians consider to be one of the main unresolved problems of algebraic geometry. The conjecture was an unproven statement about the relationship between the two different ways of mathematically describing a given curve. Most of the earlier work on conjecture, and Larson's proof, used an inductive argument that relates the truth of conjecture to a given curve to the truth of conjecture for simpler curves.

"This requires an extremely complicated induction," said Joe Harris, professor of mathematics at Harvard, who chaired Larson's dissertation committee.

Hertz Fellow Thomas Weaver (1971), who recommended Larson's thesis to the Hertz Foundation's Ph.D. dissertation reviewers, predicted that Larson's methods would have a significant impact on pure mathematics.

"Surprisingly often, this eventually turns out to be applied to the physical world," Weaver said. One of the most famous examples, he noted, concerns 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 idea.

Larson's thesis focused on the relationship between two different types of formulas representing the shape of a curve. A simple example is a curve drawn on a piece of paper. A type of formula can generate all the points of the curve as it is drawn on the paper. The other type of formula can determine whether or not a given point on the paper is on the curve.

"You could say, let me tell you how to draw the curve," said Larson, today an assistant professor of mathematics at Stanford University. "Start here, move that way, do this, then you will get this curve. Or, I could give you a rule that says: "This point is not on the curve, but this point is."

Both types of formulas yield accurate results, but are based on different principles. They were like two different languages ​​that could not be translated one into the other. Larson has now connected both types of formulas. He discovered that he could randomly generate a curve using the first type of formula, and then predict the shape of the corresponding formula of the second type for the same curve.

The maximum-rank conjecture now represents a bridge between two worlds, Harris said. "Being able to commute between the two is a powerful tool."

Larson began thinking about maximum rank conjecture in the summer after his first year at Harvard University. He quickly decided that solving the problem of interpolation – how to jump from one curve to a set of points in a multidimensional space – could help solve the conjecture.

Larson spent his first two years of study working on the problem of interpolation and generated useful results. "I still have not solved the problem in its entirety and would like to solve it," he said. However, he ended up realizing that what he knew about the interpolation was enough to solve the maximum-rank guess. So he started working on the conjecture again and managed to solve it.

The Hertz Foundation generally awards its Ph.D. prize, created in 1981, for work in areas such as electrical engineering, chemistry, applied physics and biomedical engineering, said Carol Burns (1983), Hertz Scholar, who presided the thesis prize committee.

"Eric's work is certainly more purely mathematical than for the thesis award winners, but his contribution to the demonstration of maximum-rank conjecture was so fundamental and his contribution so unique that it reached the summit, "said Burns, deputy director of science, technology and engineering at the Los Alamos National Laboratory.

Weaver, a veteran interviewer on the Hertz Scholarship, agreed. "Winning a thesis in mathematics with a Hertz PhD award is extremely special – it's never happened before," Weaver said. "For me, it's the most outstanding thesis in pure mathematics I've ever seen produce with a Hertz Fellow."

Larson's thesis officially recognizes the support he has received from 2013 to 2018 as a Hertz Fellow and a Fellow in Defense Science and Engineering from the US Department of National Defense.

The Hertz fellowship, in particular, allowed him to devote all the time he needed to his research and allowed him to consult frequently and exchange ideas with other Hertz members.

"It certainly made a difference in my graduate career," Larson said.

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Larson continues to work on the problem of interpolation with his wife, Isabel Vogt, who has completed his PhD. in mathematics at MIT. They will both have postdoctoral positions at Stanford next year, and then begin permanent faculty appointments at the University of Washington in 2020.

ABOUT THE FANNIE AND JOHN HERTZ FOUNDATION

The Fannie Foundation and John Hertz is dedicated to advancing groundbreaking applied science with tangible benefits for all humanity. There are more than 1,200 Hertz Fellows and, over the last six decades, the Nobel Prize, the National Science Medal, the Turing Prize, the Revolutionary Award and the MacArthur "Engineering Scholarship". They are also leaders in the world of business and industry. They have notably developed groundbreaking diagnostics and treatments for diseases, new innovations for creating and storing energy, new tools for exploring the Earth and space and creating new models of supercomputers. Fellows have founded more than 200 companies and hold more than 3,000 patents. Forty-one members are members of the National Academies of Science, Engineering and Medicine.

This story was published on: 2019-06-18. To contact the author, please use the contact information in the article.

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