The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree," Archimedes is credited with saying. In learning what Indiana University professor of mathematics Nets Katz's peers have to say about him, one realizes that Katz works in a very beautiful world, where mathematics directly influences fields such as computer science, drug discovery, and robotics. A college graduate at age 17, a Ph.D. recipient three years later and currently Indiana University's latest Guggenheim fellow, Katz is considered one of the world's leading researchers in the field of combinatorics, according to four winners of the Fields Medal, considered by many mathematicians as the Nobel Prize of their field.
"Until about three years ago, I would say that Katz was thought of as a very good mathematician indeed," says Sir Timothy Gowers, one of those four Fields Medal winners and the Royal Society Research Professor at the University of Cambridge. "With the last two results, he has suddenly jumped up a couple of levels and joined the ranks of the truly exceptional." Gowers is referring to some of Katz's most recent work, specifically, new discoveries with Katz's former student Michael Bateman on the cap set problem in additive combinatorics, and also the resolution of a 65-year-old problem - the Erdös distance problem - in combinatorial geometry that sought to determine the minimum number of distinct distances between any finite set of points in a plane. UCLA mathematics professor Terence Tao, another Fields Medal winner, says that Katz's work on the cap set problem "has attracted the interest of many of the leading experts in the area and is likely to lead to further advances. This is first-class work, and is a testament to Nets' mathematical strength," Tao adds. Peter Sarnak, a member of the National Academy of Sciences and Princeton University's Eugene Higgins Professor of Mathematics, calls Katz's work on the Erdös distance problem a "stunning solution. The solution is brilliant, using elementary algebraic geometric methods in an ingenious way," Sarnak says. "It is one of the finest results in combinatorics in many years."
That Katz's work on the cap set problem was successful through a partnership with one of his students signals his strong mentorship of graduate students, and that commitment surfaced again with another former student, Natasa Pavolvic, when she and Katz obtained a novel approach to the problem of understanding the formulation of turbulence in Navier-Stokes equations. "Nets Katz is one of the most brilliant analysts in the world now," says Michigan State University Distinguished University Professor Alexander Volberg. "The crown achievement is of course the solution of the Erdös problem, but many of his previous results are very beautiful and inventive as well." Like Katz, success has also followed both Pavolvic, now a tenured professor in the first-rate mathematics department at the University of Texas at Austin, and Bateman, a rising star at the University of Cambridge. Jean Bourgain, a Fields Medal winner, editor of the prestigious Annals of Mathematics, and a mathematics professor at the Institute for Advanced Study at Princeton University, says that Katz has had a major impact on the field in several different areas, from harmonic analysis and geometric measure theory to arithmetic combinatorics and analysis of partial differential equations.
"He is an independent and original thinker and has shown the ability to come up with new, interesting ideas in different areas of mainstream research and leave a mark," Bourgain says. "Several of his papers had unexpected impacts and turned out to be influential in much wider circles." And finally, in an overview of Katz's work from another child prodigy who earned a bachelor's degree at 17, a Ph.D. at age 20, and a Fields Medal at 29, Princeton University Professor Charles Fefferman says the rank of distinguished professor is well-deserved by Katz. "He hasn't simply made an important discovery and pushed it hard," Fefferman says. "Rather, over and over again, he has changed technique completely in order to push past a seemingly insurmountable barrier."