Nets Hawk Katz is the W.L. Moody Professor of Mathematics at

Rice University William Marsh Rice University, commonly referred to as Rice University, is a Private university, private research university in Houston, Houston, Texas, United States. Established in 1912, the university spans 300 acres. Rice University comp ...

. He was a professor of mathematics at

Indiana University Bloomington Indiana University Bloomington (IU Bloomington, Indiana University, IU, IUB, or Indiana) is a public university, public research university in Bloomington, Indiana, United States. It is the flagship university, flagship campus of Indiana Univer ...

until March 2013 and the IBM Professor of Mathematics at the

California Institute of Technology The California Institute of Technology (branded as Caltech) is a private research university in Pasadena, California, United States. The university is responsible for many modern scientific advancements and is among a small group of institutes ...

until 2023. He is currently the W. L. Moody Professor of Mathematics at

. Katz earned a B.A. in mathematics from

in 1990 at the age of 17. He received his Ph.D. in 1993 under

Dennis DeTurck Dennis M. DeTurck (born July 15, 1954) is an American mathematician known for his work in partial differential equations and Riemannian geometry, in particular contributions to the theory of the Ricci flow and the prescribed Ricci curvature proble ...

at the

University of Pennsylvania The University of Pennsylvania (Penn or UPenn) is a Private university, private Ivy League research university in Philadelphia, Pennsylvania, United States. One of nine colonial colleges, it was chartered in 1755 through the efforts of f ...

, with a dissertation titled "Noncommutative Determinants and Applications". He is the author of several important results in

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

(especially

additive combinatorics Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset is small, what can we say about the structures of and ? In the case of th ...

harmonic analysis Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

and other areas. In 2003, jointly with

Jean Bourgain Jean Louis, baron Bourgain (; – ) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodi ...

and

Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician, Fields medalist, and professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins Chair in the Co ...

, he proved that any subset of

\Z/p\Z

grows substantially under either addition or multiplication. More precisely, if

A

is a set such that

\max(, A \cdot A, , , A+A, ) \leq K, A,

, then

A

has size at most

K^C

or at least

p/K^C

where

C

is a constant that depends on

A

. This result was followed by the subsequent work of Bourgain,

Sergei Konyagin Sergei Vladimirovich Konyagin (; born 25 April 1957) is a Russian mathematician. He is a professor of mathematics at the Moscow State University. His primary research interest is in applying harmonic analysis to number theoretic settings. Konyagi ...

and Glibichuk, establishing that every approximate field is almost a field. Somewhat earlier he was involved in establishing new bounds in connection with the dimension of

Kakeya set In mathematics, a Kakeya set, or Besicovitch set, is a set of points in Euclidean space which contains a unit line segment in every direction. For instance, a disk of radius 1/2 in the Euclidean plane, or a ball of radius 1/2 in three-dimensiona ...

s. Jointly with

Izabella Łaba Izabella Łaba (born 1966) is a Polish-Canadian mathematician, a professor of mathematics at the University of British Columbia. Her main research specialties are harmonic analysis, geometric measure theory, and additive combinatorics. Professio ...

and

he proved that the upper Minkowski dimension of Kakeya sets in 3 dimensions is strictly greater than 5/2, and jointly with

he established new bounds in large dimensions. In 2010, Katz along with

Larry Guth Lawrence David Guth (; born 1977) is a professor of mathematics at the Massachusetts Institute of Technology. Education and career Guth graduated from Yale University in 2000 with a BS in mathematics. In 2005, he received his PhD in mathemati ...

published the results of their collaborative effort to solve the

Erdős distinct distances problem In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly-linear number of distinct distances. It was posed by Paul Erdős in 1946 and almost proven by Larry Guth and Nets Katz in 2015. ...

, in which they found a "near-optimal" result, proving that a set of

N

points in the plane has at least

cN / \log N

distinct distances. In early 2011, in joint work with Michael Bateman, he improved the best known bounds in the

cap set In affine geometry, a cap set is a subset of the affine space \mathbb_3^n (the n-dimensional affine space over the three-element field) where no three elements sum to the zero vector. The cap set problem is the problem of finding the size of the ...

problem: if

A

is a subset of

(\Z/3\Z)^n

of cardinality at least

3^n/n^

, where

c > 0

, then

A

contains three elements in a line. In 2012, he was named a

Guggenheim fellow Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation, endowed by the late Simon and Olga Hirsh Guggenheim. These awards are bestowed upon individuals who have demonstrated d ...

. During 2011–2012, he was the managing editor of the

Indiana University Mathematics Journal The ''Indiana University Mathematics Journal'' is a journal of mathematics published by Indiana University. Its first volume was published in 1952, under the name ''Journal of Rational Mechanics and Analysis'' and edited by Zachery D. Paden and Cl ...

. In 2014, he was an

invited speaker at the International Congress of Mathematicians An invitation system is a method of encouraging people to join an organization, such as a Club (organization), club or a website. In regular society, it refers to any system whereby new members are chosen; they cannot simply apply. In relation to w ...

at Seoul and gave a talk ''The flecnode polynomial: a central object in incidence geometry''. In 2015, he received the

Clay Research Award __NOTOC__ The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievements in mathematical research. The following mathematicians have received the award: See als ...

.Clay Research Award 2015
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Work

* *

References

External links

Nets Katz's personal web page, including info on research, teaching, etc.
{{DEFAULTSORT:Katz, Nets Year of birth missing (living people) Living people Rice University alumni Indiana University faculty University of Pennsylvania alumni California Institute of Technology faculty Rice University faculty