Nets Hawk Katz is the IBM Professor of Mathematics at the

California Institute of Technology The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasional ...

. He was a professor of Mathematics at

Indiana University Bloomington Indiana University Bloomington (IU Bloomington, Indiana University, IU, or simply Indiana) is a public university, public research university in Bloomington, Indiana. It is the flagship university, flagship campus of Indiana University and, with ...

until March 2013. Katz earned a B.A. in mathematics from

Rice University William Marsh Rice University (Rice University) is a Private university, private research university in Houston, Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranke ...

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

University of Pennsylvania The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universitie ...

, 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 an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

(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 ''A'' + ''B'' is small, what can we say about the structures of A ...

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

and other areas. In 2003, jointly with

Jean Bourgain Jean, 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, ergodic t ...

and

Terence Tao Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...

, 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 (russian: Серге́й Владимирович Конягин; born 25 April 1957) is a Russian mathematician. He is a professor of mathematics at the Moscow State University. Konyagin participated in the Internat ...

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-dimensional ...

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. Professi ...

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 in 2000, with BS in mathematics. In 2005, he got his PhD in mathematics from the Massach ...

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 \mathbb_3^n (an n-dimensional affine space over a three-element field) with no three elements in a line. The cap set problem is the problem of finding the size of the largest possible cap set, as a func ...

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 to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

. 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 ...

. In 2014, he was an

invited speaker at the International Congress of Mathematicians This is a list of International Congresses of Mathematicians Plenary and Invited Speakers. Being invited to talk at an International Congress of Mathematicians has been called "the equivalent, in this community, of an induction to a hall of fame." ...

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 achievement in mathematical research. The following mathematicians have received the award: {, class=" ...

.Clay Research Award 2015
/ref>

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