Nicholas Michael Katz (; born December 7, 1943) is an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

, working in

arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...

, particularly on ''p''-adic methods,

monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...

and moduli problems, and

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

. He is currently a professor of Mathematics at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

and an editor of the journal ''

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

''.

Life and work

Katz graduated from

Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland, United States. Founded in 1876 based on the European research institution model, J ...

(BA 1964) and from

, where in 1965 he received his master's degree and in 1966 he received his doctorate under supervision of

Bernard Dwork Bernard Morris Dwork (May 27, 1923 – May 9, 1998) was an American mathematician, known for his application of ''p''-adic analysis to local zeta functions, and in particular for a proof of the first part of the Weil conjectures: the rationality ...

with thesis ''On the Differential Equations Satisfied by Period Matrices''. After that, at Princeton, he was an instructor, an assistant professor in 1968, associate professor in 1971 and professor in 1974. From 2002 to 2005 he was the chairman of faculty there. He was also a visiting scholar at the

University of Minnesota The University of Minnesota Twin Cities (historically known as University of Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Twin Cities of Minneapolis and Saint ...

, the

University of Kyoto , or , is a national research university in Kyoto, Japan. Founded in 1897, it is one of the former Imperial Universities and the second oldest university in Japan. The university has ten undergraduate faculties, eighteen graduate schools, and t ...

, Paris VI, Orsay Faculty of Sciences, the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

and the IHES. While in France, he adapted methods of

scheme theory In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different s ...

and

category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...

to the theory of

modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...

s. Subsequently, he has applied geometric methods to various

exponential sum In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typi ...

s. From 1968 to 1969, he was a NATO Postdoctoral Fellow, from 1975 to 1976 and from 1987–1988

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 ...

and from 1971 to 1972 Sloan Fellow. In 1970 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 ...

in Nice (''The regularity theorem in algebraic geometry'') and in 1978 in

Helsinki Helsinki () is the Capital city, capital and most populous List of cities and towns in Finland, city in Finland. It is on the shore of the Gulf of Finland and is the seat of southern Finland's Uusimaa region. About people live in the municipali ...

(''p-adic L functions, Serre-Tate local moduli and ratios of solutions of differential equations''). Since 2003 he is a member of the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...

and since 2004 the

National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...

. In 2003 he was awarded with

Peter Sarnak Peter Clive Sarnak (born 18 December 1953) is a South African and American mathematician. Sarnak has been a member of the permanent faculty of the School of Mathematics at the Institute for Advanced Study since 2007. He is also Eugene Higgins ...

the

Levi L. Conant Prize The Levi L. Conant Prize is a mathematics prize of the American Mathematical Society, which has been awarded since 2001 for outstanding expository papers published in the ''Bulletin of the American Mathematical Society'' or the ''Notices of the Ame ...

of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

(AMS) for the essay "Zeroes of Zeta Functions and Symmetry" in the ''Bulletin of the American Mathematical Society''. Since 2004 he is an editor of the ''Annals of Mathematics''. In 2023 he received from the AMS the Leroy P. Steele Prize for Lifetime Achievement. He played a significant role as a sounding-board for

Andrew Wiles Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...

when Wiles was developing in secret his proof of

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

. Mathematician and

cryptographer Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More gen ...

Neal Koblitz Neal I. Koblitz (born December 24, 1948) is a Professor of Mathematics at the University of Washington. He is also an adjunct professor with the Centre for Applied Cryptographic Research at the University of Waterloo. He is the creator of hype ...

was one of Katz's students. Katz studied, with Sarnak among others, the connection of the eigenvalue distribution of large random matrices of classical groups to the distribution of the distances of the zeros of various ''L'' and zeta functions in algebraic geometry. He also studied trigonometric sums (

Gauss sum In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is ...

s) with algebro-geometric methods. He introduced the

Katz–Lang finiteness theorem In number theory, the Katz–Lang finiteness theorem, proved by , states that if ''X'' is a smooth scheme, smooth Glossary of scheme theory, geometrically connected Scheme (mathematics), scheme of finite type over a field (mathematics), field ''K'' ...

Writings

* ''Gauss sums, Kloosterman sums, and monodromy groups.'' Annals of Mathematical Studies, Princeton 1988. * ''Exponential sums and differential equations.'' Annals of Mathematical Studies, Princeton 1990
Manuscript with corrections
* ''Rigid Local Systems.'' Annals of Mathematical Studies, Princeton 1996. * ''Twisted

L

-functions and Monodromy.'' Annals of Mathematical Studies, Princeton 2002. * ''Moments, Monodromy, and Perversity. A Diophantine Perspective.'' Annals of Mathematical Studies, Princeton 2005, .
''Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms.''
Annals of Mathematical Studies, Princeton 2012. * With

Barry Mazur Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...

: ''Arithmetic Moduli of elliptic curves.'' Princeton 1985. * With

: ''Random Matrices, Frobenius Eigenvalues, and Monodromy.'' AMS Colloquium publications 1998, . * With Peter Sarnak:
Zeroes of zeta functions and symmetry
. ''Bulletin of the AMS'', Vol. 36, 1999, S.1-26.

References

External links

*
Nick Katz's web page in Princeton
{{DEFAULTSORT:Katz, Nick 1943 births Living people Arithmetic geometers Members of the United States National Academy of Sciences Jewish American scientists Baltimore City College alumni Johns Hopkins University alumni Princeton University alumni Princeton University faculty 20th-century American mathematicians 21st-century American mathematicians Fermat's Last Theorem People from Baltimore Paris-Saclay University people 21st-century American Jews