1. Introduction

Nicholas Michael Katz (born December 7, 1943) is an United States mathematician, working in the fields of algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory. He is currently a professor in the Mathematics Department at Princeton University in Princeton, New Jersey and an editor of the journal Annals of Mathematics.^[1]

2. Life and Work

Katz graduated from Johns Hopkins University (BA 1964) and from Princeton University, where in 1965 he received his master's degree and in 1966 he received his doctorate under supervision of Bernard Dwork 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 Kyoto, Paris VI, Orsay, the Institute for Advanced Study and the IHES. While in France, he adapted methods of scheme theory and category theory to the theory of modular forms. Subsequently, he has applied geometric methods to various exponential sums.

From 1968 to 1969, he was a NATO Postdoctoral Fellow, from 1975 to 1976 and from 1987–1988 Guggenheim Fellow and from 1971 to 1972 Sloan Fellow. In 1978 he was an invited speaker at the International Congress of Mathematicians in Helsinki (p-adic L functions, Serre-Tate local moduli and ratios of solutions of differential equations) and 1970 in Nice (The regularity theorem in algebraic geometry).

Since 2003 he is a member of the American Academy of Arts and Sciences and since 2004 the National Academy of Sciences. In 2003 he was awarded with Peter Sarnak the Levi L. Conant Prize of the American Mathematical Society (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.

He played a significant role as a sounding-board for Andrew Wiles when Wiles was developing in secret his proof of Fermat's last theorem. Mathematician and cryptographer Neal Koblitz 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 sums) with algebraic-geometric methods.

He introduced the Katz–Lang finiteness theorem.

3. 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 [math]\displaystyle{ L }[/math]-functions and Monodromy. Annals of Mathematical Studies, Princeton 2002.
• Moments, Monodromy, and Perversity. A Diophantine Perspective. Annals of Mathematical Studies, Princeton 2005, ISBN:0691123306.^[2]
• Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms. Annals of Mathematical Studies, Princeton 2012.^[3]
• With Barry Mazur: Arithmetic Moduli of elliptic curves. Princeton 1985.
• With Peter Sarnak: Random Matrices, Frobenius Eigenvalues, and Monodromy. AMS Colloquium publications 1998, ISBN:0821810170.
• With Peter Sarnak: "Zeroes of zeta functions and symmetry". Bulletin of the AMS, Vol. 36, 1999, S.1-26.