WorldCat Identities

Katz, Nicholas M. 1943-

Overview
Works: 104 works in 327 publications in 5 languages and 7,378 library holdings
Genres: Conference papers and proceedings 
Roles: Author, Editor, Contributor, Other, Director
Publication Timeline
.
Most widely held works by Nicholas M Katz
Arithmetic moduli of elliptic curves by Nicholas M Katz( Book )

16 editions published between 1984 and 2016 in English and held by 457 WorldCat member libraries worldwide

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld
Exponential sums and differential equations by Nicholas M Katz( Book )

13 editions published between 1990 and 2016 in English and Italian and held by 424 WorldCat member libraries worldwide

This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case
Gauss sums, Kloosterman sums, and monodromy groups by Nicholas M Katz( Book )

17 editions published between 1987 and 2016 in English and held by 404 WorldCat member libraries worldwide

The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's work on the Weil Conjectures. It now appears as a very attractive mixture of algebraic geometry, representation theory, and the sheaf-theoretic incarnations of such standard constructions of classical analysis as convolution and Fourier transform. The book is simultaneously an account of some of these ideas, techniques, and results, and an account of their application to concrete equidistribution questions concerning Kloosterman sums and Gauss sums
Random matrices, Frobenius eigenvalues, and monodromy by Nicholas M Katz( Book )

16 editions published between 1998 and 2012 in English and held by 387 WorldCat member libraries worldwide

The main topic of this book is the deep relation between the spacings between zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and L-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinit
Rigid local systems by Nicholas M Katz( Book )

14 editions published between 1995 and 2016 in English and Italian and held by 356 WorldCat member libraries worldwide

Riemann introduced the concept of a "local system" on P1-{a finite set of points} nearly 140 years ago. His idea was to study nth order linear differential equations by studying the rank n local systems (of local holomorphic solutions) to which they gave rise. His first application was to study the classical Gauss hypergeometric function, which he did by studying rank-two local systems on P1- {0,1,infinity}. His investigation was successful, largely because any such (irreducible) local system is rigid in the sense that it is globally determined as soon as one knows separately each of its local monodromies. It became clear that luck played a role in Riemann's success: most local systems are not rigid. Yet many classical functions are solutions of differential equations whose local systems are rigid, including both of the standard nth order generalizations of the hypergeometric function, n F n-1's, and the Pochhammer hypergeometric functions. This book is devoted to constructing all (irreducible) rigid local systems on P1-{a finite set of points} and recognizing which collections of independently given local monodromies arise as the local monodromies of irreducible rigid local systems. Although the problems addressed here go back to Riemann, and seem to be problems in complex analysis, their solutions depend essentially on a great deal of very recent arithmetic algebraic geometry, including Grothendieck's etale cohomology theory, Deligne's proof of his far-reaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the l-adic Fourier Transform
Twisted L-functions and monodromy by Nicholas M Katz( Book )

21 editions published between 2001 and 2002 in English and held by 334 WorldCat member libraries worldwide

Annotation
Moments, monodromy, and perversity : a diophantine perspective by Nicholas M Katz( Book )

11 editions published in 2005 in English and held by 302 WorldCat member libraries worldwide

It is now some thirty years since Deligne first proved his general equidistribution theorem, thus establishing the fundamental result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated L-functions). Roughly speaking, Deligne showed that any such family obeys a "generalized Sato-Tate law," and that figuring out which generalized Sato-Tate law applies to a given family amounts essentially to computing a certain complex semisimple (not necessarily connected) algebraic group, the "geometric monodromy group" attached to that family. Up to now, nearly all techniques for determining geometric monodromy groups have relied, at least in part, on local information. In Moments, Monodromy, and Perversity, Nicholas Katz develops new techniques, which are resolutely global in nature. They are based on two vital ingredients, neither of which existed at the time of Deligne's original work on the subject. The first is the theory of perverse sheaves, pioneered by Goresky and MacPherson in the topological setting and then brilliantly transposed to algebraic geometry by Beilinson, Bernstein, Deligne, and Gabber. The second is Larsen's Alternative, which very nearly characterizes classical groups by their fourth moments. These new techniques, which are of great interest in their own right, are first developed and then used to calculate the geometric monodromy groups attached to some quite specific universal families of (L-functions attached to) character sums over finite fields
Convolution and equidistribution : Sato-Tate theorems for finite-field Mellin transforms by Nicholas M Katz( Book )

17 editions published in 2012 in English and held by 250 WorldCat member libraries worldwide

Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a
Sommes exponentielles : cours à Orsay, automne, 1979 by Nicholas M Katz( Book )

21 editions published between 1979 and 1981 in 4 languages and held by 155 WorldCat member libraries worldwide

Groupes de monodromie en géométrie algébrique by Pierre Deligne( Book )

11 editions published between 1972 and 1973 in French and held by 69 WorldCat member libraries worldwide

The Grothendieck Festschrift : a collection of articles written in honor of the 60th birthday of Alexander Grothendieck by P Cartier( Book )

35 editions published between 1990 and 2009 in 3 languages and held by 60 WorldCat member libraries worldwide

The many diverse articles presented in these three volumes, collected on the occasion of Alexander Grothendieck's sixtieth birthday and originally published in 1990, were offered as a tribute to one of the world's greatest living mathematicians. Grothendieck changed the very way we think about many branches of mathematics. Many of his ideas, revolutionary when introduced, now seem so natural as to have been inevitable. Indeed, it is difficult to fully grasp the influence his vast contributions to modern mathematics have subsequently had on new generations of mathematicians. Many of the groundbreaking contributions in these volumes contain material that is now considered foundational to the subject. Topics addressed by these top-notch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, K-theory, category theory, and homological algebra. CONTRIBUTORS to Volume I: A. Altman; M. Artin; V. Balaji; A. Beauville; A.A. Beilinson; P. Berthelot; J.-M. Bismut; S. Bloch; L. Breen; J.-L. Brylinski; J. Dieudonné; H. Gillet; A.B. Goncharov; K. Kato; S. Kleiman; W. Messing; V.V. Schechtman; C.S. Seshadri; C. Soulé; J. Tate; M. van den Bergh; and A.N. Varchenko
La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie by Jean Cerf( Book )

11 editions published between 1970 and 1971 in 3 languages and held by 43 WorldCat member libraries worldwide

Groupes de monodromie en géométrie algébrique by Seminaire de géométrie algébrique du Bois Marie( Book )

7 editions published in 1972 in French and English and held by 31 WorldCat member libraries worldwide

Group extensions of p-adic and adelic linear groups by C. C Moore( Book )

7 editions published between 1968 and 1969 in 3 languages and held by 22 WorldCat member libraries worldwide

Introduction à la théorie de Dwork by Nicholas M Katz( Book )

4 editions published in 1965 in French and held by 12 WorldCat member libraries worldwide

Publications mathematiques de l'ihes by Jean Bourgain( Book )

1 edition published in 1989 in French and held by 8 WorldCat member libraries worldwide

Arithmetic Moduli of Elliptic Curves. (AM-108) by Nicholas M Katz( )

1 edition published in 2016 in English and held by 0 WorldCat member libraries worldwide

This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld
Exponential Sums and Differential Equations. (AM-124) by Nicholas M Katz( )

1 edition published in 2016 in English and held by 0 WorldCat member libraries worldwide

This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case
Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116) by Nicholas M Katz( )

1 edition published in 2016 in English and held by 0 WorldCat member libraries worldwide

The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's work on the Weil Conjectures. It now appears as a very attractive mixture of algebraic geometry, representation theory, and the sheaf-theoretic incarnations of such standard constructions of classical analysis as convolution and Fourier transform. The book is simultaneously an account of some of these ideas, techniques, and results, and an account of their application to concrete equidistribution questions concerning Kloosterman sums and Gauss sums
Rigid Local Systems. (AM-139) by Nicholas M Katz( )

1 edition published in 2016 in English and held by 0 WorldCat member libraries worldwide

Riemann introduced the concept of a "local system" on P1-{a finite set of points} nearly 140 years ago. His idea was to study nth order linear differential equations by studying the rank n local systems (of local holomorphic solutions) to which they gave rise. His first application was to study the classical Gauss hypergeometric function, which he did by studying rank-two local systems on P1- {0,1,infinity}. His investigation was successful, largely because any such (irreducible) local system is rigid in the sense that it is globally determined as soon as one knows separately each of its local monodromies. It became clear that luck played a role in Riemann's success: most local systems are not rigid. Yet many classical functions are solutions of differential equations whose local systems are rigid, including both of the standard nth order generalizations of the hypergeometric function, n F n-1's, and the Pochhammer hypergeometric functions. This book is devoted to constructing all (irreducible) rigid local systems on P1-{a finite set of points} and recognizing which collections of independently given local monodromies arise as the local monodromies of irreducible rigid local systems. Although the problems addressed here go back to Riemann, and seem to be problems in complex analysis, their solutions depend essentially on a great deal of very recent arithmetic algebraic geometry, including Grothendieck's etale cohomology theory, Deligne's proof of his far-reaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the l-adic Fourier Transform
 
moreShow More Titles
fewerShow Fewer Titles
Audience Level
0
Audience Level
1
  Kids General Special  
Audience level: 0.51 (from 0.07 for Convolutio ... to 0.83 for Introducti ...)

WorldCat IdentitiesRelated Identities
Arithmetic moduli of elliptic curves
Alternative Names
Katz, N.

Katz, N. 1943-

Katz, N. M. 1943-

Katz, Nicholas

Katz, Nicholas 1943-

Katz, Nicholas M.

Katz, Nick 1943-

Nicholas Katz matematico statunitense

Nicholas Katz mathématicien américain

Nicholas Katz US-amerikanischer Mathematiker

Nick Katz Amerikaans wiskundige

Nick Katz amerikansk matematikar

Nick Katz amerikansk matematiker

Nick Katz matemático estadounidense

ניקולס כץ

ניקולס כץ מתמטיקאי אמריקאי

نیکلاس کتز ریاضی‌دان آمریکایی

ニック・カッツ

Languages
Covers
Exponential sums and differential equationsGauss sums, Kloosterman sums, and monodromy groupsRandom matrices, Frobenius eigenvalues, and monodromyRigid local systemsTwisted L-functions and monodromyMoments, monodromy, and perversity : a diophantine perspectiveThe Grothendieck Festschrift : a collection of articles written in honor of the 60th birthday of Alexander Grothendieck