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 EichlerShimura, Igusa, and DeligneRapoport. 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
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 EichlerShimura, Igusa, and DeligneRapoport. 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 oneparameter 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
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 oneparameter 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 sheaftheoretic 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
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 sheaftheoretic 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 Lfunctions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the MontgomeryOdlyzko law, is shown to hold for wide classes of zeta and Lfunctions 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
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 Lfunctions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the MontgomeryOdlyzko law, is shown to hold for wide classes of zeta and Lfunctions 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 ranktwo 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 n1'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 farreaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the ladic Fourier Transform
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 ranktwo 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 n1'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 farreaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the ladic Fourier Transform
Twisted Lfunctions 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
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" algebrogeometric families of character sums over finite fields (and of their associated Lfunctions). Roughly speaking, Deligne showed that any such family obeys a "generalized SatoTate law," and that figuring out which generalized SatoTate 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 (Lfunctions attached to) character sums over finite fields
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" algebrogeometric families of character sums over finite fields (and of their associated Lfunctions). Roughly speaking, Deligne showed that any such family obeys a "generalized SatoTate law," and that figuring out which generalized SatoTate 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 (Lfunctions attached to) character sums over finite fields
Convolution and equidistribution : SatoTate theorems for finitefield 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 theorythe theory of exponential sums over finite fields and their Mellin transformsfrom 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 finitefield 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
17 editions published in 2012 in English and held by 250 WorldCat member libraries worldwide
Convolution and Equidistribution explores an important aspect of number theorythe theory of exponential sums over finite fields and their Mellin transformsfrom 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 finitefield 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
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
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 topnotch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, Ktheory, 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
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 topnotch contributors match the breadth of Grothendieck's own interests, including: functional analysis, algebraic geometry, algebraic topology, number theory, representation theory, Ktheory, 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 pseudoisotopie by
Jean Cerf(
Book
)
11 editions published between 1970 and 1971 in 3 languages and held by 43 WorldCat member libraries worldwide
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
7 editions published in 1972 in French and English and held by 31 WorldCat member libraries worldwide
Group extensions of padic 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
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
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
1 edition published in 1989 in French and held by 8 WorldCat member libraries worldwide
Arithmetic Moduli of Elliptic Curves. (AM108) 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 EichlerShimura, Igusa, and DeligneRapoport. 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
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 EichlerShimura, Igusa, and DeligneRapoport. 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. (AM124) 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 oneparameter 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
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 oneparameter 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. (AM116) 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 sheaftheoretic 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
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 sheaftheoretic 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. (AM139) 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 ranktwo 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 n1'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 farreaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the ladic Fourier Transform
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 ranktwo 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 n1'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 farreaching generalization of the original Weil Conjectures, the theory of perverse sheaves, and Laumon's work on the ladic Fourier Transform
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Related Identities
 Mazur, Barry Contributor
 Sarnak, Peter
 Laumon, Gérard Editor
 Deligne, Pierre Other Author Editor Director
 Illusie, Luc Editor
 Cartier, Pierre Author Editor
 Manin, Yuri I. Editor
 Ribet, Kenneth A. Editor
 Grothendieck, A. (Alexandre) Honoree Dedicatee Editor
 Cerf, Jean Author Contributor
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Associated Subjects
Abelian groups Abelian varieties Algebra Algebra, Homological Algebraic fields Algebraic topology Algebras, Linear Arithmetical algebraic geometry Categories (Mathematics) Cohomology operations Complexes Convolutions (Mathematics) Curves, Algebraic Curves, Elliptic Diffeomorphisms Differential equations Differential equationsNumerical solutions Diophantine analysis Eigenvalues Exponential functions Exponential sums Functional analysis Functions Functions, Zeta Gaussian sums Geometric group theory Geometry, Algebraic Geometry, Analytic Germs (Mathematics) Grothendieck, A.(Alexandre) Group theory Homology theory Hypergeometric functions Kloosterman sums Ktheory Lfunctions Limit theorems (Probability theory) Linear algebraic groups Mathematics Matrices Mellin transform Moduli theory Monodromy groups Number theory Random matrices Sequences (Mathematics) Sheaf theory Singularities (Mathematics) Spectral theory (Mathematics) Topology
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 USamerikanischer 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
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