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254
Koszul duality patterns in representation theory
 Jour. Amer. Math. Soc
, 1996
"... 2. Koszul rings 479 3. Parabolicsingular duality and Koszul duality 496 ..."
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Cited by 211 (14 self)
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2. Koszul rings 479 3. Parabolicsingular duality and Koszul duality 496
Zeroes of Zeta Functions and Symmetry
, 1999
"... Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of cur ..."
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Cited by 103 (2 self)
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Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the lowlying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and Lfunctions.
Applications of Multilinear Forms to Cryptography
 Contemporary Mathematics
, 2002
"... We study the problem of finding efficiently computable nondegenerate multilinear maps from G 1 to G 2 , where G 1 and G 2 are groups of the same prime order, and where computing discrete logarithms in G 1 is hard. We present several applications to cryptography, explore directions for building such ..."
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Cited by 51 (7 self)
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We study the problem of finding efficiently computable nondegenerate multilinear maps from G 1 to G 2 , where G 1 and G 2 are groups of the same prime order, and where computing discrete logarithms in G 1 is hard. We present several applications to cryptography, explore directions for building such maps, and give some reasons to believe that finding examples with n > 2 may be difficult.
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
"... ..."
Definable sets, motives and padic integrals
 J. Amer. Math. Soc
, 2001
"... 0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) ofitsZprational points. For every n in N, there is a natural map πn: X(Zp) → X(Z/pn+1) assigning to a Zprational point its class modulo pn+1. The image Yn,p of X(Zp) byπn is ..."
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Cited by 43 (9 self)
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0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) ofitsZprational points. For every n in N, there is a natural map πn: X(Zp) → X(Z/pn+1) assigning to a Zprational point its class modulo pn+1. The image Yn,p of X(Zp) byπn is exactly the set of
Evidence for a Spectral Interpretation of the Zeros of LFunctions
, 1998
"... By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported ..."
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Cited by 33 (7 self)
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By looking at the average behavior (nlevel density) of the low lying zeros of certain families of Lfunctions, we find evidence, as predicted by function field analogs, in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups. This is further supported by numerical experiments for which an efficient algorithm to compute Lfunctions was developed and implemented. iii Acknowledgements When Mike Rubinstein woke up one morning he was shocked to discover that he was writing the acknowledgements to his thesis. After two screenplays, a 40000 word manifesto, and many fruitless attempts at making sushi, something resembling a detailed academic work has emerged for which he has people to thank. Peter Sarnak from Chebyshev's Bias to USp(1). For being a terrific advisor and teacher. For choosing problems suited to my talents and involving me in this great project to understand the zeros of Lfunctions. Zeev Rudnick and Andrew Oldyzko for many disc...
The HarderNarasimhan system in quantum groups and cohomology of Quiver Moduli
, 2002
"... Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the HarderNarasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. ..."
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Cited by 26 (6 self)
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Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the HarderNarasimhan recursion is constructed inside the quantized enveloping algebra of a KacMoody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in ’linear algebra type’ situations.
Deformation theory and the computation of zeta functions
 Proc. London Math. Soc
, 2004
"... An attractive and challenging problem in computational number theory is to count in an e cient manner the number of solutions to a multivariate polynomial equation over a nite eld. One desires an algorithm whose time complexity is a small polynomial function of some appropriate measure of the size o ..."
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Cited by 24 (1 self)
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An attractive and challenging problem in computational number theory is to count in an e cient manner the number of solutions to a multivariate polynomial equation over a nite eld. One desires an algorithm whose time complexity is a small polynomial function of some appropriate measure of the size of the