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RankinCohen Deformations and Representation Theory
"... In this paper, we use the unitary representation theory of SL2(R) to understand the RankinCohen brackets for modular forms. Then we use this interpretation to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established. ..."
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In this paper, we use the unitary representation theory of SL2(R) to understand the RankinCohen brackets for modular forms. Then we use this interpretation to study the corresponding deformation problems that Paula Cohen, Yuri Manin and Don Zagier initiated. Two uniqueness results are established. 1
NUMBER THEORY IN PHYSICS
"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."
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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The