Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)'s Algebraic Geometry and Number Theory: In Honor of Vladimir PDF

By Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

ISBN-10: 0817644717

ISBN-13: 9780817644710

ISBN-10: 0817645322

ISBN-13: 9780817645328

One of the main artistic mathematicians of our occasions, Vladimir Drinfeld bought the Fields Medal in 1990 for his groundbreaking contributions to the Langlands software and to the speculation of quantum groups.

These ten unique articles by means of famous mathematicians, devoted to Drinfeld at the celebration of his fiftieth birthday, widely replicate the variety of Drinfeld's personal pursuits in algebra, algebraic geometry, and quantity theory.

Contributors: A. Eskin, V.V. Fock, E. Frenkel, D. Gaitsgory, V. Ginzburg, A.B. Goncharov, E. Hrushovski, Y. Ihara, D. Kazhdan, M. Kisin, I. Krichever, G. Laumon, Yu.I. Manin, A. Okounkov, V. Schechtman, and M.A. Tsfasman.

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3 By normally ordering all fermionic operators in (29) and using the estimate (31) one sees that the trace converges if |yn /q| > |x1 y1 | > |y1 | > · · · > |xn yn | > |yn | > 1. 10]: ψ(xy)ψ ∗ (y) = x 1/2 1 − x −1/2 × exp n (xy)n − y n α−n exp n n y −n − (xy)−n αn . n (33) It allows to express the operator in (29) in terms of bosonic operators αn . With respect to the action of the operators αn , the charge zero subspace of the infinite wedge space decomposes as the infinite tensor product ∞ 2 0 ∞ ∞ V ∼ = k α−n v∅ , n=1 k=0 the distinguished vector in each factor being v∅ .

10] G. Jones, Characters and surfaces: A survey, in The Atlas of Finite Groups: Ten years On (Birmingham, 1995), London Mathematical Society Lecture Note Series, Vol. 249, Cambridge University Press, Cambridge, UK, 1998, 90–118. [11] T. Józefiak, Symmetric functions in the Kontsevich-Witten intersection theory of the moduli space of curves, Lett. Math. , 33-4 (1995), 347–351. [12] V. Kac, Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge, UK, 1990, 1995. [13] M. Kaneko and D.

1. 1 Basic definitions A cluster seed , or just seed , I is a quadruple (I, I0 , ε, d), where (i) (ii) (iii) (iv) I is a finite set; I0 ⊂ I is a subset; ε is a matrix (εij ), where i, j ∈ I , such that εij ∈ Z unless i, j ∈ I0 ; d = {di }, where i ∈ I , is a set of positive integers, such that the matrix (εij ) = (εij dj ) is skew-symmetric. The elements of the set I are called vertices, the elements of I0 are called frozen vertices. The matrix ε is called a cluster function, the numbers {di } are called multipliers, and the function d on I whose value at i is di is called a multiplier function.

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Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday by Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)

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