By Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)
One of the main artistic mathematicians of our instances, Vladimir Drinfeld bought the Fields Medal in 1990 for his groundbreaking contributions to the Langlands application and to the speculation of quantum groups.
These ten unique articles by way of favorite mathematicians, devoted to Drinfeld at the get together of his fiftieth birthday, greatly mirror the diversity 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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Extra info for Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday
Math. , 151-1 (1993), 193–219.  M. Douglas, Conformal ﬁeld theory techniques in large N Yang-Mills theory, in Quantum Field Theory and String Theory (Cargèse, 1993), NATO Advanced Science Institutes Series B: Physics, Vol. 328, Plenum, New York, 1995, 119–135.  A. Eskin, H. Masur, and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. , 97 (2003), 61–179.  A. Eskin and A. Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent.
5 is a composition of mutations. 6. 5, as well as a more conceptual proof of part 4. Recall that there is a transposition antiautomorphism which interchanges Eα and α ¯ F and does not change H α . Thus the formulas α¯ α¯ → α, ¯ α¯ β¯ → β¯ α, ¯ α¯ β¯ α¯ → β¯ α¯ β, and α¯ β¯ α¯ β¯ → α¯ β¯ α¯ β¯ follow from the respective formulas for positive roots. Let us introduce a more traditional generator Hα (x) = exp(log(x)hα ). • αα → α. It is easy to show using computations with 2 × 2 matrices that Eα Hα (t)Eα = Hα (1 + t 2 )1/2 Eα Hα (1 + t −2 )−1/2 .
4 We now proceed to the computation of the matrix element (Wvλ , vλ ). We have the following. Proposition 2. We have ⎛ (Wvλ , vλ ) = ⎝2N ⎞2 2N λi −λj +j −i b(λi − i + 2N ) i=1 (λi − λj + j − i)(−1) ⎠ , i
Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday by Alex Eskin, Andrei Okounkov (auth.), Victor Ginzburg (eds.)