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

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

One of the main inventive mathematicians of our instances, Vladimir Drinfeld obtained the Fields Medal in 1990 for his groundbreaking contributions to the Langlands software and to the idea of quantum groups.

These ten unique articles by way of popular mathematicians, devoted to Drinfeld at the celebration 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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In particular a cluster X -manifold has a canonical Poisson structure and the automorphism group of this manifold acts on it by Poisson transformations. 4. Cluster transformations are given by rational functions with positive integral coefﬁcients. 5. If εij = εj i = 0, then µi µj µj µi = id. ∼ 6. If εij = −εj i = −1, then µi µj µi µj µi = id. ) 7. If If εij = −2εj i = −2, then µi µj µi µj µi µj = id. 8. If If εij = −3εj i = −3, then µi µj µi µj µi µj µi = id. Conjecturally all relations between mutations are exhausted by properties 5–8.

K (i)µk (j ) ⎧ ⎪ if i = k or j = k, ⎨−εij = εij + εik max(0, εkj ) if εik ≥ 0, ⎪ ⎩ εij + εik max(0, −εkj ) if εik < 0. A symmetry of a seed I = (I, I0 , ε, d) is an automorphism σ of the set I preserving the subset I0 , the matrix ε and the numbers di . In other words, it satisﬁes the following conditions: 1. σ (I0 ) = I0 , 2. dσ (i) = di , 3. εσ (i)σ (j ) = εij Symmetries and mutations induce (rational) maps between the corresponding seed X -tori, which are denoted by the same symbols µk and σ and given by the formulas xσ (i) = xi and xµk (i) ⎧ −1 ⎪ ⎨xk = xi (1 + xk )εik ⎪ ⎩ xi (1 + (xk )−1 )εik if i = k, if εik ≥ 0 and i = k, if εik ≤ 0 and i = k.

N − 1, we glue the maximal γ -decorated element of J (αi ) and the minimal γ -decorated element of J (αi+1 ). The seed J(D) is the amalgamated product for this gluing data of the seeds J(α1 ), . . , J(αn ). The seed J(D) has frozen vertices only: J (D) = J0 (D). To deﬁne the seed J(D) we will defrost some of them, making the set J0 (D) smaller. 1 we glue only the elements decorated by the same positive simple root. Thus the obtained set J (D) has a natural decoration π : J (D) −→ , extending those of the subsets J (αi ).