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

One of the main inventive mathematicians of our instances, Vladimir Drinfeld got 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 well known mathematicians, devoted to Drinfeld at the celebration of his fiftieth birthday, largely 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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**Example text**

The latter matrix elements are determined in the following. Proposition 1. We have ⎧ ⎪1, k ≡ l ≡ 0 mod 2, (Wek , el ) ⎨ = 0, k ≡ l ≡ 1 mod 2, ⎪ b(k)b(l) ⎩ 2/(k − l), otherwise, where b(k) = 2k k! 3 For the proof of Proposition 1, form the generating function f (x, y) = x k y l (Wek , el ). k,l From the equality exp n>0 x 2n+1 2n + 1 = 1+x 1−x and deﬁnitions, we compute f (x, y) = 1 1 − xy 1+x 1−x 1−y . 1+y The factorization x ∂ ∂ −y ∂x ∂y f (x, y) = (x + y)(1 + x)(1 − y) (1 − x 2 )3/2 (1 − y 2 )3/2 by elementary binomial coefﬁcient manipulations proves (27) for k = l.

There is a unique function c on K such that ps∗ d = d(s) for any s ∈ C. 1. The seed K := (K, K0 , ε, d) is the amalgamation of the seeds I(s) with respect to the given gluing data (i)–(iii). 2. The amalgamation of seeds commutes with cluster transformations. Proof. Thanks to (ii), for any element i ∈ I (s) − I0 (s) one has |P −1 (ps (i))| = 1. Thus when we do a mutation in the direction ps (i), we can change the values of the cluster function only on the subset ps (I (s)2 ). The lemma follows. The amalgamation map of cluster X -varieties Let us consider the following map of X -tori: XI(s) → XK , m: m∗ xi = xj .

24 12 24 84 423 3 2 2 Z ((3, 3, 3, 3), ∅) = − E2 (q 4 ) − E2 (q 4 ) E2 (q 2 ) + E2 (q 4 )E2 (q 2 ) 5 5 20 39 7 2 3 4 4 − E2 (q ) + E4 (q )E2 (q ) + E4 (q 4 )E2 (q 2 ) 10 4 Pillowcases and quasimodular forms 23 33 141 21 2 2 E2 (q 4 ) + E2 (q 4 )E2 (q 2 ) − E2 (q 2 ) 4 16 32 25 27 9 27 + E4 (q 4 ) + E2 (q 4 ) + E2 (q 2 ) + . 32 256 32 2048 − 3 2 2 Z ((5, 3, 3, 1), ∅) = 132E2 (q 4 ) − 708E2 (q 4 ) E2 (q 2 ) + 639E2 (q 4 )E2 (q 2 ) 3 − 114E2 (q 2 ) − 15E4 (q 4 )E2 (q 4 ) + 55E4 (q 4 )E2 (q 2 ) 1365 285 2 2 − 310E2 (q 4 ) + E2 (q 4 )E2 (q 2 ) − E2 (q 2 ) 4 8 175 615 85 375 + E4 (q 4 ) + E2 (q 4 ) + E2 (q 2 ) + .