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 instances, 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 well-known mathematicians, devoted to Drinfeld at the get together of his fiftieth birthday, largely replicate 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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**Additional resources for Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld’s 50th Birthday**

**Example text**

2 Amalgamation We introduce operations of amalgamation and defrosting of seeds. The amalgamation of a collection of seeds I(s), parametrised by a set S, is a new seed K = (K, K0 , εij , d). The set K is deﬁned by gluing some of the frozen vertices of the sets I (s). The frozen subset K0 is obtained by gluing the frozen subsets I0 (s). The rest of the data of K is also inherited from those of I(s). Defrosting simply shrinks the subset of the frozen vertices of K, without changing the set K. One can defrost any subset of K such that εij ∈ Z for any i, j from this subset.

To prove the rest of the theorem, we deﬁne map 7 as the amalgamation map between the corresponding X -varieties. Then it is sufﬁcient to construct the other maps for the shortest word where the maps are deﬁned, and then extend them using the multiplicativity property 7 to the general case. 8. Cluster X -varieties, amalgamation, and Poisson–Lie groups 45 It is useful to recall an explicit description of the amalgamation map XJ(A) × XJ(B) → XJ(AB) . Let {xiα }, {yiα } and {ziα } be the coordinates on XJ(A) , XJ(B) and XJ(AB) , respectively.

4 η(q 2 )4 ϑ(−1)2 (39) First, observe only even powers of (38) appear in the answer. This is because the formula (34) has a balance of minus signs in the arguments of theta functions in the Pillowcases and quasimodular forms 21 numerator and denominator. Every time we specialize yi to one of the poles in (35), the balance of minus signs changes by an even number. Inverse powers of (39) cannot appear in the answer because they grow exponentially as q → 1 and there are no other exponentially large terms to cancel this growth out.

### 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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