By victor ginzburg

The most inventive 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 via admired mathematicians, devoted to Drinfeld at the get together of his fiftieth birthday, largely replicate the variety of Drinfeld's personal pursuits in algebra, algebraic geometry, and quantity conception.

**Read or Download Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld's 50th Birthday PDF**

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**Additional info for Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld's 50th Birthday**

**Sample text**

Miwa, M. Jimbo, and E. Date, Solitons: Differential Equations, Symmetries and Inﬁnite-Dimensional Algebras, Cambridge Tracts in Mathematics, Vol. 135. Cambridge University Press, Cambridge, UK, 2000. [21] A. Okounkov and R. AG/0204305, 2002. [22] A. Okounkov and R. Pandharipande, in preparation. Cluster X -varieties, amalgamation, and Poisson–Lie groups V. V. Fock∗ and A. B. Goncharov† ∗ Institute for Theoretical and Experimental Physics B. edu To Vladimir Drinfeld for his 50th birthday. Subject Classiﬁcations: Primary 22E46, 20G42.

If H ∈ H, the map ev : XH → G is injective at the generic point. If H is the longest element of H, then the image of ev is Zariski dense in G. The map ev : XB → G is a composition π evH XB −→ XH → G, H = π(B). Remark 1. A part of the above data is axiomatized as follows. , the category of Poisson varieties with the product as monoidal structure), and for every pair s, t ∈ S a canonical morphism ms,t : Xs × Xt −→ Xst . , for every r, s, t ∈ S, the following diagram is commutative: Id ×ms,t Xr × Xs × Xt −→ Xr × Xst , mr,s ↓ ×Id ↓ mr,st , (6) mrs,t Xrs × Xt −→ Xrst .

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.