By victor ginzburg
Essentially the most artistic mathematicians of our occasions, Vladimir Drinfeld got 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 favorite mathematicians, devoted to Drinfeld at the party of his fiftieth birthday, generally replicate the diversity of Drinfeld's personal pursuits in algebra, algebraic geometry, and quantity idea.
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Extra resources for Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld's 50th Birthday
6 We further claim that (34) is, in fact, a polynomial in the coefﬁcients of (36), (37), and 1 η(q)8 1 = − = 2E2 (q) − 12E2 (q 2 ) + 16E2 (q 4 ). 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.
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 .
K (I0 ) = I0 , 2. dµk (i) = di , 3. εµk (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.
Algebraic Geometry and Number Theory: In Honor of Vladimir Drinfeld's 50th Birthday by victor ginzburg