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By Konrad Schöbel

Konrad Schöbel goals to put the rules for a consequent algebraic geometric remedy of variable Separation, that is one of many oldest and strongest the way to build precise ideas for the basic equations in classical and quantum physics. the current paintings finds a stunning algebraic geometric constitution in the back of the recognized checklist of separation coordinates, bringing jointly a very good diversity of arithmetic and mathematical physics, from the past due nineteenth century concept of separation of variables to fashionable moduli house concept, Stasheff polytopes and operads.

"I am rather inspired through his mastery of quite a few strategies and his skill to teach essentially how they have interaction to supply his results.” (Jim Stasheff)

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Extra resources for An Algebraic Geometric Approach to Separation of Variables

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11. Again, the lemma could also be deduced from the symmetry classification of Riemann tensor polynomials [FKWC92]. We have shown the equivalence of the second integrability condition to g¯ij g¯kl S ikb1 b2 S jc2 d1 d2 + S ic2 b1 b2 S jd1 kd2 S lf2 e1 e2 b2 b1 d 1 e 1 e 2 c2 d2 f2 x b 1 x b 2 x d 1 x e 1 x e 2 u c2 v d 2 w f 2 = 0 ∀x ∈ M, ∀u, v, w ∈ Tx M. 3 The 2nd integrability condition 43 As before, the restrictions on the vectors u, v, w and x can be dropped, which allows us to write this condition independently of x, u, v, w ∈ V as b2 b1 d 1 e 1 e 2 c2 d2 f2 g¯ij g¯kl S ikb1 b2 S jc2 d1 d2 + S ic2 b1 b2 S jd1 kd2 S lf2 e1 e2 = 0.

Regarded as endomorphisms, is then given The product of K and K, by ˜ γ = Ra b a b R ˜ c d c d x a 1 x a 2 x c1 x c2 ∇ α x b 1 ∇ γ x b 2 ∇ γ x d 1 ∇ β x d 2 . 4 we have ∇γ xb2 ∇γ xd1 = g b2 d1 − xb2 xd1 . As a consequence of the antisymmetry of algebraic curvature tensors in the last index pair, the term xb2 xd1 does not contribute when substituting this identity into the previous expression: ˜ γ = g b2 d 1 Ra b a b R ˜ c d c d x a 1 x a 2 x c 1 x c2 ∇ α x b 1 ∇ β x d 2 . 41) for all x ∈ M and v, w ∈ Tx M .

13. 14) the Killing-St¨ variety, or KS variety for short. This already solves part (ii) of Problem I. Exploiting the algebraic geometric structure of the Killing-St¨ ackel variety then allows us to deduce the following results for the sphere S3 . 42: An algebraic geometric description of the space I(S3 ) of integrable Killing tensors on S3 . This solves part (iii) of Problem I. 12: A set of polynomial isometry invariants characterising the integrability of an arbitrary Killing tensor on S3 . This solves part (iv) of Problem I.

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