An Algebraic Geometric Approach to Separation of Variables by Konrad Schöbel

By Konrad Schöbel

Konrad Schöbel goals to put the principles for a consequent algebraic geometric remedy of variable Separation, that's one of many oldest and strongest how you can build certain suggestions for the basic equations in classical and quantum physics. the current paintings finds a stunning algebraic geometric constitution in the back of the recognized record of separation coordinates, bringing jointly an exceptional diversity of arithmetic and mathematical physics, from the past due nineteenth century idea of separation of variables to trendy moduli house concept, Stasheff polytopes and operads.

"I am fairly inspired by means of his mastery of various suggestions and his skill to teach basically how they have interaction to provide his results.” (Jim Stasheff)

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4 Interpretation of the Killing-St¨ ackel variety . . 1 St¨ ackel systems and isokernel lines . . . . . . 2 Antisymmetric matrices and special conformal Killing tensors . . . . . . . . . . . . . . 3 Isokernel planes and integrable Killing tensors from S2 K.

This solves part (iii) of Problem III. 34: A set of polynomial isometry invariants classifying separation coordinates on S3 . This solves part (iv) of Problem III. In particular, we will prove that the space of St¨ackel systems on S3 with diagonal algebraic curvature tensors is isomorphic to the blow-up P2 #4P2 of P2 in four points and that S(S3 )/O(4) ∼ = (P2 #4P2 )/S4 . All material presented in the first two chapters is self contained, based only on St¨ackel and Eisenhart’s characterisation of orthogonal separation coordinates.

This solves part (iii) of Problem III for spheres of arbitrary dimension. 1: A labelling of the different classes of separation coordinates on spheres in terms of the combinatorics of Stasheff polytopes. 2: A description of the operad structure on orthogonal separation coordinates on spheres, resulting in an explicit construction of separation coordinates on spheres. 3: The same for St¨ackel systems. In particular, we recover in this way the classical list of separation coordinates on Sn in an independent, self-contained and purely algebraic way.

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