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| == Grunsky == | |||
| Hi R. Since the theory of Grunsky matrices involves interpreting them as operators, I will be making ] the main article and ] a redirect. There are a series of articles to write including a ] article (using the Cauchy transform for existence for sufficiently smooth curves/diffeomorphisms and the Beltrami equation more generally), and the Grunsky matrices fit into that, this time for a ''pair'' of univalent functions corresponding to the conformal welding of the diffeomorphism. In that context the Grunsky matrices become corners of a unitary matrix. They also appear as part of the smooth model of ], which is the same thing as a slight generalisation of Diff ( S<sup>1</sup>) that several authors have studied (including your friend from UCSC, who seems to have published a corrected version of the infamous paper that did not get slated in on Maths Reviews). The matrix coefficient in the projective representation of Diff ( S<sup>1</sup>) is expressed as a Fredholm determinant which arises in geometric function theory (called Fredholm eigenvalues of a planar domain) and has been calculated explicitly. Anyway, just so that you are warned (I thought quite long about what title the article should have). I liked the bio on Grunsky, which I had contemplated writing myself. Regards, A. ] (]) 18:28, 8 December 2011 (UTC) | |||
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Grunsky
Hi R. Since the theory of Grunsky matrices involves interpreting them as operators, I will be making Grunsky matrix the main article and Grunsky inequalities a redirect. There are a series of articles to write including a conformal welding article (using the Cauchy transform for existence for sufficiently smooth curves/diffeomorphisms and the Beltrami equation more generally), and the Grunsky matrices fit into that, this time for a pair of univalent functions corresponding to the conformal welding of the diffeomorphism. In that context the Grunsky matrices become corners of a unitary matrix. They also appear as part of the smooth model of universal Teichmüller space, which is the same thing as a slight generalisation of Diff ( S) that several authors have studied (including your friend from UCSC, who seems to have published a corrected version of the infamous paper that did not get slated in on Maths Reviews). The matrix coefficient in the projective representation of Diff ( S) is expressed as a Fredholm determinant which arises in geometric function theory (called Fredholm eigenvalues of a planar domain) and has been calculated explicitly. Anyway, just so that you are warned (I thought quite long about what title the article should have). I liked the bio on Grunsky, which I had contemplated writing myself. Regards, A. Mathsci (talk) 18:28, 8 December 2011 (UTC)