Discrete Schlesinger Transformations, Discrete Painleve Equations, and Birational Geometry of Algebraic Surfaces

Anton Dzhamay (University of North Colorado)

Friday 5th July, 2013 14:00-15:00 Maths 416

Abstract

I will discuss recent joint work with Hidetaka Sakai and Tomoyuki Takenawa on discrete Hamiltonian structures of Schlesinger transformations and their application to the theory of discrete Painlevé equations. One of our motivations is to see whether we can develop the isomonodromic approach to studying discrete Painlevé equations with an eye towards using the classification of Fuchsian system based on spectral type to study and classify the higher-order analogues of difference Painlevé equations, similar to the recent work of Sakai and his collaborators in the continuous case. We were able to find an explicit formula for a generating function of elementary Schlesinger transformations of Fuchsian systems in the eigenvector coordinates of the system and explained that this function should be thought of as a discrete Hamiltonian of our dynamic. We then computed some explicit examples of Schlesinger transformations using this generating function. We verified that it corresponds to the dP-V dynamic for the Fuchsian system that reduces to the usual differential P-VI equation. We then computed an example of the difference equation of the type d-P(A_{2}^{(1)*}) that has the symmetry group E_{6} and we have some preliminary results for the difference equation of the type d-P(A_{1}^((1)*)) that has the symmetry group E_{7}. To compare our dynamic with the standard dynamic of the same type we used the birational geometry of the Okamoto space of initial conditions, in the spirit of Sakai's geometric approach to the theory of Painlevé equations. One very attractive feature of this approach is the efficiency of the geometric description, which allowed us not only to compare the dynamics, but also to to find the explicit change of coordinates relating different dynamics of the same type.

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