Riemann surfaces in geometry and analysis

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Riemann surfaces are fascinating objects with rich structure that involves complex analysis, topology and algebraic geometry. The techniques developed to deal with Riemann surfaces and related objects proved to be very useful, for example, in finding and studying solutions to important model differential equations arising in physics.We are going, on one hand, to develop further the theory of functions on Riemann surfaces by establishing properties of the so-called sigma-functions defined for an arbitrary Riemann surface.On the other hand, we plan to apply existing techniques of Riemann surface theory to find new relationship between two classical problems: Painlevé equations and Poncelet theorem. Painlevé equations play an important role in classification of differential equations having certain natural properties. The Poncelet theorem, in its simplest case, gives a criterion for a billiard trajectory within an ellipse to be closed. We want to find new relationship between these two problems to then generalize it and to conclude a similar relationship between more general structures. We hope that discovering these relationships will lead us to a better understanding of the structures involved. Another project deals with the so-called cluster algebras, certain combinatorial structures related to triangulations of surfaces. These algebras have theoretical importance in mathematics but they also have links to modern physics, namely to string theory. We are going to explore further their relationship with Riemann surfaces.We also plan to use the theory of Riemann surfaces to find discrete analogues of ellipsoidal coordinate lines in three dimensional space, which can in the future lead to applications in computer graphics.