Spectral curves of polygons and triangulated tori

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We investigate an approach to discrete conformality based on the notion of holomorphic line bundles over "discrete surfaces", that is, over vertex sets of triangulated surfaces with black and white colored faces. As a special case, we give a reinterpretation of Dynnikov's and Novikov's approach to conformal maps to S2 = CP1 which reveals it as the first example of a theory of discrete holomorphicity that is at the same time Möbius-invariant and governed by linear equations.We introduce Darboux transformations for arbitrary immersions of discrete surfaces into S4 = HP1 which can be interpreted as a time discrete Davey-Stewartson flow on the space of immersions. For generic immersions of discrete tori with regular combinatorics, we show that the space of Darboux transformations can be desingularized to a compact Riemann surface (the spectral curve) thus making available powerful methods from the theory of algebraically completely integrable systems.In the second period, beyond the soliton theory of triangulated surfaces, our investigations will concentrate on developing a definition of conformality for immersions of "discrete Riemann surfaces". Moreover, we plan to study a new class of "discrete minimal surfaces" that appears naturally in the context of our investigations.