Moduli Spaces and Differential Equations

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Proposal Number DMS - 0205643 Investigator Emma Previato ep@math.bu.edu Moduli spaces and differential equationsThis project combines projective geometry and differential algebra. For a few decades now, moduli spaces of vector bundlesover curves have been prominent in mathematical physics. A few interrelated open problems are the object of this research. In projective space, models of moduli of vector bundles still don't have explicit description, by equations or by classification. The generalized theta functions (sections of higher-rank vector bundles) are not developed to the point that one can describe flows of completely integrable hierarchies or perform the necessary calculations of quantum field theory(partition function, e.g.) Finally, flows of commutative rings of partial differential operators (generalized KP flows) have not been described explicitly. In the proposed work: on the projective-geometry side, equations for moduli spaces of higher-rank bundles and dimensions of Brill-Noether loci will be calculated by methods of representation theory and correspondences between Grassmannians. On the analytic side, differential equations for the Kleinian functions (which generalize the Weierstrass p-function) will be derived and applied to integrate new Hamiltonian systems and the generalized KP flows. A theme that runs through the proposed research is the classical problem of reduction: Weierstrass' students Koenigsberger and Kowalevski, respectively, characterized the abelian integrals of genus 2, 3 respectively, that reduce with "multiplicity" 2 to an elliptic integral; since then, very little was found in general (e.g., higher genus or multiplicity). Progress on this problem is now under way, partly due to results that revisit the Kleinianfunctions, and partly to the aid of computer algebra. The problem of reduction is linked with the problem of curves with automorphisms, and the project includes applications of the results to differential Galois theory, monodromy of ordinary differential equations, and decoding algorithms for algebraic (Goppa) codes. Elliptic functions have an "unreasonable effectiveness",to borrow E.P. Wigner's phrase. They occur when modeling harmonic oscillators, shooting billiards, measuring the amplitude of ocean waves, computing the partition functions of quantum field theory. The word "elliptic" refers to the number of periods of the functions (one over the real, two over the complex numbers). Theta functions are the multi-periodic analog of elliptic functions. Although Old Masters such as Klein, H.F. Baker and O. Bolza,obtained equations for genus-2 (4 complex periods) theta functions, most properties of theta functions are still inexplicit, including their dependence on the period lattice. New impetus in the study of such functions came from the theory of integrable PDEs, such as the Kadomtsev-Petviashvili equation, and the attendantalgebraically completely integrable systems that have been intensively studied since the 1970s. This project will combine projective geometry and differential algebra to identify differential equations satisfied by the (Kleinian) theta functions, and apply them to find exact solutions of Hamiltonian systems and non-linear PDEs.At the same time the project will pursue the other major, "non-abelian" generalization of theta functions, by embedding moduli spaces into projective space. Theta functions that can be reduced to expressions in elliptic functions will be characterized geometrically and used in effective decoding algorithms.