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"Applying Fundamental Mathematics in Discrete Mathematics, Optimization, and Algorithmics"

Final Report Summary - AFMIDMOA (Applying Fundamental Mathematics in Discrete Mathematics, Optimization, and Algorithmics)

The project has solved problems of De la Harpe and Jaeger concerning counting graph homomorphisms, of Szegedy concerning edge-colouring models, of Lovász concerning graph limits, of Chmutkov et al. concerning the weight system from the exceptional Lie algebra G_2, and of Erdös and Pach in Ramsey theory.
Moreover, the application of algebraic geometry and representation theory to partition functions was extended so as to include the symplectic group, R-matrices, virtual link invariants, and weight systems for the Vassiliev knot invariants.
With an extended semidefinite programming method based on Young tableaux several bounds on error-correcting codes (including constant-weight codes and Lee codes) were improved.
With an improved technique with combinatorial groups a polynomial-time algorithm was designed for finding partially disjoint paths in a directed planar graph.
A Tutte polynomial for embedded graphs was introduced and shown to extend counting substructures like quasi-trees.
A new lower bound on the Shannon capacity of C_7 was found.
Extension of zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs.