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We give linear equations that characterize the Fourier Jacobi expansions of paramodular forms from among all convergent series of Jacobi forms. We suspect that these linear equations in fact characterize the Fourier-Jacobi expansions of paramodular forms from among all formal series of Jacobi forms. We use these linear equations to compute small eigenvalues of possible weight two paramodular cusp forms up to level 1000. We compare this data with the Paramodular Conjecture for modularity in genus two using the work on rational abelian surfaces of A. Brumer and K. Kramer.

We describe an abelian group structure on the set of classes of modular categories modulo some equivalence relation. The resulting Witt group of modular categories resembles (and contains) the Witt group of finite abelian groups with quadratic forms. The conjecture of Moore and Seiberg, that all chiral rational conformal field theories come from reductive groups via WZW, coset and orbifold constructions, can be interpreted as a statement about generators of this Witt group.

This is a joint work with Rina Anno (U Chicago). We show how to construct a network of functors which correspond to `generalized braid diagrams', between derived categories of coherent sheaves on cotangent bundles of full and partial flag varieties. For a subclass of these diagrams (which includes all the ordinary braids) we prove that isotopic diagrams correspond to isomorphic functors. We then outline our strategy for proving the general case.