The global geometry of moduli spaoes

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Nakamura proved a new theorem on McKay correspondence of a simple singularity C2/G, G being a finite group of SL(2,C). The theorem shows certain natural modules V and Vdagger have simple structures. Among others, the structure of Vdagger explains completely the known bijetive correspondence of the extended Dynkin diagram and all the irreducible representations of G.. The paper is now in print. He also proved an important vanishing theorem for degenerate quasi-abelian varieties.Weng is constructing a new important theory of geometric class field theory, which is modeled after Mumford's stability of GIT, Seshadri-Narashiman's theory of unitary vector bundles. In this program he defined nonabelian L-functions, and in some cases he proved a theorem analogous to Riemann hypothesis.Kato studied with Fujiwara the fundamental theory of p-adic geometry and rigid geometry. He proved also that Mumford fake projective plane and the other known fake projective planes are among Shimura varieties.Kondo constructed a uniformization by a 5-dimensional complex ball of the moduli of ordered 8 points of the projective line, by using Borcherds modular forms.Matsumoto gave a very precise description of a link and its complement in S3 by using real theta functions.Yoshioka proved a formula of counting the number of instantons on certain complex surfaces with Nakajima.