Global Analysis of the heat kernel and Green kernel of an Infinite Graph

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We have obtained the following results:(1)We constructed the theory of Yang-Mills connections over compact symplectic manifolds.(2)We estimated the Cheeger constant, the heat kernel and the Green kernel for an infinite graph in terms of the volume growth, growth of in and out degree.(3)We determined the stiffness and mass matrices of the finite element method for the Dirichlet eigenvalue problem for a plane domain.(4)We calculated the Cheeger constant, the heat kernel and Green kernel of semi-regular infinite graphs and gave the explicit comparison theorem for every infinite graph.(5)We extended Yang-Mills theory to Weyl structure, and established Atiya-Hitchin-Singer theory to Weyl manifolds, and to affine connections.(6)We formulated discrete improper affine surface theory and show its loop group description.(7)We showed the relation in affine differential geometry, Weyl geometry, Yang-Mills theory.(8)We defined the notion of pseudoharmonic maps from CR manifolds to a Riemanninan manifold, and showed the first variation formula and the second variation formula.(9)We clarified the relation of each Yang-Mills theory on Kaehler manifolds, CR manifolds, and symplectic manifolds, and characterized the minimizers of the Yang-Mills functional over compact symplectic manifolds.