GEOMETRY OF NUMBERS AND CODING THEORY

Urakawa定义了图的调和态射，给出了无限树的格林核的一个很好的估计。他发现了无限图的离散拉普拉斯谱的一个很好的估计。他在CR流形上建立了向量束的厄密连接，证明了非齐次Yang-Mills方程解的存在唯一性。他证明了伪凸CR流形的CR-映射是伪调和的当且仅当它们是伪厄米特的。他还引入了映射稳定性的概念，并证明了负弯曲黎曼流形中的伪调和映射是稳定的。他发展了没有方程Dh=0的杨-米尔斯理论，用于黎曼流形上向量束的连接，并将该理论应用于爱因斯坦-魏氏几何和仿射微分几何。Taya在假设利奥波德猜想的情况下，利用p进zeta函数的值，得到了表示循环zp扩展的中间域的p类数的公式。他还证明存在无穷多个实数二次域，其中素数。3分裂使得不变量为0。他估计了这些磁场的密度。Shimokawa考虑对产生晶状体间隙的强可逆结进行Dehn手术。他发现了圆盘上的图包含一些特征子图的条件。他证明了压缩体中平凡弧线的任何高度分裂都是标准的。他引入了（M， T）对的heegard分裂的概念，其中M是紧致可定向的3流形，T是1子流形。

Urakawa defined the harmonic morphisms of graphs, and gave good estimate of the Green kernel of an infinite tree. He found good estimate of the spectrum of the discrete Laplacian of an infinite graphs. He built Hermitian connections of a vector bundle on a CR manifold, then showed existence and uniqueness of the solution of inhomogeneous Yang-Mills equation. He proved that CR-maps of pseudo convex CR manifolds are pseudoharmonic if and only if they are pseudohermitian. He also introduced the notion of stability of the maps, and proved pseudoharmonic maps into negatively curved Riemannian manifolds are stable. He developed the Yang-Mills theory without the equation Dh=0 for connections in a vector bundle over a Riemannian manifold, and applied this theory to Einstein-Wey1 geometry and to affine differential geometry.Taya found the formula representing p-class numbers of intermediate fields of the cyclic Zpextension by the values of p-adic zeta functions, assuming Leopoldt conjecture. He also showed there exist infinitely many real quadratic fields in which the prime.3 splits such that lambda invariants are 0. He estimated the density of such fields.Shimokawa considered Dehn surgeries on strongly invertible knots yielding lens spaces. He found conditions for a graph in the disc to contain some characteristic subgraphs. He showed any Heegaard splitting of trivial arcs in a compression body is standard. He introduced the notion of Heegaard splittings of the pair (M, T), where M is a compact orientable 3-manifold and T is a 1-submanifold.