GEOMETRY OF NUMBERS AND CODING THEORY

Urakawa定义了图的调和态射，并给出了无限树的Green核的良好估计。他发现了无限图的离散拉普拉斯谱的很好的估计。他建立了CR流形上向量丛的Hermite联络，然后证明了非齐次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。他还引入了映射的稳定性的概念，并证明了到负曲黎曼流形的伪调和映射是稳定的。他发展了关于黎曼流形上向量丛中连通的Yang-Mills理论，没有方程dh=0，并将该理论应用于Einstein-Wey1几何和仿射微分几何。Taya在Leopoldt猜想的假设下，找到了用p-进Zeta函数的值来表示循环Zp扩张的中间场的p类数的公式。他还证明了存在无穷多个实二次域，其中素数3分裂使得Lambda不变量为0。他估计了这种磁场的密度。Shimokawa考虑在产生透镜空间的强可逆结上进行Dehn手术。他发现了圆盘中的一个图包含某些特征子图的条件。他指出，压缩物体中平凡圆弧的任何Heegaard分裂都是标准的。他引入了(M，T)对的Heegaard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.