Hodge-Type Decompositions for Piecewise Constant Vector Fields on Simplicial Surfaces and Solids with Boundary
Hodge-Type Decompositions for Piecewise Constant Vector Fields on Simplicial Surfaces and Solids with Boundary
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单纯曲面和有边界固体上分段常向量场的Hodge型分解
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发表时间：
2017
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通讯作者：
Konstantin Poelke
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作者：
Konstantin Poelke
在无边界的紧致可定向光滑流形上，调和k型空间可以看作是第k次实系数奇异上同的具体表示，这是de Rham经典定理的一个结果。然而，在存在边界的情况下，这个空间变成了无限维，并且失去了与拓扑结构的联系。然而，有有限维的狄利克雷和诺伊曼形式的子空间，它们分别同构于相对上同构和绝对上同构。deTurck， Gluck和Shonkwiler最近的结果确定了这两个空间之间的主角-创造了庞卡罗对偶角-作为具有边界的流形的重要内在特征，它将边界分量的影响与流形的“内部”拓扑联系起来。在这次演讲中，我将介绍和讨论具有边界的三角曲面上调和诺伊曼场和狄利克雷场的离散版本的庞加莱对偶角，并附有一些数值例子来说明这一概念。《死亡之争》（Die dispute）是一篇关于“死亡之争”（Die dispute）的论文。波尔蒂耶博士教授
On a compact, orientable smooth manifold without boundary the space of harmonic k-forms can be considered as a concrete representation of the k-th singular cohomology with real coefficients, which is a consequence of a classical theorem by de Rham. In the presence of a boundary, though, this space becomes infinite-dimensional and the linkage to the topology is lost. Still, there are finite-dimensional subspaces of socalled Dirichlet and Neumann forms, isomorphic to the relative and absolute cohomology, respectively. A recent result by deTurck, Gluck and Shonkwiler identifies the principal angles between these two spaces -coined the Poincaré duality angles -as a significant intrinsic characteristic of manifolds with boundary, which relates the influence of the boundary components to the "inner" topology of the manifold. In this talk I will introduce and discuss the Poincaré duality angles for a discrete version of harmonic Neumann and Dirichlet fields on triangulated surfaces with boundary, accompanied by some numerical examples which illustrate this concept. Die Disputation besteht aus dem o. g. Vortrag, danach der Vorstellung der Dissertation einschließlich jeweils anschließenden Aussprachen. Interessierte werden hiermit herzlich eingeladen Der Vorsitzende der Promotionskommission Prof. Dr. K. Polthier