The Large-Scale Fractal Geometry of Riemannian Moduli Space
The Large-Scale Fractal Geometry of Riemannian Moduli Space
复制标题
黎曼模空间的大尺度分形几何
DOI：
10.2307/j.ctv17db3tj.8
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
S. Weinberger
中科院分区：
文献类型：
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作者：
S. Weinberger
这本书是第一次提出一个新的数学研究领域，结合拓扑，几何和逻辑。Shmuel Weinberger试图解释和说明由Alex Nabutovsky首先强调的一般原则的含义，即逻辑复杂性产生几何复杂性。他提供了应用问题的封闭测地线，理论的子流形，结构的模空间的等距类黎曼度量与曲率界对一个给定的流形。最终，模空间的几何复杂性迫使定义在该空间上的函数具有许多临界点，并且关于极值或平衡点的存在性的新结果随之而来。出现的主要算法问题是识别：所呈现的对象是否等同于某个标准对象？如果很难确定问题是否可解，那么原始对象就有了二重体--也就是说，很难与它区分开来的其他对象。许多新的问题出现了，关于已知几何定理的算法性质，关于"二分法问题"，以及关于模空间的度量熵。温伯格使用群论、可计算性、微分几何和拓扑学的工具来研究它们，所有这些他都在使用前进行了解释。由于几个例子已经制定出来，总体原则是在一个明确的救济，超越了任何一个问题的细节。
This book is the first to present a new area of mathematical research that combines topology, geometry, and logic. Shmuel Weinberger seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity. He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow. The main sort of algorithmic problem that arises is recognition: is the presented object equivalent to some standard one? If it is difficult to determine whether the problem is solvable, then the original object has doppelgangers--that is, other objects that are extremely difficult to distinguish from it. Many new questions emerge about the algorithmic nature of known geometric theorems, about "dichotomy problems," and about the metric entropy of moduli space. Weinberger studies them using tools from group theory, computability, differential geometry, and topology, all of which he explains before use. Since several examples are worked out, the overarching principles are set in a clear relief that goes beyond the details of any one problem.