Random Simplicial Complexes and Associated Spaces

PI建议研究随机单纯复形的各种模型的性质和应用。这些对象与随机图密切相关，50多年来，随机图一直是数学领域的研究热点，并且在计算机科学和工程中被广泛用于各种网络和互联网的建模。将图看作一维单纯复形，研究高维随机单纯复形是很自然的。除了这些对象的基本代数、组合和拓扑属性外，PI还将研究一些相关的对象，如机器人应用中潜在感兴趣的对象，包括随机配置空间和矩-角复数。随机图理论是离散数学中一个迅速发展的分支，集合了图论、组合学和概率论的思想。随机图在计算机科学、工程学、物理学、生物学、化学和社会科学中都有应用。这些对象还在数学中充当其他更复杂的随机结构的可访问模型。随机单纯复形提供了这样一种结构，它将代数和拓扑学中的技术和思想与前述领域的技术和思想相结合。

The PI proposes to investigate the properties and applications of various models for random simplicial complexes. These objects are closely related to random graphs which have been of interest in mathematics for over fifty years, and have been widely used in computer science and engineering for the modeling of various networks and the internet. Viewing a graph as a one-dimensional simplicial complex, it is natural to investigate random simplicial complexes of higher dimension. In addition to basic algebraic, combinatorial, and topological properties of these objects, the PI will study a number of related objects of potential interest in applications such as robotics, including random configuration spaces and moment-angle complexes.The theory of random graphs is a rapidly growing branch of discrete mathematics, bringing together ideas from graph theory, combinatorics, and probability theory. Random graphs have found use in computer science, engineering, physics, biology, chemistry, and the social sciences. These objects also serve within mathematics as accessible models for other, more complex random structures. Random simplicial complexes provide such a structure, one that will combine techniques and ideas from algebra and topology with those from the aforementioned fields.