Random Simplicial Complexes and Associated Spaces

PI计划研究随机简单配合物的各种模型的性质和应用。这些对象与随机图密切相关，随机图在数学中引起了50多年的兴趣，并广泛应用于计算机科学和工程中，用于各种网络和互联网的建模。把一个图看作一个一维的简单复合体，研究高维的随机简单复合体是很自然的。除了这些对象的基本代数、组合和拓扑性质外，PI还将研究一些在机器人等应用中有潜在兴趣的相关对象，包括随机构型空间和矩角复合体。随机图理论是离散数学中一个快速发展的分支，它汇集了图论、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。这些对象在数学中也可以作为其他更复杂的随机结构的可访问模型。随机简单复合体提供了这样一种结构，它将把代数和拓扑学的技术和思想与上述领域的技术和思想结合起来。

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.