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.