Global Effects of Local Structure

Ziel

A fundamental question in many areas of mathematics is how to understand the relationship between local and global structure. For instance, what can you say about the local structures of a large, complicated network? How is global structure constrained by what we see locally? How can we piece together local structure to obtain global information? Problems of this type have become urgent in many parts of mathematics and related areas, including model theory, geometric group theory, topology, combinatorics, computer science, statistical physics, biology, probability, network science, and data analysis. Concrete examples include DNA sequencing, where fragments of a genetic sequence need to be assembled to a global sequence, and problems in big data, where the global structure is too large to access directly and its properties must be inferred from “local” information.

The goal of this project is to investigate the relationship between the local and global structure of discrete objects. The proposal draws on methods from combinatorics and other areas of mathematics and computer science. It has four strands: (1) asymptotic structure, with the goal of developing an analogue of the Graph Minors Project in the new area of coarse graph theory; (2) global structure and induced substructure, with the goal of developing a rough structural theory for graphs with forbidden induced subgraphs; (3) reconstruction from local information, with the goal of building a coherent body of theory for reconstructing a wide variety of mathematical objects; and (4) structure and noise, with the goal of developing rigorous methods for identifying hidden structure in the presence of noise.

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