Discrete Geometry, Extremal Graph Theory, and Additive Combinatorics

This project concerns fundamental problems about the emergence of hidden structure in large and complex systems. One setting of interest is geometric configurations, where many basic questions about arrangements of elementary geometric objects (e.g., points, lines, and surfaces) will require new mathematical tools to resolve. There are many promising connections between these abstract problems and concrete questions involving lattice structure, efficient communication, and quantum information science. For example, the design of highly efficient quantum sensors is intrinsically linked to understanding the structure of highly symmetric arrangements of flat surfaces. This project will also study abstract networks, which model biological and neural systems, social media networks, and logistics networks, as well as arithmetic systems, which are closely tied to basic research in number theory. Such number-theoretic problems provide the foundation for the design of secure cryptosystems and the development of extremely efficient algorithms for analyzing large datasets. This project will also involve the training of students in the skills necessary to tackle these fundamental problems. In discrete geometry, this project will develop new polynomial method tools to study high-dimensional incidence geometry, including applications to the Erdős distinct distances problem in dimensions three and higher. The project will also study equi-isoclinic arrangements of flats, with direct applications to Zauner's conjecture in quantum information science. Connections to extremal graph theory will be studied, focusing specifically on the existence of geometric separators and their applications to efficient algorithms in computational geometry. Further work in extremal graph theory will include the study of hypergraph Ramsey numbers and their relationship to hypergraph sparsity. This part will build upon the PI's resolution of the hypergraph Burr-Erdős conjecture. Connecting extremal graph theory and additive combinatorics, this project will study problems in arithmetic Ramsey theory that prior work of the PI has linked to foundational questions in higher-order Fourier analysis. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2554092 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Jonathan Tidor | Institution: Princeton University, PRINCETON, NJ | Award Amount: $150,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554092 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554092.html