This project lies at the interface of two fundamental mathematical fields: combinatorics and algebra. The research pursues a powerful and novel approach to establish connections between the two areas, and the interplay between mathematical structures in common to both areas is investigated in depth. As a general theme, many problems in mathematics and physics can be recast as finding the largest size of a mathematical structure subject to prescribed constraints on the substructures. It is often the case that the largest structures organize themselves in a highly algebraic way, which opens the door to algebraic methods. These so-called extremal structures have a remarkable number of applications to other areas of mathematics and science, including to theoretical computer science, statistical physics and coding theory. A number of recent breakthroughs in the area reveal an exciting future for the proposed work. The PI will work closely with both undergraduate and graduate students in the research project. In addition, the PI is preparing comprehensive educational materials on the new techniques, whose target audience will be both students and experts in the area.
There are several problems currently known to be at the interface of extremal combinatorics and algebra. Some examples are explicit constructions of Ramsey graphs, constructions of large graphs not containing a specific subgraphs, bounds on spherical codes, zero-error capacities of communication channels, set families with restricted intersections. The PI is an expert in creating novel algebraic and geometric methods for application to such combinatorial problems, and these methods are central to many of the most importance advances and new results in the area.
