Combinatorics is the area of mathematics concerned with finite structures and their properties. This subject is enormously diverse and has connections to many different areas of science: for example, objects of study include networks, sets of integers, error-correcting codes, voting systems, and arrangements of points in space.
Increasingly, randomness has come to play an inseparable role in combinatorics. Indeed, non-constructive probabilistic arguments are a powerful way to prove the existence of various kinds of combinatorial objects (this is the so-called probabilistic method, pioneered by Paul Erdős), and the study of random discrete structures has illuminated nearly all fields of combinatorics. The purpose of this project is to achieve a deeper understanding of this role of randomness in combinatorics, emphasising the relationship between “structured” objects, and random or “random-like” objects.
In particular, special emphasis is placed on the subjects of Ramsey theory (which studies how "disordered" it is possible for an object to be) and design theory (which studies combinatorial "arrangements" with very strong regularity properties). These two subjects sit at opposite ends of the structure-vs-randomness divide: objects with good Ramsey properties are easy to obtain via randomness, but it is difficult to easily specify them, while combinatorial designs are most naturally obtained by exploiting symmetry/regularity properties of algebraic structures.
More concretely, the goals of this project are to build connections across fields (via perspectives related to randomness), develop general probabilistic and combinatorial tools related to the structure-vs-randomness dichotomy, and make decisive progress on a number of important conjectures.
