Periodic Reporting for period 1 - RANDSTRUCT (Randomness and structure in combinatorics)
Reporting period: 2023-05-01 to 2025-10-31
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.
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.
Some of the main achievements so far have been in the direction of the so-called polynomial Littlewood-Offord problem, which studies "smoothness" properties of polynomials of independent random variables, and is closely related to some statistical questions about graphs (particularly some questions arising from Ramsey theory). In particular, we were able to prove a longstanding conjecture in this area, obtaining square-root bounds for the quadratic Littlewood-Offord problem. We have also uncovered some connections to phenomena in number theory (specifically, about counting lattice points on varieties).
Another key achievement is a comprehensive study of the smoothed analysis paradigm in the context of the graph isomorphism problem. The graph isomorphism problem is one of the most important problems in computer science, and a classical theorem of Babai, Erdős and Selkow tells us that very simple algorithms are very effective "on average", but we know of examples with very intricate structure where such algorithms perform poorly. We extended the Babai-Erdős-Selkow theorem to the so-called smoothed analysis setting, in which one starts with an object with arbitrary structure and randomly perturbs it.
In addition, we have made some important methodological advances to the theory of random designs, showing how to combine methods based on switching with methods based on enumeration to resolve at least one old conjecture in the area.
We have also found successes in a number of unexpected directions, including the resolution of a longstanding conjecture of Nielsen in quantum information theory, and the solution of some important questions on the rank and matching number of random graphs and random matrices.
Another key achievement is a comprehensive study of the smoothed analysis paradigm in the context of the graph isomorphism problem. The graph isomorphism problem is one of the most important problems in computer science, and a classical theorem of Babai, Erdős and Selkow tells us that very simple algorithms are very effective "on average", but we know of examples with very intricate structure where such algorithms perform poorly. We extended the Babai-Erdős-Selkow theorem to the so-called smoothed analysis setting, in which one starts with an object with arbitrary structure and randomly perturbs it.
In addition, we have made some important methodological advances to the theory of random designs, showing how to combine methods based on switching with methods based on enumeration to resolve at least one old conjecture in the area.
We have also found successes in a number of unexpected directions, including the resolution of a longstanding conjecture of Nielsen in quantum information theory, and the solution of some important questions on the rank and matching number of random graphs and random matrices.
Already, in our work on the quadratic Littlewood-Offord problem and on random designs, we have developed new general methods which we believe can be applied to many more problems (including some particular problems outlined in the proposal, but also some problems in quite different areas, such as the analysis of Boolean functions). We have also uncovered connections between fields that warrants significant further investigation (in particular, a link between the polynomial Littlewood-Offord problem and analytic number theory)
In general, techniques and perspectives developed in probabilistic combinatorics have a history of leading to important breakthroughs in related areas of mathematics and computer science.
In general, techniques and perspectives developed in probabilistic combinatorics have a history of leading to important breakthroughs in related areas of mathematics and computer science.