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Randomness and structure in combinatorics

Project description

Investigating the impact of randomness on structured and random combinatorial objects

Combinatorics concerns the study of discrete structures, such as networks, permutations or set systems. Randomness has been shown to play an inseparable role in combinatorics. Probabilistic arguments that are non-constructive offer a powerful way to prove the existence of various kinds of combinatorial objects. The ERC-funded RANDSTRUCT project will seek to improve the understanding of the role of randomness in combinatorics, emphasising the relationship between structured (explicit) objects and random or random-like objects. RANDSTRUCT will investigate several concrete combinatorial problems, particularly in the areas of Ramsey theory and design theory.

Objective

Randomness plays an inseparable role in combinatorics. Indeed, non-constructive probabilistic arguments are a powerful way to prove the existence of various kinds of combinatorial objects, and the study of random discrete structures has illuminated nearly all fields of combinatorics. I propose a program to achieve a deeper understanding of this role of randomness in combinatorics, emphasising the relationship between “structured” (hence explicit) objects, and random or “random-like” objects.

A) There are many situations in combinatorics where probabilistic arguments demonstrate that “almost all” objects satisfy a certain property, but it is difficult to explicitly specify an object with the property. The most notorious examples are in Ramsey theory, which studies how “disordered” it is possible for an object to be. I plan to investigate the structure of Ramsey graphs, with the goals of unifying the area and making decisive progress on important conjectures.

B) Conversely, certain areas of combinatorics have been slower to benefit from the probabilistic method; particularly areas in which algebraic constructions play a major role. Design theory is the study of combinatorial “arrangements” with very strong regularity properties, most naturally obtained by exploiting symmetry/regularity properties of algebraic structures. I plan to investigate probabilistic aspects of design theory, and in particular to build a theory of random designs.

C) Actually, structure and randomness often come together, due to the “structure vs pseudorandomness dichotomy” elucidated by Tao. Indeed, there are many important problems in combinatorics for which it is known how to solve both random instances and “structured” instances; in such cases we hope to decompose general instances into structured and pseudorandom parts, handled by different means. I describe several concrete problems in this vein, whose study will advance our general understanding of this phenomenon.

Host institution

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA
Net EU contribution
€ 1 343 890,00
Address
Am Campus 1
3400 Klosterneuburg
Austria

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Region
Ostösterreich Niederösterreich Wiener Umland/Nordteil
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 1 343 890,00

Beneficiaries (1)