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Randomness and structure in combinatorics

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 structur... ver más

30/04/2028

IST AUSTRIA

1M€

Presupuesto del proyecto: 1M€

Líder del proyecto

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Financiación concedida El organismo HORIZON EUROPE notifico la concesión del proyecto el día 2022-12-19

ERC-2022-STG: ERC STARTING GRANTS Scope:Objectives

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Cerrada hace 2 años

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1 Participantes

IST AUSTRIA

1.34M€ | Lider

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Duración del proyecto: 64 meses Fecha Inicio: 2022-12-19
Fecha Fin: 2028-04-30

Líder del proyecto

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 1M€

Descripción del proyecto 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.

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