SubsidieMeestersRandomness and structure in combinatorics
The project aims to deepen the understanding of randomness in combinatorics by exploring the relationship between structured and random objects, focusing on Ramsey graphs and design theory.
Projectdetails
Introduction
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, emphasizing the relationship between “structured” (hence explicit) objects, and random or “random-like” objects.
Investigating Ramsey Graphs
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
- Making decisive progress on important conjectures.
Exploring Design Theory
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.
Structure vs Pseudorandomness
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.
Financiële details & Tijdlijn
Financiële details
| Subsidiebedrag | € 1.343.890 |
| Totale projectbegroting | € 1.343.890 |
Tijdlijn
| Startdatum | 1-5-2023 |
| Einddatum | 30-4-2028 |
| Subsidiejaar | 2023 |
Partners & Locaties
Projectpartners
- INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIApenvoerder
Land(en)
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