Random Combinatorial Designs

Projektinformationen

RAND-COMB-DESIGN

ID Finanzhilfevereinbarung: 101106032

DOI

10.3030/101106032

Projekt abgeschlossen

EK-Unterschriftsdatum 17 April 2023

Startdatum 1 April 2024

Enddatum 31 März 2026

Finanziert unter

• Marie Skłodowska-Curie Actions (MSCA)

Gesamtkosten

Keine Daten

EU-Beitrag

€ 181 152,96

Koordiniert durch

UNIVERSITAT POLITECNICA DE CATALUNYA
Spain

Periodic Reporting for period 1 - RAND-COMB-DESIGN (Random Combinatorial Designs)

Berichtszeitraum: 2024-04-01 bis 2026-03-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

Experiments are used across many scientific fields to test various factors, and mathematics provides the tools to ensure these tests are both fair and efficient. Randomness is employed to make experiments fair by avoiding unintended biases, while combinatorial designs (highly regular patterns like those in a completed Sudoku square) make experiments efficient by testing every variable in the most resource-effective way. Purely random layouts can be wasteful, while ordered designs may have symmetries that reintroduce bias. This project investigated random designs to capture the benefits of both approaches. The main objectives were to advance the understanding of the properties of large designs from a probabilistic perspective and to develop new mathematical pathways toward efficient ways to sample these designs. The project also explored other high-level directions in discrete mathematics, including graph bootstrap percolation, which studies how information spreads through a network, and canonical Ramsey theory in random environments, which identifies inescapable regular patterns. A secondary goal was the professional development of the researcher to ensure a successful transition to an independent research career in Europe.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

The project was carried out through a combination of technical research, international collaboration, and academic leadership. The researcher conducted technical investigations that resulted in nine peer-reviewed publications and preprints in leading mathematics journals. These results were disseminated to the global scientific community through talks at major international congresses, such as the European Congress of Mathematics and the Random Structures and Algorithms conference, as well as through seminar visits to institutions in Brazil, Taiwan, and the United States. The researcher played a key role in animating the international research community by initiating the Barcelona Discrete Mathematics Meeting and co-organising a workshop at the Banff International Research Station. Academic mentorship was a significant part of the project, including the delivery of a specialised graduate course for the Barcelona Graduate School of Mathematics and the successful supervision of a Master’s thesis that resulted in a new research article. Public outreach was performed through popularisation articles written for general audiences to explain the impact of randomness and design theory on scientific experimentation. A major outcome of the project was the researcher successfully securing a five-year tenure-track Ramón y Cajal position at the Universitat Politècnica de Catalunya.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

The project has achieved significant progress across a diverse range of problems in discrete mathematics, delivering results that move beyond the current state of the art in several fields. In a result published in the journal Combinatorics, Probability and Computing, the researcher along with Amarja Kathapurkar and Guillem Perarnau solved a prominent case of a meta-conjecture regarding spanning structures in hypergraphs, establishing conditions under which rainbow loose Hamilton cycles must emerge in bounded colourings of dense hypergraphs. Another advance, published in the Journal of Combinatorial Designs, established a deterministic polynomial-time algorithm for finding large transversals in mathematical arrays, providing a constructive method for identifying balanced configurations. The project also established a new theoretical framework for analysing the time complexity of iterative processes on networks. This work with David Fabian and Tibor Szabó, summarised in the survey article 'Graph bootstrap percolation — a discovery of slowness' for the British Combinatorial Conference 2026, provides the mathematical community with rigorous benchmarks to calculate how long certain mathematical processes take to stabilise on large-scale networks. Furthermore, the researcher established new thresholds for regularity in random environments through canonical Ramsey theorems for even cycles and arithmetic progressions, appearing in journals such as the Proceedings of the American Mathematical Society. Finally, collaborative work with Jack Allsop on random Latin squares has generated new probability bounds that address methodological barriers previously preventing researchers from proving the efficiency of sampling methods used in bias-free experimental designs.