Combinatorial Construction

Informazioni relative al progetto

CC

ID dell’accordo di sovvenzione: 647678

DOI

10.3030/647678

Progetto chiuso

Data della firma CE 23 Marzo 2015

Data di avvio 1 Gennaio 2016

Data di completamento 30 Giugno 2021

Finanziato da

• EXCELLENT SCIENCE - European Research Council (ERC)

Costo totale

€ 1 706 729,00

Contributo UE

€ 1 706 729,00

Coordinato da

THE CHANCELLOR, MASTERS AND SCHOLARS OF THE UNIVERSITY OF OXFORD
United Kingdom

Periodic Reporting for period 4 - CC (Combinatorial Construction)

Periodo di rendicontazione: 2020-07-01 al 2021-06-30

Sintesi del contesto e degli obiettivi generali del progetto

Combinatorial Construction is a mathematical challenge with many applications. Examples include the construction of networks that are very sparse but highly connected, or codes that can correct many transmission errors with little overhead in communication costs. For a general class of combinatorial objects, and some desirable property, the fundamental question in Combinatorial Construction is to demonstrate the existence of an object with the property, preferably via an explicit algorithmic construction. Thus it is ubiquitous in Computer Science, including applications to expanders, sorting networks, distributed communication, data storage, codes, cryptography and derandomisation. In popular culture it appears as the unsolved "lottery problem" of determining the minimum number of tickets that guarantee a prize. In 2014 I proved the Existence Conjecture for combinatorial designs, via a new method of Randomised Algebraic Constructions; this result has already attracted considerable attention in the mathematical community. The significance is not only in the solution of a problem posed by Steiner in 1852, but also in the discovery of a powerful new method, that promises to have many further applications in Combinatorics, and more widely in Mathematics and Theoretical Computer Science. This project has achieved its aim of developing this idea into a general method, thus resolving several other related longstanding open problems.

Lavoro eseguito dall’inizio del progetto fino alla fine del periodo coperto dalla relazione e principali risultati finora ottenuti

Significant progress has been achieved on all four objectives of the project, resulting in several publications which have been disseminated by team members at academic conferences. The first objective is to develop the method of Randomised Algebraic Construction. This has been achieved by two papers, one refining and simplifying my original proof of the Existence Conjecture, and the second greatly generalising it to encompass many new applications, including generalised forms of two conjectures of Ringel: the Oberwolfach Problem and his tree packing conjecture. The second paper also makes progress on the second objective, which is a unifying framework for the existence of perfect matchings in sparse graphs. Further progress on this objective has been achieved by a series of papers culminating in improved bounds for Ryser's Conjecture on transversals in latin squares. Within the third objective, which is the theory of random independent sets and matchings in hypergraphs, we have developed new techniques for approximations of partition functions. The fourth objective, a structural characterisation of isoperimetry in Hamming spaces, has been very substantially addressed by several papers, including stability results for the vertex-isoperimetric and edge-isoperimetric inequalities.

Progressi oltre lo stato dell’arte e potenziale impatto previsto (incluso l’impatto socioeconomico e le implicazioni sociali più ampie del progetto fino ad ora)

In each of the four objectives, the work described above goes beyond the previous state of the art.

An octahedral move for triangle decompositions