Tassio Naia starts his Marie Curie Postdoctoral Fellowship at CRM

Tassio Naia got a MSCA-PF at Centre de Recerca Matemàtica, hosted by Guillem Perarnau. The prestigious 2-year grant, will allow him to develope a project on Extremal and Probabilistic Combinatorics.

Title: Rainbow partitions and oriented structures in random graphs

Abstract: Combinatorial designs are some of the most fascinating and elusive objects in mathematics. For over 200 years their study has generated many sophisticated methods, and the theory has found applications in biological experiment design, construction of strong error correcting codes, and network fault detection. The celebrated proof of the existence of designs has been an impressive achievement of the last decade. At the core of this landmark result lies a combination of powerful contemporary tools: the semi-random and absorption methods. The PARTIORI project seeks to utilise and extend these ideas to demonstrate the existence of certain transversal partitions and large oriented structures of, respectively, coloured an oriented random graphs. At the intersection of combinatorial design theory, Ramsey theory, and probabilistic combinatorics, PARTIORI lies at the edge of active sub-areas, exploring rainbow decompositions of edge-coloured random graphs and thresholds for the emergence of an Ramsey property of graph orientations. More precisely, the project aims at leveraging powerful methods from extremal and probabilistic combinatorics (the semi-random, absorption, regularity and container methods) to make significant contributions to the following two areas: - obtaining decompositions of edge-coloured complete graphs into few rainbow paths; - establishing universality results for the containment of oriented structures in arbitrary orientations of random graphs. The research is planned for two years and takes as a basis prior results obtained by the applicant while also taking advantage of approaches developed by the host. A secondment at the University of Warwick (U.K.) is also planned, hosted by Prof. Richard Montgomery, a leading figure in mathematics and expert in both topics covered by the project.