Design Theory: Extremal and probabilistic perspectives

• Applicant: Patrick Morris, Ph.D.
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 504502205

Combinatorics aims to understand discrete mathematical structures such as graphs, set systems and integers, often using techniques from other fields of mathematics including probability theory, topology, algebra, analysis and even statistical physics. In recent years, the field has thrived in its own right and has established itself as a core pillar in the modern mathematical landscape. Indeed, the area has earned some of the highest honours in mathematics, including 3 of the last 13 Abel prizes being awarded to Szemerédi, Lovász and Wigderson, all of whom work in combinatorics. Despite its fundamental nature, research in this area has many applications as discrete structures can describe a wealth of key notions arising in all corners of scientific research, from large biological systems to complex travel networks. Most significantly, the digital world is built entirely upon discrete structures and computer science and combinatorics have become intrinsically linked as both areas have flourished. The focus of this research project is a certain class of discrete structures with strong symmetrical properties, known as designs. These objects have fascinated pure mathematicians for over 200 years and have also found many applications, for example in biological experiment design or in the design of strong error correcting codes in order to transmit data securely. The majority of research in combinatorial design theory has focused on proving the existence of certain designs and finding ways to construct them. However recent breakthroughs have opened up exciting new perspectives, allowing for a much deeper understanding of the properties of these objects.The aim of this project is to develop these new directions, focusing on the existence of large substructures in designs. On the one hand, I will explore extremal aspects, addressing which large substructures necessarily exist in an arbitrary design. On the other hand, I will adopt a probabilistic stance, investigating the behaviour of a typical design. I will also address questions that merge the two settings and variations on the theme, for example looking at large substructures in randomly perturbed designs. By exploring these topics and tackling new and unsolved questions in design theory, this project will lead to insight on some of the cornerstone conjectures and open problems in the field. In order to do this, I will build upon a range of powerful novel methods developed in graph and hypergraph theory, drawing on my own expertise in pseudorandom and spanning structures and mastering techniques in random and rainbow settings. The latter will be facilitated through collaboration with my host, who is an expert in those areas. This project thus provides a pivotal opportunity for the development of my career as a young scientist, enabling me to push this exciting branch of design theory forward, broaden my knowledge and cement myself as a prominent researcher in combinatorics.

• DFG Programme: WBP Fellowship
• International Connection: Spain
• Host: Professor Guillem Perarnau, Ph.D.