The main project area is probabilistic and extremal combinatorics, with applications to algorithms. This is a rapidly expanding field of Mathematics, which has connections to many areas, e.g. probability, analysis, number theory, theoretical computer science and statistical physics.
Indeed, the study of combinatorial structures such as graphs provides the theoretical foundation for the analysis of many large networks arising in theoretical computer science, scheduling and communication. In the simplest case, these networks give rise to a graph, which consists of vertices, with suitable pairs of these vertices connected by edges. Hypergraphs arise when modelling non-binary relationships: instead of pairs we can also connect triples or larger vertex sets by a single hyperedge. Hypergraphs turn out to be much more challenging to investigate than graphs, so that much less is known about their properties.
A central theme of extremal combinatorics is the interplay and relationship between the parameters of combinatorial objects. The first and most immediate question which arises in this context is that of the (i) existence of objects with a given set of parameters. Once this has been answered, the next natural step is to ask for (ii) the number of such objects - i.e.to ask for a counting result. This is of fundamental importance in the context of many combinatorial questions arising in statistical physics. A very fruitful approach here is to ask for asymptotic results rather than exact formulas. This is sufficient for many applications and allows for the use of powerful techniques e.g. from probabilistic combinatorics. Moreover, this asymptotic approach sometimes makes it possible to go even further and ultimately uncover the (iii) typical structure of the objects in such a given class.
In this project, we consider the above perspective with a focus on inter-related topics involving combinatorial designs, decompositions of large structures into small objects, Latin squares, spanning cycles, matchings and colourings in graphs as well as hypergraphs.
These include longstanding open problems of central importance as well as promising new directions that are currently emerging. Moreover, the project themes have close connections e.g. to probability and theoretical computer science.
