Beijing Institute of Mathematical Sciences and Applications

Benjamin Sudakov

Peter Keevash

Tuesday, April 16, 2024 5:05 PM - 6:15 PM

Online

Zoom 787 662 9899 (BIMSA)

Many combinatorial objects can be thought of as a hypergraph decomposition, i.e. a partition of (the edge set of) one hypergraph into (the edge sets of) copies of some other hypergraphs. For example, a Steiner Triple System is equivalent to a decomposition of a complete graph into triangles. In general, Steiner Systems are equivalent to decompositions of complete uniform hypergraphs into other complete uniform hypergraphs (of some specified sizes). The Existence Conjecture for Combinatorial Designs, which I proved in 2014, states that, bar finitely many exceptions, such decompositions exist whenever the necessary `divisibility conditions' hold. I also obtained a generalisation to the quasirandom setting, which implies an approximate formula for the number of designs; in particular, this resolved Wilson's Conjecture on the number of Steiner Triple Systems. A more general result that I proved in 2018 on decomposing lattice-valued vectors indexed by labelled complexes provides many further existence and counting results for a wide range of combinatorial objects, such as resolvable designs (the generalised form of Kirkman's Schoolgirl Problem), whist tournaments or generalised Sudoku squares. In this talk, I plan to illustrate these results and discuss some recent and ongoing developments.

Peter Keevash is a Professor of Mathematics at the University of Oxford and a Fellow of Mansfield College. He has also held positions at Queen Mary University of London and California Institute of Technology, and received degrees from Cambridge and Princeton. His research is in Combinatorics and is best known for his solution to the Existence Conjecture for Combinatorial Designs. He received the European Prize in Combinatorics in 2009 and the Whitehead Prize in 2015, and was a speaker at the 2018 International Congress of Mathematicians.