Weekly Bulletin

<p>After introducing monomial ideals and their connections to combinatorial objects such as graphs and simplicial complexes, we investigate some of their invariants and their asymptotic behaviour. In particular, we use Newton polyhedra to give an explicit description of the minimal generating sets of large powers of bivariate monomial ideals. Moreover, we establish bounds on the power beyond which the sets of associated primes of monomial ideals in three or more variables stabilize.</p>

<p>Interesting combinatorial structures can often be characterised as special subsets of association schemes. In his PhD thesis, Philippe Delsarte developed powerful linear programming techniques to prove non-existence and uniqueness results for such structures. In particular, this applies to error-correcting codes. Ideas of this type were fundamental in the work for which Maryna Viazovska was awarded the Fields medal in 2022. This talk will begin with an overview of Delsarte theory.</p> <p>We will use this theory to study perfect matchings. We show that 1-factorisations of the complete graph are special subsets in the sense of Delsarte. The same is true for hyperfactorisations and other generalisations of 1-factorisations. The characterisation of these structures as a special subset of an association scheme gives rise to divisibility conditions and non-existence results. We also give a construction of hyperfactorisations using finite geometry.</p> <p>No particular knowledge of association schemes will be required to appreciate this talk.</p>