Friday, June 15, 2012
Peter's PhD thesis, entitled "Exponentially Dense Matroids", tackled some fundamental problems in extremal matroid theory and was written under the supervision of Professor Jim Geelen. In his thesis, Peter proved results that precisely control the extremal behaviour of minor-closed classes of matroids from surprisingly weak hypotheses. For example, he proved that, if M is a simple matroid with sufficiently large rank and M does not contain an n-point line as a minor, then the size of M is at most (qr(M)-1)/(q-1) where q is the largest prime-power that is at most n-2. The proofs combine techniques from extremal combinatorics with geometric intuition and clever connectivity reductions.