Monday, March 29, 3pm ET (8pm BST, 8am Tue NZDT)
Charles Semple, University of Canterbury
Matroids with wheel- and whirl-like properties
Tutte showed that wheels and whirls are precisely the $3$-connected matroids in which every element is contained in a $3$-element circuit and a $3$-element cocircuit. As a consequence, wheels and whirls are exactly the $3$-connected matroids in which there is a circular ordering of the ground set such that every two consecutive elements is contained in a $3$-element circuit and a $3$-element cocircuit. More recently, Miller proved that sufficiently large (tipless) spikes are precisely the matroids in which every two elements is contained in a $4$-element circuit and a $4$-element cocircuit. In this talk, we investigate matroids satisfying generalisations of these properties and discuss some recent results. This is joint work with Nick Brettell, Deborah Chun, Tara Fife, James Oxley, Simon Pfeil, Gerry Toft, and Geoff Whittle.
Monday, March 22, 3pm ET (7pm GMT, 8am Tue NZDT)
Peter Nelson, University of Waterloo
Like in many areas of mathematics, long and technical proofs in combinatorics are becoming more common. When we consider the refereeing process, the unpleasant screeds of case-analysis with which many of us are familiar, and our high standards for mathematical truth, it is natural to have uncomfortable doubts due to simple human fallibility. A potential panacea is to use proof assistants to formally verify the correctness of our theorems. I will describe efforts I have made in recent months to formalize parts of matroid theory using the lean theorem prover, and a modest few results about matroids that are now formalized, including Edmonds’ Matroid Intersection Theorem. The talk is particularly aimed towards combinatorialists that are curious about this area; I will discuss both the bigger picture as well as the day-to-day experience of using a theorem prover, assuming no prior knowledge. This is joint work with Edward Lee and Mathieu Guay-Paquet.
Monday, March 15, 3pm ET (7pm GMT, 8am Tue NZDT)
Benjamin Moore, University of Waterloo
A density bound for triangle free 4-critical graphs
Password: email rose.mccarty ~at~ uwaterloo ~.~ ca if you need the password
Carsten Thomassen showed that every girth 5 graph embeddable in the torus or projective plane is 3-colourable. A complementary result of Robin Thomas and Barrett Walls shows that every girth 5 graph embedded in the Klein bottle is 3-colourable.
I’ll show neither the embeddability condition nor the girth 5 condition is needed in the above theorems by showing that every triangle-free 4-critical graph has average degree slightly larger than 10/3.
This is joint work with Evelyne Smith Roberge.
Monday, March 8, 3pm ET (8pm GMT, 9am Tue NZDT)
Nick Brettell, Victoria University of Wellington
The excluded minors for 2-regular matroids, and for Hydra-5-representable matroids
The 2-regular matroids are a generalisation of regular matroids (which are representable over all fields), and near-regular matroids (which are representable over all fields of size at least three). Hydra-5-representability characterises matroids with precisely six inequivalent representations over GF(5). I will present the following recent result: any excluded minor for the class of 2-regular matroids, or for Hydra-5-representable matroids, has at most 17 elements. In fact, we can say more about potential excluded minors with 16 or 17 elements. This leads us tantalisingly close to an excluded-minor characterisation for these two classes. In this talk, I will give some background to why these classes are of interest, discuss the long road towards proving this result, give some key ideas from the argument, and discuss where to from here.
Joint work with James Oxley, Charles Semple, and Geoff Whittle.