I attended the excellent matroid theory sessions and some of the structural graph theory talks. There are powerful developments abroad in matroid theory! I'll try to give the flavor.
This talk illustrated a recurrent theme of the close relationship between matroid and graph structure.
Some talks concerned clones, such as "On close sets of GF(q)-representable matroids", presented by James Reid.
One that caught my attention was "Binary matroids with no M(K3,3) minor". Dillon Mayhew reported on a 100-page proof that these matroids are
I (and Dan Slilaty) noticed that, since M = M*(G)+e0) is dual to the coextension M(G)×e0, M* is the complete lift matroid L0(Σ) of a signed graph. Furthermore, we can find this signed graph explicitly by projective-planar duality. Perhaps this will lead to a simpler proof?
The solution to excluding M(K3,3) was suggested by noticing subtle patterns in a search of the data base for up to 9 points.
"Inductions for '4-connected' graphs and matroids", delivered by Xiangqian Zhou, and "Chain-type and splitter-type theorems for cocircuits and hyperplanes in 3-connected matroids", by Rhiannon Hall, presented the philosophy and examples of such theorems. "On minor-minimally 3-connected binary matroids" by Loni Delaplane et al. also used a chain theorem.
There was also a chain talk in the graph theory session: "Chain theorems for 4-prime graphs" by Sang-il Oum, which I couldn't attend.
My talk was a very fast survey, omitting 2/3 to 3/4 of the subject. (I may give a short version in the seminar, later.)
Dan's talk was about when the frame matroid G(Σ) of a signed graph is binary (representable over GF(2)) or quaternary (over GF(4)). (This extends the unpublished work of my former student Steve Pagano that assumed 3-connectedness.)
Xujin Chen talked about "An excluded minor characterization of box-Mengerian matroid ports". In a matroid with distinguished element e0, a "port" is C\e0 where C is any circuit that contains e0. (M,σ) is box Mengerian if any linear program whose coefficient matrix is the incidence matrix A of C(M,σ) has an integral dual solution, given an integral objective function (this matrix is called a "TDI" matrix), even when constrained by arbitrary integral upper and lower bounds (that's called "box TDI"). This property depends on L0(M,σ)'s not having certain forbidden minors, thus on (M,σ)'s not having certain forbidden signed minors.
Guoli Ding, from L.S.U., also talked about box TDI and binary clutters, but his talk was called "On Quasi-bipartite Graphs". Take A = the unoriented incidence matrix of a graph G. (This is an oriented incidence matrix of −G, i.e., G with all negative edges.) G is "quasi-bipartite" if the polyhedron Ax ≥ 1, x ≥ 0 has integral solutions. Graph theoretically, G is quasi-bipartite iff deleting any odd circle leaves at least one isolated vertex.