Paths to Matroid Representations

In this talk, I will first review previous works on matroid representations initiated by Tutte in 1958, generalized by Dress and Wenzel in the 1980s, and refined by Baker, Bowler, and Lorscheid more recently using the language of F1-algebra. It turns out that one of the key ingredients of a "universal" proof of many representation theorems is a connectivity (or homotopy) statement of the lattice of flats of a matroid that confirms the existence of a certain path of hyperplanes. Next, I will extend this theory to orthogonal matroids. Orthogonal matroids appear in many natural ways to generalize a matroid, such as type D_n Coxeter matroids or tight 2-matroids. I will also describe the "lattice" of an orthogonal matroid and give a connectivity statement for the orthogonal matroid that produces as well many nice theorems of their representations.