Skip to main

SAFE RETURN TO CAMPUS

As part of Mason's Safe Return to Campus Plan, all classes and associated instructional activities—including final exams—will be conducted virtually beginning November 30, while most campus facilities will remain open. Visit Mason’s Safe Return to Campus Plan for COVID-19 updates.

Math equations

Generalized Graphic Matroids in Projective Geometry

Speaker: Thomas Zaslavsky, Binghamton University

Title: Generalized Graphic Matroids in Projective Geometry

Abstract: A graph Γ has a vector representation in which an edge eij = vivj corresponds to the vector bj − bi, where b1, b2, . . . are the standard unit basis vectors. Such a representation is
a vector representation of the graphic matroid M(Γ). We may restate this as a mapping of E(Γ) into projective space, giving a projective representation of M(Γ). Properties of graphs
imply that this representation is unique up to projective transformations. A gain graph Φ is a graph with something more: an orientable “gain function” ϕ : E → H where H is a group. Φ has a “frame matroid” M(Φ) that generalizes the graphic matroid. If H ≤ F ×, the multiplicative group of a field, then Φ has a vector representation (e, g) 7→ bj − ϕ(eij )bi that represents the frame matroid. We can restate this as a representation of M(Φ) in projective space (which may not be unique). What if H is not in any F ×? Even then we may be able to represent M(Φ) projectively
by using synthetic geometry. I will explain how this works, first in any dimension, and then for rank-3 matroids in projective planes. The special feature of the latter is that, while all higher-dimensional projective spaces have coordinates in a field (or skew field), most projective planes do not; this means one needs a more combinatorial treatment of representation involving quasigroups and ternary rings, which I will explain. This report is on joint work with Rigoberto Fl´orez.

Time: Friday, April 13, 2018, 3:30-4:20 p.m.

Place: Exploratory Hall, room 4106

Refreshments will be served at 3:00 p.m.