- Series
- Graph Theory Seminar
- Time
- Thursday, October 25, 2018 - 12:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Matthew Baker – Math, GT
- Organizer
- Robin Thomas

We present an algebraic framework which simultaneously
generalizes the notion of linear subspaces, matroids, valuated matroids,
oriented matroids, and regular matroids. To do this, we first
introduce algebraic objects which we call
pastures; they generalize both hyperfields in the sense
of Krasner and partial fields in the sense of Semple and Whittle. We
then define matroids over pastures; in fact, there are at least two
natural notions of matroid in this general context,
which we call weak and strong matroids. We present ``cryptomorphic'’ descriptions of each kind of matroid. To a (classical) rank-$r$ matroid $M$ on $E$, we can associate a
universal pasture (resp. weak universal pasture)
$k_M$ (resp. $k_M^w$). We show that morphisms from the universal
pasture (resp. weak universal pasture) of $M$ to a pasture $F$ are
canonically in bijection with strong (resp. weak) representations
of $M$ over $F$. Similarly, the sub-pasture $k_M^f$ of $k_M^w$
generated by ``cross-ratios'', which we call the
foundation of $M$, parametrizes rescaling classes of
weak $F$-matroid structures on $M$. As a sample application of these
considerations, we give a new proof of the fact that a matroid is
regular if and only if it is both binary and orientable.