Stefan van Zwam (Eindhoven): A stroll through partial fields

Partial fields are sets equipped with multiplication and addition. The difference between partial fields and ordinary ones is that addition can be undefined for some elements. They were introduced by Semple and Whittle to capture certain classes of matroids. In this talk we propose a small correction to the original definition, and give some motivating examples of partial fields. Furthermore we discuss the result by Vertigan that every partial field can be obtained as the restriction of a ring to a subgroup of its units. We show how to define a matroid over a partial field, and how partial field homomorphisms can be used to prove some matroid-theoretic results, including Tutte's characterization of totally unimodular matrices. Note that this talk will contain very few new results. However, I hope to convince the audience that partial fields are interesting and useful objects of study.