Algebra and geometry of matroids

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Matroids are one of the basic structures in combinatorics, like graphs and partially ordered sets. As such they appear throughout combinatorics and associated areas, including optimization, operations research, coding and information theory, computer science, algebraic geometry, and have connections to biology, physics, and statistics. A matroid is built on a set of objects, and picks out certain smaller collections of those objects as being somehow compatible in a way that represents _independence_. (For example, if the objects are vectors and two of them sum to a third, the three together are not independent.)In roughly the last decade, much progress has been made on old questions pertaining to the structure of matroids by new methods drawn from the techniques of algebra and geometry: instead of just considering the structure of a matroid from scratch, start by associating it to a collection of polynomial equations; or instead of just remembering the independence relations as above, perhaps, make a richer structure that remembers more algebra of the original set of objects. My projects consist of attempts to systematise, unify, and extend these approaches, and in so doing build on the aforementioned progress.