I will begin with a gentle introduction to hyperrings and hyperfields (originally introduced by Krasner for number-theoretic reasons), and then discuss a far-reaching generalization, Oliver Lorscheid’s theory of ordered blueprints.   Two key examples of  hyperfields are the hyperfield of signs S and the tropical hyperfield  T.  An ordering on a field K is the same thing as a homomorphism to S,  and a (real) valuation on K is the same thing as a homomorphism to T.   In particular, the T-points of an ordered blue scheme over K are  closely related to Berkovich’s theory of analytic spaces.

I will discuss a common generalization, in this language, of Descartes' Rule of Signs (which involves polynomials over S) and the  theory of Newton Polygons (which involves polynomials over T).  I will  then introduce matroids over hyperfields (as well as certain more general kinds of ordered blueprints).  Matroids over S are classically called oriented matroids, and matroids over T are also known as  valuated matroids.  I will explain how the theory of ordered  blueprints and ordered blue schemes allow us to construct a "moduli  space of matroids”, which is the analogue in the theory of ordered blue schemes of the usual Grassmannian variety in algebraic geometry.

This is joint work with Nathan Bowler and Oliver Lorscheid.