Matroid theory is a combinatorial theory of independence which has its origins in linear algebra and graph theory, and turns out to have deep connections with many other fields. With time, the geometric roots of the field have grown much deeper, bearing many new fruits.
Optimization and algebraic geometry, in particular, have provided very useful geometric models for matroids. These models have played a central role in the development of fascinating mathematics, and in the solution to longstanding questions. This talk will survey some recent successes.
I will discuss the work of many researchers, including my joint work with Caroline Klivans and with Graham Denham and June Huh. I will assume no previous knowledge of matroids. 

The active bijection in hyperplane arrangements (or more generally oriented matroids) consists in a framework of various interrelated results and constructions that appear when the ground set of the structure is linearly ordered. It notably involves canonical bijections betweens simplices and regions (or more generally bases and reorientations), a canonical decomposition of regions into bounded regions of minors, several expressions of the Tutte polynomial, as well as some strengthening of real linear programming. Joint work with Michel Las Vergnas. 
