From Oil Fields to Hilbert Schemes

New techniques for dealing with problems of numerical stability in computations involving multivariate polynomials allow a new approach to real world problems. Using a modeling problem for oil field production optimization as a motivation, we present several recent developments involving border bases of polynomial ideals. After recalling the foundations of border basis theory in the exact case, we present a number of approximate techniques such as the eigenvalue method for polynomial system solving, the AVI algorithm for computing approximate border bases, and the SOI algorithm for computing stable order ideals. To get a deeper understanding for the algebra underlying this {\it approximate world}, we present recent advances concerning border basis and Gr\"obner basis schemes. They are open subschemes of Hilbert schemes and parametrize flat families of border bases and Gr\"obner bases. For the reader it will be a long, tortuous, sometimes dangerous, and hopefully fascinating journey from oil fields to Hilbert schemes.