Well-quasi-ordering for binary matroids

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In Matroid Representation Theory we study natural geometric questions such as: "When can we embed a set of points in a space subject to prescribed dependencies?" These are similar to the questions that occupied classical Greek geometers such as Euclid and Pappus.  While the questions are theoretical in nature, the motivation is to answer fundamental problems in Combinatorial Optimization and in Theoretical Computer Science.  The goal of the proposed research is to provide structural results concerning configurations of points in "finite projective geometries". The research program is at a very exciting phase at the moment. There is a good chance that in the near future we will be able to answer some of the oldest and most fundamental problems in the area; namely Rota's Conjecture and the Well-Quasi-Ordering Conjecture.  If successful, the results will have algorithmic applications in combinatorial optimization, solving, for example, certain multicommodity transportation problems and certain integer programming problems. The results may also find application in other areas that utilize finite projective geometries, such as coding theory and cryptograpy.