In the 2010-2011 academic year I am leading a research focus group of matroids and their applications.
Matroids were introduced by Hassler Whitney in a seminal paper as a generalization of the properties of linear independence in a vector space. They are beautiful objects, with a strikingly simple definition that admits many non-obvious reformulations (cryptomorphisms).
In this RFG we will study matroids with an eye towards how ideas in invariant theory influenced their development (as Gian-Carlo Rota envisioned the subject), and how matroids can be used as a unifying and simplifying language in algebra and geometry.
Fall 2010: The primary activity of the RFG in the fall will be a series of introductory lectures on matroids, in the form of a graduate course. First year graduate students and advanced undergraduates are particularly encouraged to take this class.
We meet Tuesdays and Thursdays at 3:10 in MSB 2112.
A tentative list of covered topics is: Axiomatics and cryptomorphisms, realizability results (relations to the Nullstellensatz, universality, topological realization theorems), constructions and morphisms, Tutte-Grothendieck invariants, Orlik-Solomon algebras, cohomology of complex hyperplane complements, other algebraic invariants, some aspects of the geometry of the Grassmannian.
Winter 2011: I organize a "reading seminar" for credit. It will differ from typical reading seminars in that it will only meet a handful of times. We will probably have an organizational meeting at the begining of the quarter and then meet at the end of the quarter for several (~3?) days to have an intensive study of a particular topic. I am open to suggestions for the topic; my own suggestions are Coxeter matroids or positroids.
Spring 2011: We are doing some readings from the book Topics in hyperplane arrangements, polytopes and box splines, by De Concini and Procesi.