John Stembridge provided a planar network interpretation for the pfaffian of a skew-symmetric matrix. This gave the original motivation for Thomas Lam and the PI to introduce and study the skew version of total positivity. Further investigation in this direction, and in particular the relation of skew total positivity to the theory of total positivity developed by George Lusztig, is the subject of the first project. The second project grew out of an attempt to reinterpret the often used standard bases of the coordinate ring of Grassmanians, and to find different kinds of bases occurring naturally. The question is closely related to the question of triangulating the cone of Gelfand-Tsetlin patterns, which motivated David Speyer and the PI to introduce the notion of driving rules as a way to locally determine the triangulation. Properties of the simplicial complexes and the bases which arise are the subject of the investigation. The third project deals with K-homology of Grassmanian, studied so far to a smaller extend than the dual notion of K-theory. The first step in this direction was taken by Thomas Lam and the PI who introduced dual stable Grothendieck polynomials as representatives of Schubert classes in the K-homology ring. Further questions to be studied include the K-theoretic version of the Robinson-Schensted correspondence, K-homology of complete flag varieties and more.
The unifying theme of all three projects is the study of nice varieties, such as Grassmanians and more general flag varieties. A Grassmanian as the variety of all k-dimensional subspaces of an n-dimensional space. An old and yet active research area is Schubert calculus, which - very roughly - studies how subspaces of a space intersect. A simple example would be the following question: in a three-dimensional space four lines in generic position are chosen. What is the number of lines that intersect all four of them? Schubert calculus was recently enriched by the work of George Lusztig, who defined and studied a special part of flag varieties, called totally the positive part. Real numbers naturally come with a notion of positive and negative, and when we consider flag varieties over real numbers there is a way to identify a positive part in them. The study of the totally positive part has lead to a number of beautiful recent developments.
