Total Positivity: Combinatorics and Applications

Total positivity studies mathematical spaces and their positive parts. For example, the positive part of a sphere is a triangle bounded by three great circles. More general spaces such as pyramids and polyhedra also arise as the positive parts of more complicated objects, and they turn out to have interesting boundary structures: vertices connected by edges, which are in turn contained in faces, etc. The boundaries of various totally positive spaces model diverse phenomena such as outcomes of a scattering experiment in particle physics or different ways to sort a list of numbers. In turn, these spaces are amenable to concrete study using techniques from combinatorics, which concerns discrete objects such as graphs and lists. The project will use the lens of total positivity to reveal the underlying combinatorial structure of spaces of polynomials and hyperplanes. It will also support K-12 outreach activities. Lorentzian polynomials were recently introduced to resolve several outstanding log-concavity conjectures. They also serve as geometric realization spaces of matroids, which are combinatorial abstractions of linear spaces. The project will determine the topology of various spaces of Lorentzian polynomials. A second direction concerns Schubert calculus, which studies intersection problems involving linear subspaces. Since the formulation of the Shapiro-Shapiro conjecture in the 1990's it has been known that total positivity can be employed to construct Schubert intersection problems whose solutions over the complex numbers are all real. This project will broaden this phenomenon with applications to real algebraic geometry. A third direction is to study the totally positive parts of Springer fibers, certain subvarieties of flag varieties which play a significant role in representation theory and algebraic combinatorics. The project will develop a new approach to studying the geometry of components of Springer fibers by relating them to Richardson varieties. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.