Combinatorial analysis of continuous problems
Many objects in mathematics, at first sight, seem to belong to the domain of continuous mathematics. These objects are continuous, smooth and infinite, far different from the discrete and finite objects that are the classical domain of combinatorics. Objects of the former type are, for instance, determinants of matrices (which can take on every complex value), Grassmannians (which are smooth manifolds), and the eigenvalues of matrices (which take on any tuple of complex values). In the latter class lie objects such as paths in graphs, finite groups and generating functions. Applications of the study of such finite objects to the continuous ones would seem unlikely, or at least, trivial. For example, one may count the number of minors of a matrix, but that's about it. As we will demonstrate, however, this is not the case. The field of combinatorics has developed into a mature field of study, and it is the author's view that combinatorics can be used as a toolbox to obtain interesting and deep information on all areas of mathematics, continuous especially.
In this work, we will demonstrate this by studying three different continuous problems using the techniques of combinatorics. The first problem concerns the study of symmetric matrices and their principal- and almost-principal minors. Here the main result is a proof of a conjectural combinatorial formula of Kenyon and Pemantle (2014) for the entries of a square matrix in terms of its connected principal and almost-principal minors. The second problem is the study of Bruhat interval polytopes. These polytopes arise as the moment-map images of Richardson varieties of flag varieties. Their study is motivated in part by the integrable system called the Toda lattice. Information obtained about these polytopes can be readily translated to information about the Richardson varieties. For instance, the dimension of the polytope will be used to determine when the Richardson variety is toric. The third problem will pertain to the study of the spectral theory of tensors via tropical methods. We show that an elegant theory in which there is a unique tropical eigenvalue arises. We describe briefly how the corresponding eigenvalue informs us of the asymptotic behavior of the corresponding classical eigenvalues.