| This thesis explores topics in polyhedral geometry, point set triangulations, tropical geometry, and applications to biology.;The first chapter examines the problem of finding minimal tropical bases of linear ideals. A set of polynomials p1,..., pn is a tropical basis for the ideal they generate if and only if T (⟨p1,...,pn⟩) = T (⟨p1⟩) ∩···∩ T (⟨pn⟩), where T denotes taking the tropical variety. In the case of ideals of tropical linear forms, one can relate the constant coefficient linear ideals to matroids. In this case, it is known that the circuits of the matroid form a tropical basis, however this tropical basis will not be minimal in general. We describe unique minimal bases for graphic matroids, cographic matroids, the matroid R10, and the Fano matroid. We describe minimal tropical bases for uniform matroids and give computational results and bounds on the cardinalities of their minimal tropical bases (which are not unique).;In the second chapter, we use new computational methods to find the hyperdeterminant of format 2 x 2 x 2 x 2. It is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube, a closely related object. The 87959448 regular triangulations of the 4-cube are classified into 25448 D-equivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant.;The third chapter uses polyhedral geometry and point set triangulations in an application to biological disease modeling. A disease model is given by a set of probabilities that individuals with various genotypes will acquire a certain disease. There are infinitely many such models, and our goal is to categorize them. Our categorization is based on the theory of regular polyhedral subdivisions, and the resulting categories reflect epistasis, or interactions between genes, in the models. Epistasis is of great interest to biologists, and is the key feature of our classification system. We construct 69 categories of disease models, corresponding to the regular triangulations of a certain point configuration, up to symmetry. We also explore the secondary polytope of this point configuration, and give its faces a biological interpretation. |
