Tropical geometry is an area of mathematics that has enjoyed a quick development in the last 15 years. It can be seen as a tool for translating problems in algebraic geometry to combinatorial problems in convex polyhedral geometry. In this way, tropical geometry has proved to be very succesful in different areas of mathematics like enumerative algebraic geometry, phylogenetics, real algebraic geometry, mirror symmetry, and computational algebra.
One of the most basic tropical varieties are tropical linear spaces, which are obtained as tropicalizations of classical linear subspaces of projective space. They are polyhedral complexes with a very rich combinatorial structure related to matroid polytopes and polytopal subdivisions.
In Chapter 1 we give a basic introduction to tropical geometry and tropical linear spaces, and review the basic theory of tropical linear spaces that was developed by Speyer.
In Chapter 2 we study a family of functions on the class of matroids, which are ``well behaved'' under matroid polytope subdivisions. In particular, we prove that the ranks of the subsets and the activities of the bases of a matroid define valuations for the subdivisions of a matroid polytope into smaller matroid polytopes.
The pure spinor space is an algebraic set cut out by the quadratic Wick relations among the 2^n principal subPfaffians of an n x n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In Chapter 3 we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes. We also examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical linear spaces and valuated matroids to the class of Coxeter matroids of type D.
In Chapter 4 we study tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the set of all vectors v in L that make B a basis of maximal v-weight. The tropical linear space L is the union of all its local tropical linear spaces, which we prove are homeomorphic to Euclidean space. We also study the combinatorics of local tropical linear spaces, and we prove that they are combinatorially dual to mixed subdivisions of a Minkowski sum of simplices. We use this duality to produce tight upper bounds on their f-vectors. We introduce a certain class of tropical linear spaces called conical tropical linear spaces, and we give a simple proof that they satisfy the f-vector conjecture. Along the way, we give an independent proof of a conjecture of Herrmann and Joswig.
In Chapter 5 we introduce the cyclic Bergman fan of a matroid M. This is a simplicial polyhedral fan supported on the tropical linear space T(M) of M, which is amenable to computational purposes. It slightly refines the nested set structure on T(M), and its rays are in bijection with flats of M which are either cyclic flats or singletons. We give a fast algorithm for calculating it, making some computational applications of tropical geometry now viable. We develop a C++ implementation, called TropLi, which is available online. Based on it, we also give an implementation of a ray shooting algorithm for computing vertices of Newton polytopes of A-discriminants.