Tropical geometry is an emerging mathematical discipline at the crossroads of algebraic geometry and geometric combinatorics. It arose simultaneously and independently from enumerative geometry, phylogenetic inference, computational algebraic geometry, symplectic geometry, and control theory, and it has led to insights and new results in these areas. It may be regarded as the geometry that one obtains upon replacing the arithmetical operations of addition and multiplication by maximum and sum.

    In this course, I will develop tropical geometry, with a particular emphasis on basic objects such as tropical linear spaces and polytopes, relations to matroids, tropical curves and their Jacobians. This will give a basis for the use of tropical geometry in phylogenetic inference, in computational algebraic geometry, and in enumerative geometry.

    The course will use a draft of a text book by Maclagan-Sturmfels, as well as other material including notes from an short course on tropical enumerative geometry and material derived from recent research papers, some of which will be presented by students for their grade. My goal will be to introduce students to some of the current research featured in the Semester at the MSRI on this topic next Autumn. We will also have several guest speakers from that semester.