# SMALL 2020

In the summer of 2020, I’ll be leading a group on Tropical and Algebraic Geometry as part of Williams’ SMALL REU.  This website contains possible projects the group could work on; in order to apply for this group, you should visit the posting on mathprograms.org.  (By the way:  there are WAY more projects here than could ever be covered in one REU.  Once we figure out what sorts of questions the group members are interested in, we’ll start narrowing down, or figuring out related topics.)

Algebraic geometry is the study of solution sets of polynomial equations, dating back at least to Descartes’ introduction of coordinate systems in the 1500s.  Tropical geometry is a new field of mathematics that studies combinatorial, piecewise-linear analogs of the objects in algebraic geometry, often with a view towards understanding the original algebro-geometric objects themselves.  For instance, an algebraic curve is a one-dimensional solution set to polynomial equations; and a tropical curve is a union of line segments and rays, embedded in Euclidean space in a balanced way and having the underlying structure of a metric graph.  And as an algebraic geometer studies poles and zeros of rational functions on the algebraic curve, a tropical geometry studies configurations of chips on the metric graph, which move around according to chip-firing rules.

The projects for this group will be based in algebraic geometry, tropical geometry, and the interplay between these two fields.  Here are a few topics and resources to get a flavor of the sorts of questions we might be thinking about.  I’ll frequently refer to this book chapter I wrote, which gives you some background on tropical geometry as well as possible research projects.  I’ll also refer to different papers I’ve coauthored, many with undergrads; you can also find all my papers here.

• Tropical theorems from algebraic ones:  there are many results from algebraic geometry that also hold in tropical geometry.  For instance, Bezout’s theorem tells us how many points two plane curves should intersect in; an almost identical result holds tropically (see Theorem 3).  Occasionally we have theorems of a similar flavor, but with a different result:  algebraically, a smooth plane curve of degree 4 has 28 bitangent lines; tropically, it has only 7 as shown by me and my coauthors.  There are many results from the classical world of algebraic geometry that have not been explored in the tropical setting, and would be perfect for projects for this group (see Project 4 for a few examples).