Tropical Geometry
At some high level, for me Tropical Geometry is a relatively new (and arbitrarily named) field of mathematics studying the behaviour of algebraic varieties near coordinate hyperplane intersections. It's also the study of the logarithmic transformation of \(\mathbb{C}\) algebra, as \(+\) maps to \(\max\), and as \(\times\) maps to \(+\). An absolutely fascinating topic.

My first foray into doing computations in Tropical Geometry came with the invitation to work with Jon Hauenstein and Cynthia Vinzant on the development of an algorithm to compute real tropical curves.

The resulting algorithm has been submitted to the Arxiv. Paper available here.
Dimensions of real and complex algebraic varieties whose tropicalizations I can currently decompose numerically:

News

I have written Matlab code calling Bertini implementing the tropical curve decomposition algorithm Jon, Cynthia, and I developed. The code appears on the code page on this site under the curve section. The code is licensed using the GPL 3 license.

Some examples of using the code to compute tropical curves, both real and complex, are on the curve examples page.