Foundations of Applied Algebra: Tropical Varieties

Applied Algebra, also known as Nonlinear Algebra, is an area of mathematics that focuses on fundamental problems in algebra, geometry, and combinatorics, that arise from applications such as biology, economics, or physics. It uses tools from areas that are commonly regarded as pure mathematics and tackles problems arising in applied sciences and industry.

One important tool for applied algebra is tropical geometry, which studies piecewise linear objects arising from polynomial equations. These so-called tropical varieties appear naturally in many applications. They describe:

• tree spaces in phylogenetics and are useful for data science with phylogenetic trees

• loci of indifference in auction theory are used to analyse competitive equilibria

• they describe graphs of neural networks and measure their complexity

For a quick glimpse into the very basics of tropical geometry, see this 8-minute introduction into tropical polynomials and hypersurfaces by Madeline Brandt. For a deeper look into tropical geometry, check out some of the lectures in this lecture series by Bernd Sturmfels.

Despite its importance, no software is capable of computing tropical varieties in full generality. Currently, two libraries exist for tropical varieties:

1. Gfan

2. Singular's tropical package

However, both are restricted to very specific cases, consist of unreadable C code, and have not been updated in seven years. This project aims to remedy the situation and work towards providing the applied algebra community a modern implementation for present and future applications.

We will work together to develop a basic understanding what tropical geometry is, what tropical varieties are, and how they can be computed in full generality. You will then each focus on an individual aspect of the computation, and study its intricacies, its bottlenecks, and possible improvements. Key to our implementation is the computer algebra system OSCAR, which provides easy access to several crucial features.

Requirements

The project has no strict module requirements, but Algebra II (MATH2581) would be beneficiary. The project will involve coding, and experience in python or julia would be very useful.