Advances in Applied Algebra:
Tropical Geometry and Polynomial System Solving

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.

In this project, we will be researching problems in tropical geometry that arise from applications in polynomial system solving. Polynomial system solving is the task of (numerically) computing the solutions of a system of polynomial equations, which are equally ubiquitous in many applications. This could be:

• computing the number of solutions of polynomial optimization problems (relates to computer vision)

• computing the realisation of rigid graphs (relates to material science)

• computing tropical varieties arising from coupled oscillators (relates to physics and engineering)

• and many more

The concrete topic will depend on your background and interest, as well as current open research questions in applied algebra.