Tropical geometry as a tool in algebraic geometry and beyond

Tropical geometry is a field in mathematics that lies at the interface of algebraic geometry and combinatorics. One of the main goals in tropical geometry is to uncover the combinatorial patterns hidden in algebraic geometry. This basic principle can already be found in Bézout’s theorem, which counts the roots of sufficiently generic polynomial equations, and its generalization in the form of the BKK theorem, both of which can be proven tropically. The abstract combinatorial ideas that come out of this study have since also seen applications in economics, machine learning, chemical reaction networks and mathematical physics, among others.

In this talk I will give an introduction to tropical geometry and I will discuss some of the main results. I will also discuss some of my latest work on finding the topology of an algebraic variety using tropical methods. In particular, I will discuss how this gives rise to a CW complex structure on a K3 surface. This structure for instance gives us a quick way to see various phenomena from mirror symmetry such as the monodromy of integral affine structures.