Year 3 Project 2027 - Basics of applied algebra: tropical linear spaces
Tropical linear spaces arise naturally in many applications, such as phylogenetics, chemical reaction networks, and neural networks.
Basics of Applied Algebra:
Tropical linear spaces
Description
Algebraic Geometry is an area of mathematics which uses algebraic tools to study geometric problems. One problem of particular interest is solving (systems of) polynomial equations. Dating back to the ancient Greeks, this task is still at the heart of many abstract open problems in pure mathematics as well as concrete challenges in industry.
Tropical Geometry is a piecewise linear version of algebraic geometry. It studies so-called tropical varieties, which arises naturally in many applications, such as phylogenetics in biology, auction theory in economics, and machine learning in computer science. It allows researchers to use the toolkit of algebraic geometry in real problems in the applied sciences.
Tropical linear spaces are the tropical equivalent of linear spaces in algebraic geometry. They are a vital component in tropical geometry and have been played a crucial role in recent breakthroughs in abstract mathematics. Unlike their algebraic counterparts however, they are combinatorial objects of almost infinite intricacy, which is why their utility in concrete applications has been limited so far.
The goal of this project is to make a first foray into exploring tropical linear spaces from a numerical point of view. Because while abstract mathematics is only interested in the absolute truths, in applications being very close is often good enough, and maybe tropical linear spaces aren't nearly as difficult if your eyesight is bad enough.
Group Project
Content
The group project will revolve around learning the basics of tropical geometry, matroids, and tropical linear spaces:
(Realizable) Matroids
Tropical varieties and the Structure Theorem
Tropical linear spaces
Tropical Grassmannians
Moreover, there will be a technical component, where we will use the computer algebra system OSCAR.
Mode of Operation and Evidence of Learning
We will have a weekly group meeting in which we discuss the maths. This will include you giving informal mini-presentations and me asking you spontaneous questions on the material (this is not marked but done in anticipation of the presentations and oral exam).
Prerequisites and Co-requisites
Programming I, Algebra II
Resources
Diane Maclagan, Bernd Sturmfels: Introduction to Tropical Geometry (link)
Michael Joswig: Essentials of tropical combinatorics (link)
Bernd Sturmfels: Twelve lectures on tropical geometry (youtube link)
Individual project
Content
The individual projects will revolve around using the knowledge you have gained and they will be driven by current research. As of writing, potential topics follow in two strands:
Tropical distance in the ambient space vs intrinsic distance on the tropical Grassmannian.
What is the maximal discrepancy between both distances?
What is the expected discrepancy between both distances for random tropical linear spaces?
"Low-corank" approximations of tropical linear spaces. Here, "low-corank" refers for example to tropical linear spaces that arise as stable intersection of corank 2 tropical linear spaces.
How dense is the set of "low-corank" tropical linear spaces inside the tropical Grassmannian?
Given a tropical linear space, how can we find such a “low-corank” approximation?
Given a tropical linear space, how does its intersection number differ from its low-corank approximation?
Mode of operation and evidence of learning
We will continue our regular meetings individually or in smaller groups should individual topics align. We will continue with the mini-presentations and questions, but with a focus on your final report.