The algebra, geometry, and combinatorics of Total Positivity

This is an area which has seen a ton of interest in the last 30 or so years, but there is no canonical textbook.

Possible topics include:

• the remarkable combinatorial/topological structure of totally positive spaces (e.g., https://arxiv.org/abs/math/0609764 and https://arxiv.org/abs/1904.00527);
• connections with canonical bases and cluster algebras (e.g., https://math.mit.edu/~gyuri/papers/pod.ps and https://arxiv.org/abs/1005.1086);
• applications in physics (e.g., https://arxiv.org/abs/1212.5605).