Lethbridge Number Theory and Combinatorics (Mini-course): Cristhian Garay

Idempotent semirings have been relevant in several branches of applied mathematics, like formal languages and combinatorial
optimization.

They were brought recently to pure mathematics thanks to its link with tropical geometry, which is a relatively new branch of mathematics that has been useful in solving some problems and conjectures in classical algebraic geometry.

However, up to now we do not have a proper algebraic formalization of what could be called “Tropical Algebraic Geometry”, which is expected to be the geometry arising from idempotent semirings.

In this mini-course we aim to motivate the necessity for such theory, and we recast some old constructions in order theory in terms of commutative algebra of semirings and modules over them.