Fundamentals and applications of tropical schemes

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Tropical geometry is geometry over the tropical semiring, where multiplication is replaced by addition, and addition is replaced by minimum. This turns an algebraic variety (the solution to polynomial equations) into an object from polyhedral geometry called a tropical variety. The goal is to translate algebraic geometry questions about the original variety into more tractable questions about the tropical variety.Since its inception two decades ago, tropical geometry has rapidly expanded as a field, with applications both inside and outside mathematics (including to Economics, Machine Learning, Phylogenetics, Optimization and Statistics). Most applications, however, do not exploit the full strength of modern algebraic geometry. This is because tropical versions of these tools are still being developed.The main goal of this project is to continue the development of tropical scheme theory, and apply this to problems in algebraic geometry. Scheme theory was introduced to algebraic geometry in the 1960s by the work of Grothendieck and collaborators, and underpins all of modern algebraic geometry. By integrating this into tropical geometry we will greatly expand the reach and application of the field. Potential applications include both open problems in classical algebraic geometry, and also tropical versions of fundamental moduli spaces in algebraic geometry.