Foundations and Applications of Tropical Geometry

点击翻译按钮获取中文摘要

Tropical geometry is geometry over the tropical semiring, which is the usual real numbers with addition replaced by minimum, and multiplication replaced by addition. While the tropical semiring has had applications in optimization and computer science for several decades, the connection to algebraic geometry was first made only at the start of this century. The subject has expanded rapidly in the past decade.Classically algebraic geometry studies the geometry of the sets of solutions to polynomial equations, called varieties. The fundamentals of algebraic geometry changed dramatically in the 60s with Grothendieck's introduction of schemes. In the first decade of tropical geometry, however, only tropical versions of varieties were considered. This has changed in the past few years, beginning with the work of the Giansiracusas, and of the PI with Rincon, which together introduced a scheme theory into tropical geometry. This allows much more of the power of modern algebraic geometry to be used in tropical geometry.The primary aim of this project is to further develop the theory of tropical schemes, and apply this technology to problems in algebraic geometry.The first goal is to develop more of the basic commutative algebra and algebraic geometry of the new theory of tropical schemes. This will then be used to construct tropical versions of important moduli spaces in algebraic geometry, starting with the Hilbert scheme, and use this to address fundamental open questions about the Hilbert scheme. It will also be used to address realizability questions in tropical geometry, which have applications to birational geometry.