Combinatorial Algebraic Geometry

点击翻译按钮获取中文摘要

The guiding problem in algebraic geometry is to understand the structure of the geometric objects defined by polynomial equations. Of particular interest, combinatorial varieties are loosely defined as those spaces for which the defining collection of equations (or other fine geometric structure) has a concrete combinatorial interpretation. As the central objects at the interface between algebra, combinatorics, and geometry, these varieties have a broad range of theoretical, industrial, and applied applications. Indeed, they account for a disproportionally large number of the geometric objects arising in algebraic statistics, commutative algebra, mathematical physics, and representation theory. This research program illuminates the subtle properties of combinatorial varieties and expands their connections with other areas of science. Given their centrality within the mathematical sciences, any progress on these fundamental problems will impact and influence a large community of scientists. The explicit long-term goals aim to enlarge the class of combinatorial spaces and to enhance our knowledge about specific members of this class.  As short-term objectives, this proposal concentrates on three innovative problems: (a) identify and analyze those Hilbert schemes (the prototyical parameter spaces in algebraic geometry) whose irreducible components are all smooth, (b) refine our expectations for the number of real solutions to a sparse system of polynomial equations, (c) create new sources of artinian rings that behave like the cohomology ring of smooth projective variety, and explain both their geometric and combinatorial significance. This research will produce new mathematical results and new open-source computational tools. The vast majority of the funds will be used to train of highly qualified personnel (HQP). The undergraduate students, graduate students, and postdoctoral fellows supported by this grant will all advance the overarching research program; they will make direct contributions to our scientific knowledge by proving new theorems and creating new mathematical software.  Nonetheless, the intellectual involvement of HQP will also develop key research abilities such as independence, critical thinking, problem solving, and communication skills. Given their exceptional training, the personnel will be well-positioned to move on to highly impactful careers in natural sciences and engineering (NSE).