Combinatorial Algebraic Geometry

组合代数几何

• 批准号: RGPIN-2020-05724
• 负责人: Smith, Gregory
• 金额: $ 2.26万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759126
• 关键词: Combinatorial; Algebraic; Geometry

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).

代数几何中的指导问题是理解由多项式方程定义的几何对象的结构。特别有趣的是，组合簇被松散地定义为定义方程集合(或其他精细几何结构)具有具体组合解释的那些空间。作为代数、组合学和几何之间的中心对象，这些变体具有广泛的理论、工业和应用应用。事实上，它们解释了在代数统计、交换代数、数学物理和表示理论中出现的几何对象的不成比例的大量。这项研究计划阐明了组合变种的微妙特性，并扩大了它们与其他科学领域的联系。鉴于它们在数学科学中的中心地位，在这些基本问题上取得的任何进展都将影响和影响一大批科学家。明确的长期目标旨在扩大组合空间的类别，并增强我们对这类特定成员的了解。作为短期目标，这个建议集中在三个创新问题上：(A)识别和分析那些不可约分量都是光滑的Hilbert格式(代数几何中的原型参数空间)，(B)改进我们对稀疏多项式方程组的实解的数量的期望，(C)创建表现得像光滑射影变种的上同调环的Artin环的新来源，并解释它们的几何和组合意义。这项研究将产生新的数学结果和新的开源计算工具。绝大多数资金将用于培养高素质人才(HQP)。这笔资金支持的本科生、研究生和博士后研究员都将推进总体研究计划；他们将通过证明新定理和创造新的数学软件来直接贡献我们的科学知识。尽管如此，HQP的智力参与也将培养关键的研究能力，如独立性、批判性思维、解决问题和沟通技能。考虑到他们的特殊培训，这些人员将处于有利地位，可以在自然科学和工程(NSE)领域从事具有高度影响力的职业。