Combinatorics and Geometry of Moduli Spaces

模空间的组合学和几何

• 批准号: RGPIN-2021-04169
• 负责人: Levinson, Jake
• 金额: $ 1.68万
• 依托单位: Simon Fraser University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741938
• 关键词: Combinatorics; Geometry; Moduli; Spaces

Long-term goal. Many computational and qualitative problems can be distilled down to mathematical questions about classification and enumeration. The first question asks: what are all the possible curves, surfaces, and other mathematical objects? The second asks: how many of them have a given property, or solve a given problem? In algebraic geometry, we address these questions using moduli spaces: parameter spaces that describe all geometric objects of a given type. My research program focuses on classification and enumeration related to linear spaces and curves, two of the most ubiquitous mathematical objects. The overarching goal of my research is to develop combinatorial tools to count, classify and compute solutions to algebraic and geometric problems involving planes and curves, and to apply these tools to explicitly understand the geometry of the associated moduli spaces. Short-term goals. In the next 5 years, my program will focus on the following: 1. Establish new connections between complex and real geometry involving linear spaces tangent to curves. 2. Develop new combinatorial tools to solve enumerative problems involving moduli of curves. 3. Produce new geometric spaces predicted from combinatorial constructions in representation theory. Approach. Moduli problems are typically approached by focusing on limiting features and by recursively passing to special and degenerate boundary cases. These degeneration techniques usually simplify the geometry while introducing a great deal of combinatorial complexity. As such, tools from combinatorics are essential for the analysis and frequently shed light on the underlying geometry. My research program will establish formal connections between geometry and combinatorics, generally by showing that the geometric processes under examination follow the same recursive patterns as do simpler (discrete) combinatorial models. In most cases, one of either the geometry or the combinatorics is better understood than the other, so the scope of this research also includes developing new combinatorial tools (Objective 2) and constructing and analysing new geometric spaces (Objective 3). Impact. Moduli theory and enumeration are central topics in algebraic geometry, related to deformation theory, birational geometry and broad classification problems. This research program will produce crossover results relating geometry, combinatorics and representation theory. These results will allow researchers to borrow the techniques of other fields, as well as their intuitions, goals and avenues of inquiry. Moreover, the new foundational tools developed for linear spaces and curves will contribute to the development of new techniques for engineering and the natural sciences. This proposal will also support the training of highly qualified personnel in mathematics, particularly algebraic geometry and combinatorics, and will contribute to enhancing Canada's standing in these mathematical fields.

长期目标。 许多计算和定性问题可以归结为有关分类和枚举的数学问题。第一个问题是：所有可能的曲线、曲面和其他数学对象是什么？第二个问题是：他们中有多少人拥有给定的属性，或者解决了给定的问题？在代数几何中，我们使用模空间来解决这些问题：描述给定类型的所有几何对象的参数空间。我的研究项目侧重于与线性空间和曲线（两种最普遍的数学对象）相关的分类和枚举。我研究的总体目标是开发组合工具来计算、分类和计算涉及平面和曲线的代数和几何问题的解决方案，并应用这些工具来明确理解相关模空间的几何形状。短期目标。 在接下来的五年中，我的计划将重点关注以下内容： 1. 在涉及与曲线相切的线性空间的复杂几何和真实几何之间建立新的联系。 2. 开发新的组合工具来解决涉及曲线模的枚举问题。 3. 产生从表示论中的组合构造预测的新几何空间。方法。模问题通常通过关注限制特征并递归地传递到特殊和简并的边界情况来解决。这些退化技术通常会简化几何形状，同时引入大量的组合复杂性。因此，组合学工具对于分析至关重要，并且经常揭示底层几何结构。我的研究计划将在几何学和组合学之间建立正式的联系，通常是通过表明所检查的几何过程遵循与更简单（离散）组合模型相同的递归模式。在大多数情况下，几何学或组合学中的一种比另一种更容易理解，因此本研究的范围还包括开发新的组合工具（目标 2）以及构建和分析新的几何空间（目标 3）。影响。模理论和枚举是代数几何的中心主题，与变形理论、双有理几何和广泛的分类问题相关。该研究项目将产生与几何、组合学和表示论相关的交叉成果。这些结果将使研究人员能够借用其他领域的技术，以及他们的直觉、目标和探究途径。此外，为线性空间和曲线开发的新基础工具将有助于工程和自然科学新技术的发展。该提案还将支持数学领域高素质人才的培训，特别是代数几何和组合数学，并将有助于提高加拿大在这些数学领域的地位。