Combinatorics and Geometry of Moduli Spaces

长期目标。许多计算和定性问题可以归结为关于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：有多少人具有给定的性质，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。我的研究项目主要集中在与线性空间和曲线相关的分类和枚举，这是两个最普遍的数学对象。我的研究的首要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。在未来的5年里，我的计划将集中在以下几个方面：1。在复杂几何和真实的几何之间建立新的连接，包括与曲线相切的线性空间。2.开发新的组合工具来解决涉及曲线模的枚举问题。3.从表示论中的组合结构中预测新的几何空间。Approach.模问题通常通过关注限制特征和递归地传递到特殊和退化边界情况来处理。这些退化技术通常简化了几何结构，同时引入了大量的组合复杂性。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形状。我的研究计划将建立正式的几何和组合学之间的联系，一般通过显示，在检查的几何过程遵循相同的递归模式做简单的（离散）组合模型。在大多数情况下，几何学或组合学中的一个比另一个更好理解，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）。冲击模理论和计数是代数几何的中心课题，与变形理论、双有理几何和广义分类问题有关。这项研究计划将产生交叉的结果，涉及几何，组合学和表示理论。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，目标和调查途径。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。这项建议还将支持培训数学、特别是代数几何和组合学方面的高素质人才，并将有助于提高加拿大在这些数学领域的地位。

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.