Algebraic combinatorics and its applications to algebraic geometry

自20世纪80年代以来，（现代）几何学在与数学物理学的深刻联系的推动下，经历了一段漫长而激烈的活动。随着研究在这些领域的进展，越来越明显的是，在某些问题的深处，隐藏着非常复杂的新问题，即受复杂关系约束的新对象的集合。这些新问题本质上是组合的，因为它们可以用纯粹的组合术语来表达，而不需要参考几何和数学物理中的原始设置。本研究提案的目的是推进以这种方式出现的组合问题的解决方案。为了做到这一点，我建议研究代数几何中的三个非常复杂的问题，它们本身就值得注意，因为我相信隐藏在它们之中的是丰富的结构，一旦理解了，将导致更广泛适用的强大的数学方法。 我建议使用组合构造来转换新的相互关联的对象，然后使用精心构造的代数和进一步的数学变换的性质，以引出有关原始几何问题本身的有形和具体的信息。这将需要发展代数和分析方法。 由于这些抽象概念所允许的一般性程度，这里提出的方法将适用于这一领域的其他问题。 这种方法的基础是我在代数组合学方面已经积累的研究和经验。(For数学家：我将研究的问题是：i）结构的双重Hurwitz数的分歧覆盖的领域，ii）费伯的交叉数猜想的模空间的光滑曲线，和iii）某些不变量的3流形。

Since about the 1980's, there has been a long period of remarkable and intense activity in (modern) geometry, spurred on by the deep connexions with mathematical physics. As research progressed in these areas, it became increasingly apparent that deeply within certain questions lay new questions of great complexity about aggregates of new objects constrained by complex relations. These new questions are essentially combinatorial in the sense that they can be expressed in purely combinatorial terms, without reference to the original setting in geometry and mathematical physics. The purpose of this Research Proposal is to make advances into the solution of combinatorial questions that arise in this way. To do so, I propose to examine three questions of great  complexity in algebraic geometry, that are notable in their own right, in the conviction that hidden within them is rich structure that, once understood, will lead to powerful mathematical methodology of  wider applicability.  I propose to use combinatorial constructions to transform the new interrelated objects, and then to use properties of carefully constructed algebras and further mathematical transformations to elicit tangible and concrete information about the original geometric questions themselves. This will require the development of the algebraic and analytic methodology.  Because of the level of generality that these abstractions permit, the methodology developed here will be applicable to other questions in this area.  At the foundation of this approach is the research and experience that I have already accumulated in algebraic combinatorics. (For mathematicians: The questions I shall examine are: i) the structure of the double Hurwitz numbers for ramified covers of the sphere, ii) Faber's intersection number conjecture for the moduli space of smooth curves, and iii) certain invariants of 3-manifolds.)