Algebraic combinatorics and its applications to algebraic geometry

自20世纪80年代S以来，在与数学物理的深刻联系的推动下，(现代)几何学经历了一段相当长的显著而激烈的活动时期。随着这些领域研究的进展，越来越明显的是，在某些问题的深层，隐藏着关于受复杂关系约束的新对象的集合的新的极其复杂的问题。这些新问题本质上是组合性的，因为它们可以用纯粹的组合术语来表示，而不涉及几何和数学物理中的原始背景。本研究方案的目的是为解决以这种方式产生的组合问题而取得进展。为此，我建议研究代数几何中三个极其复杂的问题，这三个问题本身就是值得注意的，因为我相信，隐藏在其中的是丰富的结构，一旦理解，将导致更广泛适用的强大数学方法论。我建议使用组合结构来变换新的相互关联的对象，然后使用精心构造的代数的性质和进一步的数学变换来得出关于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论。由于这些抽象允许的一般性水平，这里开发的方法论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。(对于数学家：我将研究的问题是：i)球的分支覆盖的双Hurwitz数的结构，ii)光滑曲线的模空间的Faber交数猜想，iii)3-流形的某些不变量。

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.)