Curve counting, moduli, and logarithmic geometry

研究复杂空间几何的一个有效方法是研究其他空间如何能够坐在它们里面。例如，在三维空间中存在精确地包含27条直线的表面。因此，我们可以得出这样的结论：这种表面必须具有与熟悉的平面二维空间根本不同的几何形状，在平面二维空间中，人们可以在任何两点之间画一条线。几个世纪以来，这样的计算吸引了数学家的想象力，但在20世纪90年代，理论物理学催生了这种想法的一种强大的新形式。物理学家们认识到，这些简单的“曲线计数”问题是弦论中某些物理模型的相关和可计算的不变量。在此后的几十年里，不变量对纯数学世界的无数遥远角落产生了影响。这个提议试图理解这些曲线计数不变量的现代化身。所讨论的空间将是多项式方程组的解集，称为代数簇。该建议的方法在于两个年轻的学科称为对数和热带几何的联系。求解多项式方程组的过程可分为两步。我们可以首先找到具有正确数量级的解，或者准确地说，找到解的可能大小的集合。作为一个类比，而不是精确计算212和330的乘积，人们可以目测答案是大约60000。虽然这是错误的答案，但它为许多目的提供了足够好的估计。热带几何试图将这种逻辑应用于几何本身，通过寻找简单的几何结构，但反映了真实几何的有用近似。对数几何是一座技术桥梁，它让人们回到多项式系统的微妙世界。热带几何学本身有着最优化理论和理论物理学的基础，其应用范围远达统计学和拍卖理论。这项研究的基本目标是了解这些热带几何结构如何控制曲线计数不变量，并寻求建立和利用这两个方向的数学探究之间的桥梁。具体目标将是解决几个长期存在的问题，关于曲线计数不变量的结构，并使用热带方法，使完整和有效的计算代数几何，超越了没有热带输入已经取得的成就。

An effective method to study the geometry of complicated spaces is to examine how other spaces are able to sit inside them. For instance, there exist surfaces in three-dimensional space that contain precisely 27 straight lines. We might therefore conclude that such surfaces must have a fundamentally different geometry than familiar flat 2-dimensional space, where one can draw a line between any two points. Calculations such as these captured the imagination of mathematicians for centuries, but in the 1990s, theoretical physics gave birth to a powerful new form of this idea. The physicists recognised that these simple minded "curve counting" questions were relevant and computable invariants of certain physical models in string theory. In the decades since, the invariants have had impacts on countless faraway corners of the pure mathematics world. This proposal seeks to understand the modern avatars of these curve counting invariants. The spaces in question will be solution sets to systems of polynomial equations, known as algebraic varieties. The methods of the proposal lie at the nexus of two young subjects known as logarithmic and tropical geometry. The process of solving a system of polynomial equations can be broken up into two steps. One can first find solutions that have the right order of magnitude, or precisely, the set of possible sizes of solutions. As an analogy, rather than calculating the product of 212 and 330 exactly, one can eyeball that the answer is about 60000. While this is the wrong answer, it gives a good enough estimate for many purposes. Tropical geometry seeks to apply this logic to geometry itself, by finding geometric structures that are simple, but reflect a useful approximation of a true geometry. Logarithmic geometry is the technical bridge that allows one to return to the subtle world of polynomial systems. Tropical geometry itself has roots in optimisation theory and theoretical physics, and applications reaching as far as statistics and auction theory. The fundamental goal of this research proposal is to understand how these tropical geometric structures control curve counting invariants, and seeks to build and exploit a bridge between these two directions of mathematical inquiry. Concrete objectives will be to address several long standing questions concerning the structure of curve counting invariants, and to use tropical methods to make complete and effective calculations in algebraic geometry, that go beyond what has been achieved without tropical input.