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Curve counting, moduli, and logarithmic geometry

曲线计数、模数和对数几何

基本信息

批准号：
EP/V051830/1
负责人：
Dhruv Ranganathan
金额：
$ 17.27万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV051830%2F1
关键词：
Curve counting moduli logarithmic geometry

项目摘要

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.

研究复杂空间几何学的一个有效方法是研究其他空间是如何能够位于其中的。例如，三维空间中存在恰好包含27条直线的曲面。因此，我们可能会得出结论，这样的曲面必须具有与人们熟悉的平面二维空间完全不同的几何图形，在平面二维空间中，人们可以在任意两点之间画一条线。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，理论物理学催生了一种强大的新形式的这种想法。物理学家们认识到，这些头脑简单的“曲线计数”问题是弦理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中无数遥远的角落产生了影响。这项提议试图理解这些曲线计数不变量的现代化身。所讨论的空间将是称为代数族的多项式方程组的解集。该提案的方法在于两门年轻学科的结合点，即对数几何和热带几何。求解多项式方程组的过程可以分为两个步骤。人们可以首先找到具有正确数量级的解，或者准确地说，具有可能的解的集合。打个比方，人们可以看到答案大约是60000，而不是精确地计算212和330的乘积。虽然这是一个错误的答案，但对于许多目的来说，它给出了一个足够好的估计。热带几何试图将这一逻辑应用于几何本身，通过寻找简单但反映真实几何的有用近似的几何结构。对数几何是一座技术桥梁，它允许人们回到多项式系统的微妙世界。热带几何学本身植根于最优化理论和理论物理，应用范围远至统计学和拍卖理论。这项研究计划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，并试图在这两个数学探索方向之间建立和开发一座桥梁。具体目标将是解决与曲线计数不变量的结构有关的几个长期存在的问题，并使用热带方法在代数几何中进行完整和有效的计算，这些计算超出了没有热带输入的情况。