Classical algebraic geometry and modern combinatorics

枚举法是对数学对象进行计数的研究。一些最重要的计数问题起源于代数几何。这些“枚举代数几何”问题具有双重性质：它们有组合(离散)边和几何(连续)边。在某些情况下，代数可以在这两方面之间架起一座桥梁。然而，有许多计算问题是许多数学领域的基础，我们没有像这样的完整图景；事实上，我们知道所有部分是什么的实例相对较少，更不用说它们如何组合在一起了。在某些情况下，我们只有神秘的组合程序，似乎是偶然地给出了正确的答案，几乎不了解它们为什么起作用。这种类型的知识很少提供如何解决新的相关问题的线索。这个研究提案中的主要工具之一是最近证明的Mukhin-Tarasov-Varchenko定理：关于实数上某些方程的解的一个看似无害的结果，事实证明这在计数几何中具有深远的意义。Mukhin-Tarasov-Varchenko定理为以更简单的方式解释这一领域中出现的许多奇怪现象提供了一个框架，也为发现新的相关定理提供了工具。这一领域的许多几何问题可以追溯到1880年的S，多年来已经得到了广泛的研究。然而，直到最近，数学家们才开发出在一些更难、更精细的问题上取得进展所需的工具。这项提案中的研究将有助于开发新的方法来回答列举几何中长期存在的基本问题。

Enumeration is the study of counting mathematical objects. Some of the most important enumeration problems have their orgins in algebraic geometry. These "enumerative algebraic geometry" problems have a dual nature: they have a combinatorial (discrete) side and geometric (continuous) side. In some cases, algebra can provide a bridge between these two sides. However, there are many counting problems that are fundamental to a number of areas of mathematics, where we do not have a complete picture like this; in fact, there are relatively few instances where we know what all the pieces are, let alone how they fit together. In some cases we have only mysterious combinatorial procedures that seem to give the right answer, as if by accident, with little understanding of why they work. This type of knowledge provides few clues as to how to solve new, related problems.One of the main tools in this research proposal is the recently proved Mukhin-Tarasov-Varchenko Theorem: a seemingly innocuous result about solutions to certain equations over the real numbers, which turns out to have profound consequences in enumerative geometry. The Mukhin-Tarasov-Varchenko theorem provides a framework for explaining many of the strange phenomena that arise in this area in a simpler way, as well as a vehicle for discovering new, related theorems.Many of the geometry problems in this field date back to the 1880's, and have been studied extensively over the years. However, it is only recently that mathematicians have developed the tools needed to make progress on some of the harder, more refined questions. The research in this proposal will help to develop new approaches for answering fundamental, longstanding questions in enumerative geometry.