Classical algebraic geometry and modern combinatorics

枚举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：它们具有组合（离散）方面和几何（连续）方面。在某些情况下，代数可以在这两个方面之间提供桥梁。然而，有许多计数问题是数学的许多领域的基础，我们没有这样一个完整的画面;事实上，我们知道所有碎片是什么的情况相对较少，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，而对它们为什么起作用却知之甚少。这类知识对如何解决新的相关问题提供了一些线索。本研究提案中的主要工具之一是最近证明的Mukhin-Tarasov-Varchenko定理：一个关于真实的数上某些方程的解的看似无害的结果，结果在枚举几何中产生了深远的影响。Mukhin-Tarasov-Varchenko定理提供了一个框架，以更简单的方式解释这一领域出现的许多奇怪现象，以及发现新的相关定理的工具。这一领域的许多几何问题可以追溯到19世纪80年代，多年来一直被广泛研究。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。在这个建议的研究将有助于开发新的方法来回答基本的，长期存在的问题，枚举几何。

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.