Real algebraic geometry and combinatorics

枚举是对数学对象进行计数的研究。一些最重要的枚举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）的一面和几何（连续）的一面。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，我们没有这样一个完整的画面；事实上，我们知道如何将所有的碎片组合在一起的例子相对较少。总的来说，这项研究的目的是解决这些知识差距。******这项研究的主要工具之一是Mukhin-Tarasov-Varchenko定理，这是一个关于某些实数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定理为解释列举组合学中出现的许多奇怪现象提供了一个框架。本建议的主要目的是找到将这一重要定理与实际代数几何和代数组合学的其他部分联系起来的新联系。******这项研究涉及的数学对象是许多纯数学领域（几何、拓扑、组合学、代数、复分析）的中心，也应用于物理学（统计力学）和工程学（控制理论）。

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 many counting problems that are fundamental for 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 how all the pieces fit together. Broadly, this aim of this research is to address these knowledge gaps.******One of the main tools in this research is the 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 enumerative combinatorics. The main objective of this proposal is to find new connections linking this important theorem with other parts of real algebraic geometry and algebraic combinatorics.******This research concerns mathematical objects that are central to many areas of pure mathematics (geometry, topology, combinatorics, algebra, complex analysis), and also has applications to physics (statistical mechanics) and engineering (control theory).