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，我们知道所有碎片如何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。 本研究的主要工具之一是Mukhin-Tarasov-Varchenko定理，这是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来解释枚举组合学中出现的许多奇怪现象。这个建议的主要目标是找到新的连接这一重要定理与其他部分的真实的代数几何和代数组合。 这项研究涉及的数学对象是纯数学的许多领域（几何，拓扑，组合学，代数，复分析）的核心，也有应用物理（统计力学）和工程（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).