Real algebraic geometry and combinatorics

实代数几何和组合数学

基本信息

批准号：
RGPIN-2018-04741
负责人：
Purbhoo, Kevin
金额：
$ 2.62万
依托单位：
University of Waterloo
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2022
资助国家：
加拿大
起止时间：
2022-01-01 至 2023-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=749577
关键词：
Real algebraic geometry combinatorics

项目摘要

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).

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，代数可以在这两者之间架起一座桥梁。然而，有许多计数问题是许多数学领域的基础，但我们并没有像这样的完整图景；事实上，我们知道所有部分如何组合在一起的情况相对较少。从广义上讲，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，这是一个关于实数上某些方程的解的看似无害的结果，结果证明：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。该提案的主要目标是找到将这一重要定理与实代数几何和代数组合学的其他部分联系起来的新联系。这项研究涉及纯数学许多领域（几何、拓扑、组合、代数、复数）的核心数学对象。分析），并且还应用于物理学（统计力学）和工程（控制理论）。