Real algebraic geometry and combinatorics

实代数几何和组合数学

• 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

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).

枚举是计算数学对象的研究。一些最重要的枚举问题在代数几何形状中具有Orgins。这些“枚举代数几何”问题具有双重性质：它们具有组合（离散）侧和几何（连续）侧。在某些情况下，代数可以在这两个侧之间提供桥梁。但是，对于许多数学领域，我们没有这样的完整图片，这是许多数学领域的基本问题。实际上，在相对较少的情况下，我们知道所有零件如何结合在一起。从广义上讲，这项研究的目的是解决这些知识差距。这项研究中的主要工具之一是Mukhin-Tarasov-Varchenko定理，这是对某些方程式在实数方面的解决方案的看似无害的结果，事实证明，这些方程式在枚举几何学中产生了深远的影响。 Mukhin-Tarasov-Varchenko定理提供了一个框架，用于解释列举组合中出现的许多奇怪现象。该提案的主要目的是找到新的联系，将这一重要定理与真实代数几何形状和代数组合的其他部分联系起来。这项研究涉及数学对象，这些对象对于纯数学，几何学，拓扑，拓扑，拓扑，组合，组合学，代数，复杂分析）以及对物理学的应用程序（统计学机制）（以及机制）以及机制（统计机制）以及统计学的机制（以及机制）。