Combinatorial Hopf Algebras and Applications

组合 Hopf 代数及其应用

• 批准号: RGPIN-2018-05821
• 负责人: Bergeron, Nantel
• 金额: $ 2.55万
• 依托单位: York University
• 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=758907
• 关键词: Combinatorial; Hopf; Algebras; Applications

Prof. Bergeron works is in Algebraic Combinatorics, that is, the interplay between algebras and the art of describing families of discrete objects. He is primarily interested in solving algebraic problems using combinatorial tools. Moreover, the problems he considers often come from other areas of mathematics and science such as mathematical physics, computer science, and more. For example, combinatorial models for some special functions can be applied to design perfect quantum state transfer, a basic task in quantum computing. Prof. Bergeron's work is also motivated by a need for better combinatorial algorithms in communication systems and in data analysis. To investigate the combinatorial structure of many algebraic objects he focuses his attention on the operations that can be performed on the underlying combinatorial structure and on the compatibility between these operations. His studies use computers to obtain a comprehensive understanding of the mechanisms involved and this allows him to discover new relationships between the objects. His group has been actively involved in the development of SAGE, an open source environment dedicated to symbolic programming. Since parts of the problems remain elementary and stimulating to work on, this type of investigation is ideal for training young researchers. Prof. Bergeron has a large group of collaborators worldwide with the program involving various combinations of research groups. His proposal is centered on the following general aspects.(A) Combinatorial Hopf algebras (CHA) and polytopes: Recently, Prof. Bergeron and his collaborators have discovered that certain operations in some families of algebraic structures (CHA) is directly related to a family of convex bodies with flat facets (polytopes). This interaction provides a better understanding of both the algebras and the geometrical objects. It is important to find the properties of general families of CHA. This deeply improves our understanding of these structures. The geometry can also be used to encode the structure constants of different bases of the CHA.(B) Special functions: Certain CHAs contain special functions of great interest to physics. For example, the structure constants of some special functions can be used to encode the possible interactions of elementary particles in quantum physics. Prof. Bergeron and his collaborators have discovered new special functions that he envisions will play a similar role in new physical models.(C) Categorification: A fundamental problem is to link a given CHA to the representations of spaces, that is, a classification of the different symmetries of spaces. This is very important for science as it gives us tools to manipulate and understand the symmetries of the laws of physics, chemistry, biology and others. Prof. Bergeron's groups have classified many CHA representations and will continue this study.

Bergeron教授的作品是代数组合的，即代数与描述离散物体家庭的艺术之间的相互作用。他主要有兴趣使用组合工具解决代数问题。此外，他认为的问题通常来自数学和科学的其他领域，例如数学物理，计算机科学等。例如，某些特殊功能的组合模型可以应用于设计完美的量子状态传输，这是量子计算中的基本任务。 Bergeron教授的作品也是出于需要在通信系统和数据分析中更好的组合算法的动机。为了研究许多代数对象的组合结构，他将注意力集中在可以在基础组合结构上进行的操作以及这些操作之间的兼容性。他的研究使用计算机来获得对所涉及机制的全面理解，这使他能够发现对象之间的新关系。他的小组一直积极参与Sage的发展，Sage是一个专门针对象征性编程的开源环境。由于部分问题仍然是基本的和刺激的工作，因此这种研究非常适合培训年轻研究人员。 Bergeron教授在全球范围内拥有大量合作者，该计划涉及各种研究小组的组合。他的建议集中在以下一般方面。（a）组合霍普夫代数（CHA）和多面有：最近，伯杰隆教授及其合作者发现，某些代数结构（CHA）的某些操作与一个具有平面相（Polytopes）的凸面的代数结构（CHA）直接相关。这种相互作用可以更好地理解代数和几何对象。重要的是要找到CHA的一般家族的特性。这深刻提高了我们对这些结构的理解。几何形状也可以用来编码CHA不同基础的结构常数。（b）特殊功能：某些CHA包含物理学引起的特殊功能。例如，某些特殊功能的结构常数可用于编码量子物理学中基本粒子的可能相互作用。 Bergeron教授和他的合作者发现了他所设想的新特殊功能将在新的物理模型中扮演类似的角色。（c）分类：一个基本问题是将给定的CHA与空间的表示，即对不同空间的不同对称的分类。这对科学非常重要，因为它为我们提供了操纵和了解物理，化学，生物学和其他人定律的对称性的工具。 Bergeron教授的小组已将许多CHA表示归类，并将继续这项研究。