Combinatorial Hopf Algebras and Applications

Bergeron教授的研究方向是代数组合学，即代数与描述离散对象族的艺术之间的相互作用。他主要对使用组合工具解决代数问题感兴趣。此外，他考虑的问题往往来自数学和科学的其他领域，如数学物理、计算机科学等。例如，一些特殊函数的组合模型可以用于设计完美的量子态转移，这是量子计算的基本任务。Bergeron教授的工作还受到通信系统和数据分析中对更好的组合算法的需求的推动。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底层组合结构上执行的操作以及这些操作之间的兼容性上。他的研究使用计算机来全面了解所涉及的机制，这使他能够发现对象之间的新关系。他的团队一直积极参与SAGE的开发，SAGE是一个致力于符号编程的开源环境。由于部分问题仍然是初级的和令人兴奋的工作，这种类型的调查是培养年轻研究人员的理想选择。Bergeron教授在世界各地有大量的合作者，包括各种研究小组的组合。他的建议主要集中在以下几个方面。(A)组合Hopf代数（CHA）和多面体：最近，Bergeron教授和他的合作者发现某些代数结构族（CHA）中的某些操作与具有平面的凸体族（polytopes）直接相关。这种相互作用提供了对代数和几何对象的更好理解。找出CHA一般族的性质是很重要的。这大大提高了我们对这些结构的理解。该几何图形还可用于对CHA的不同碱基的结构常数进行编码。(B)特殊功能：某些CHAs包含物理学非常感兴趣的特殊功能。例如，在量子物理学中，一些特殊函数的结构常数可以用来编码基本粒子之间可能的相互作用。Bergeron教授和他的合作者已经发现了新的特殊功能，他认为这些功能将在新的物理模型中发挥类似的作用。(C)分类：一个基本问题是将给定的CHA与空间的表示联系起来，即对空间的不同对称性进行分类。这对科学来说非常重要，因为它为我们提供了操纵和理解物理、化学、生物等定律的对称性的工具。Bergeron教授的小组已经对许多CHA表示进行了分类，并将继续这项研究。

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.