Combinatorial Hopf Algebras and Applications

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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.