Representation theory, the mathematical study of symmetry and its manifestations in various systems, has its roots in two key developments: the late nineteenth-century examination of groups of linear transformations and the early twentieth-century advent of quantum mechanics. Its influence has become an essential tool in mathematics, physics, and chemistry. Furthermore, representation theory has been enriched by concepts from theoretical physics, culminating in a comprehensive synthesis of diverse mathematical fields by the end of the 20th century. This includes Lie algebras and quantum groups, automorphic forms, finite groups, and the topology of knots and 3-varieties. A key component of this synthesis is a category of algebraic structures known as vertex algebras. The project seeks to further the understanding of vertex algebras and elucidate the connections between the aforementioned areas.

Physics, a realm of intricate phenomena and fundamental laws, is a rich source of mathematical structures interwoven with each other and the domain we call pure mathematics. Vertex algebras not only provide a powerful language for describing quantum field interactions but have also found applications in various areas of pure mathematics, such as representation theory, integrable systems, and algebraic geometry.

With the backing of the Serrapilheira Institute, I am pursuing my exploration of vertex algebras and their multiple representations.  Building on the foundations established in the project’s initial phase to shed light on the composition of certain braid tensor categories related to knot and braid theory, I am poised to delve deeper into the connections with simplicial geometry and the enigmatic Ramarujan formulas.