This project was an investigation into a type of mathematical structure called a Batalin-Vilkovisky algebra. This type of structure arises naturally in many branches of mathematics; this project's applications are to Batalin-Vilkovisky algebras satisfying a special condition. The examples focused on were Batalin-Vilkovisky algebras related to quantum field theory, which all satisfy the special condition. In previous work, I had begun studying a construction, called the Barannikov-Kontsevich construction, whose input was a Batalin-Vilkovisky algebra satisfying the special condition and whose output was a different kind of mathematical structure. In this project, I wrote a series of three papers, two alone and one in collaboration, which furthered this study. The first paper, written in collaboration with Bruno Vallette, extended the Barannikov-Kontsevich construction. The extension constructed in the paper allows more general input and produces more refined output. The extension also captures the meaning of the Barannikov-Kontsevich construction in terms of well-understood mathematical tools. In this paper we demonstrated that in some cases, this more refined output cannot be reduced to the original Barannikov-Kontsevich construction. The second paper, written alone, further extended the results of the first paper to apply to even more general examples of a very different nature. The first paper could be called an algebro-homotopical analysis of the Barannikov-Kontsevich construction; the second paper is then a topological-homotopical analysis of the same construction. This second paper implies certain algebro-homotopical results not contained within the first paper. The idea behind this paper was suggested by Kontsevich in a talk some years earlier, but no one had ever publicly verified the details, which turned out to be quite intricate. In the third paper, I applied the tools of the first paper to certain classes of Batalin-Vilkovisky algebras of interest in quantum field theory. I concluded that in these particular cases the more refined output was not interesting and in fact could be reduced to the original Barannikov-Kontsevich construction. The same methods suggest where to look in further research for more interesting refined output. These three papers were published in reputable peer-reviewed academic journals. Together, they further our understanding of the Barannikov-Kontsevich construction and through it quantum field theory and several other related fields.
