Multi-Novikov algebras, multi-differential algebras and their free constructions by multi-indices

Historically, differential algebras and Novikov algebras have diverse applications in mathematics and physics. Recently, multi-differential algebras and multi-Novikov algebras have emerged in the Fields Medal work of M. Hairer on regularity structures in stochastic PDEs. In these developments, a critical role is played by the induction of the Novikov-type algebras from the differential-type algebras, which can be traced back to the famous work of S. Gelfand.Motivated by these fundamental works and recent rapid advances, this dissertation provides a uniform framework for the related structures. It first introduces variations of multi-differential algebras and multi-Novikov algebras by relaxing the conditions on the derivations and multiplications. Doing so leads to the notions of noncommuting multi-differential commutative algebras, commuting multi-differential noncommutative algebras, noncommuting multi-differential noncommutative algebras, and their Novikov counterparts. Properties and examples are provided. In particular, Poisson algebras are shown to be naturally noncommuting multi-Novikov algebras. We also explore further into the connections between various multi-differential algebras and multi-Novikov algebras, by setting up a general framework of companion structures using the language of operads. This way, the various multi-Novikov algebras are shown to be precisely the companion structures of the various multi-differential algebras. Finally, free objects in the categories of multi-Novikov algebras and noncommuting multi-Novikov algebras are explicitly constructed by their relations to multi-indices in the corresponding multi-differential algebras utilizing rooted trees.