The PI is planning to focus on several problems which are related with the procedure of quantization in quantum field theory and quantum mechanics.
These problems include:
1) quantization of coisson algebras;
2) mathematically precise formulation of Batalin-Vilkovitski formalism;
3) Action of Grothendieck-Teichmueller group on various formality and quantization morphisms.
The first two problems arise in the quantum field theory. The problem of quantization of coisson algebras is posed by A. Beilinson and V. Drinfeld; they found a rather non-trivial solution to this problem in a particular case of linear coisson brackets. The general quantization problem is much harder and is not accessible by similar methods. An appropriate tool may be the deformation theory and introduction of an additional structure on an appropriate deformation complex.
The Batalin-Vilkovitsky formalism is one of the most powerful tools in quantization of systems with sophisticated gauge symmetries. This formalism implies an extensive use of path integrals, whence the lack of mathematical meaning of the most important ingredients of the construction, such as the operator $Delta$ and the $BV$-bracket. The difficulty in defining path integrals are well known, a straightforward extension of usual (finitely-dimensional) integration rules necessarily leads to divergencies. It is only a careful analysis of Batalin-Vilkovitski formalism involving a theory of D-modules and homological algebra that can allow one to construct a mathematically meaningful theory.
The action of Grothendieck-Teichmueller group is known to be present on the set of quantization functors of Lie bialgebras as well as on the set of formality quasi-isomorphisms from Kontsevich's formality theorem. This fascinated subject was originated by V. Drinfeld and was further developed by P.Etingof-D. Kazhdan, M. Kontsevich and other authors. Yet there are several open questions concerning this action on the space of formality quasi-isomorphisms, namely, whether it is transitive or free in a certain homotopical sense. I hope that studying these problems should deepen our understanding of algebro-geometric and motivic aspects of quantization.
