This project is devoted to the understanding of several new algebraic structures which are deformations of classical notions. The first aim is the description of the new algebraic structures that have appeared in string topology and graphs. Here, the solution is expected to be given in terms of dioperads using their recently developed theory of Koszul duality. The second aim is the description of the structure and representation theory of a new noncommutative algebra which arises from quantization of Slodowy slices. Here, it is hoped that the methods in the commutative case can be extended to the noncommutative situation. The third aim is the construction of a conjectural deformed Harish-Chandra homomorphism that gives a quantum Hamiltonian reduction description of the spherical subalgebras of symplectic reflection algebras. Here, the idea is to generalize the radial part construction of invariant differential operators and the construction of Dunkl representations to the new situation.
Broader impacts: This project will hopefully build a common algebraic framework for the discovery and study of operations in various branches of mathematics and physics such as topology, graph theory, string theory, and quantum field theory. It also hopes that working out some interesting concrete examples in noncommutative algebra will shed light on the general theory that one can hope to develop.
