The PI will work on a series of interrelated problems for quantum groups and quantum cluster algebras. Firstly, he will work on proving that each member of a very general axiomatically defined class of quantum nilpotent algebras admits a canonical quantum cluster algebra structure. Based on this, he will attempt to construct a unified categorification for all algebras in the class. Many important families arise as special cases of algebras in this large class, most notably the quantum double Bruhat cell algebras and the quantum Schubert cell algebras. The former case will lead to a proof of the Berenstein-Zelevinsky quantum cluster algebra conjecture. In both cases the quantum cluster algebra structure will be used to study the topology of the spectra of quantum groups and quantum Schubert cell algebras. In the former case he will attempt to prove bicontinuity of his recently constructed Dixmier map from the symplectic foliation on a simple Lie group equipped with the so called standard Poisson structure to the primitive spectrum of a quantum group. The PI and his graduate students will apply these ideas for proving the existence of quantum foldings and for building quantum cluster algebras from them. He will also apply his recent results on rigidity of quantum tori to the classification of automorphism groups of interesting (quantum) cluster algebras. Via the work of Gekhtman, Shapiro and Vainshtein a certain large class of classical cluster algebras can be approached using Poisson geometry. The PI will work on Poisson analogs of the above projects using a notion of Poisson unique factorization domains.
Noncommutative and Poisson algebras arise in many different aspects of mathematics (functions on geometric objects) and physics (observables in classical and quantum mechanical systems). These objects are studied using many different techniques on the basis of algebraic, geometric, analytic and combinatorial methods. The PI will study these objects via two different methods. The first one is a classical one, based on studying the presentations of these algebras as collections of operators (representations). This method uses techniques from algebra and geometry. The second method is based on the recent combinatorial notion of cluster algebras invented by Fomin and Zelevinsky. It leads to a very concrete combinatorial structure on the objects. The idea of mutation is then used to study various parts of the objects which are not seen by the previous methods (they focused on a particular "initial side" of these algebras). Using his recent rigidity results, the PI will also study and classify the symmetries of the objects in the above classes. The motivation for this is that symmetries reduce the complexity of an abject and the full description of the collection of symmetries provides an understanding of the complexity of the object.
