CAREER: Cluster algebras, combinatorics and representation theory

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The project focuses on cluster algebras and their relation to the representation theory of finite-dimensional algebras and combinatorics. Cluster algebras are commutative algebras with a special combinatorial structure, which are related to various fields in mathematics and physics. The PI will pursue several investigations. For general cluster algebras, he plans to prove the positivity conjecture, which is the oldest unsolved conjecture in the field. For cluster algebras of surface type, the PI will study the properties of canonical bases introduced by him and his collaborators. Moreover, he will use the bases to gain a better understanding of the cluster algebra itself as well as its relationship with representation theory and Teichmueller theory. The Pi will also study automorphisms of cluster algebras which will lead to a better comprehension of the intrinsic symmetries of cluster algebras. The PI will integrate education and research at several levels. He will organize a graduate summer school on cluster algebras at the University of Connecticut. The PI will continue to develop an international network at institutions in Argentina, Canada and the US, which will allow advanced graduate students and postdocs to go abroad for two-week long research visits in order to exchange ideas with local researchers. Moreover, the PI will continue to run the cluster algebra seminar in his department.When cluster algebras were introduced by Fomin and Zelevinsky in 2002, their original motivation came from representation theory, which is a branch of modern algebra. Studying the symmetries of a model is often more fruitful than studying the model directly, and representation theory has found many applications in physics and chemistry as well as in other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns which occur throughout representation theory. Surprisingly, these patterns are also observed in various other branches of science which, a priori, are not related to representation theory. This motivates a further development of the theory of cluster algebras to which this project will contribute.