The project focuses on cluster algebras and their relation to the representation theory of finite dimensional algebras. Cluster algebras are commutative algebras with a special combinatorial structure. The theory of cluster algebras is a fast developing field which is related to many areas of mathematics. The PI will continue his study of the special type of cluster algebras associated to Riemann surfaces; he will introduce generalized cluster variables and use them to construct canonical bases for these cluster algebras, and he will develop combinatorial models for explicit computations inside the cluster algebra. The PI will also investigate cluster-tilted algebras, which are certain finite dimensional algebras, whose modules correspond to elements of the cluster algebra. Cluster-tilted algebras provide a new point of view on tilting theory over hereditary algebras, and their module categories carry interesting information about the cluster algebra.
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 that is concerned with the study of symmetries of scientific models. 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.
The project's focus in on the theory of cluster algebras. 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 that is concerned with the study of symmetries of scientific models. Studying the symmetries of a model is often more fruitful than studying the model directly, and representation theory has found many applications in other mathematical fields, as well as in physics and chemistry. 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. Each cluster algebra is determined by a distinguished set of elements which are called cluster variables. These cluster variables themselves are constructed in a recursive way, involving algebraic computations with rational functions in several variables as well as combinatorial operations on graphs, called quiver mutations. Since the rational functions involved in this procedure can be very complex, it is in general highly difficult to compute cluster variables explicitly. One of the outcomes of the project is a systematic study of the cluster variables for cluster algebras of small rank. In particular, it yields an explicit formula for cluster variables of rank 2 and a proof of Fomin-Zelevinsky's longstanding positivity conjecture in ranks 2 and 3, as well as a proof of non-commutative version of the conjecture formulated by Fields Medalist Maxim Kontsevich. Furthermore, the project investigates the interior symmetries of the cluster algebra by introducing the concept of cluster automorphism, and lays the foundation for the study of cluster algebras from a categorial point of view by introducing cluster homomorphisms. In certain special types of cluster algebras the cluster variables can be interpreted algebraically in terms of representation theory of algebras and in other types there is a geometric interpretation in terms of intersection patterns of curves on surfaces. For the surface type, the project provides a canonical basis for the cluster algebra in terms of curves on the surface, as well as a graphical calculus of the relations among the cluster variables. These are two essential steps towards a better conceptual understanding of the cluster algebra as well as towards efficient explicit computation. For the cluster algebras related to representation theory, the project improves the understanding of the homological features of the algebras involved, by computing the so-called Hochschild cohomology of the algebras. As a part of this project, two students have completed their PhD and another one a Master's thesis. The PI and his students have presented the results at international and national conferences in Argentina, Canada, France, Germany, Mexico, Turkey and the US. The PI has given invited short courses on the topic to graduate students at institutions in Argentina, Canada and the US. Furthermore, he has developed new graduate courses at the University of Connecticut and organized a special session on cluster algebras at the 2012 Summer Meeting of the Canadian Mathematical Society,
