Some questions in total positivity and cluster algebras

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The proposed project focuses on cluster algebras and total positivity. Fomin and Zelevinsky have repeatedly encountered in their work a new kind of combinatorics, which they axiomatized as so-called cluster algebras. Cluster algebras have found applications in numerous areas of mathematics in recent years, such as quiver representations, non-commutative geometry, discrete integrable systems, Teichmuller theory and even string theory. In joint work with Sergey Fomin, the PI will explore a model for certain cluster algebras in terms of tensor diagrams. The PI will also study a closely related topic of total positivity in joint work with Thomas Lam. Totally positive parts of algebraic groups are closely related to cluster algebras and Lusztig's canonical bases. The proposed research continues the study of affine total positivity, which is a new and fruitful direction in this area.The proposed project is rooted in two areas of mathematics: representation theory and combinatorics. Representation theory is an important area that is fundamental for the study of quantum mechanics and particle physics. Cluster algebras provide a new approach to representation theory and related disciplines. The PI's research focuses on cluster algebras that lie just beyond the current frontier of our understanding (the so called infinite mutation type). The notion, closely related to cluster algebras, of total positivity first appeared in the literature at the beginning of previous century. It is a major tool for understanding oscillations in mechanical systems. The PI's work concentrates on extending this theory to the affine (or infinite dimensional) case.