Polyhedral Combinatorics in Representation Theory and Algebraic Geometry

Northeastern University, Boston MA

Investigators

Abstract

The project focuses on cluster algebras recently discovered by the investigator in collaboration with S.Fomin. Cluster algebras are integral domains of a special kind designed to provide an algebraic framework for the study of total positivity and canonical bases in semisimple groups and their representations. The investigator studies structural properties of cluster algebras and their quantum deformations. This study uncovers unexpected connections with such diverse subjects as the structural theory of Kac-Moody algebras, thermodynamic Bethe ansatz, quiver representations, and integrable systems. One of the main instruments of the study is polyhedral combinatorics, more specifically, an interplay between piecewise-linear and subtraction-free birational transformations based on the tropical calculus. The main motivation for this project comes from two classical areas of mathematics: representation theory and the theory of total positivity. Representation theory is a mathematical approach to studying symmetry; more specifically, it encodes the symmetry properties of various physical and biological systems that occur in nature. Total positivity is a remarkable property of matrices (arrays of numbers) that generalizes the familiar notion of positive numbers. Both theories find numerous applications in physics, chemistry and other sciences, as well as in other mathematical disciplines. In fact, representation theory serves as the mathematical foundation of quantum mechanics, while total positivity is a major tool for explaining oscillations in mechanical systems. During the last decade, deep connections were found between the two fields, and the scope of their applications was greatly extended. This project explores the modern framework of representation theory and total positivity, with the goal of making its formalism much more explicit and understandable.