Research in Noncommutative Algebra

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Many models in the sciences and engineering lead to mathematical problems about solutions of equations in commuting variables. However, starting with quantum mechanics, a great number of models emerged that led to mathematical problems involving variables that no longer commute. Noncommutative algebra is one of the areas of mathematics that studies those structures. This research project investigates the interrelations between the commutative and noncommutative settings, and the possible ways to study the latter via geometry and combinatorics. The geometric approach attaches structures, called Poisson manifolds, to noncommutative objects. The combinatorial approach studies intricate internal transformations of the objects, called cluster mutations. Both approaches will be used to carry out a detailed study of the properties and symmetries of noncommutative objects. In the opposite direction, the noncommutative objects will be shown to exhibit various forms of rigidity, which in turn will be used to address algebraic, geometric, and combinatorial problems of independent interest. These research activities will be used as the foundation for the training of graduate and undergraduate students and for mentoring mathematics postdocs. This research project investigates problems in noncommutative algebra and the relations of these structures to Poisson geometry, representation theory, combinatorics, homological algebra, and noncommutative geometry. The program aims at using techniques from those areas to settle problems in noncommutative algebra, and in the opposite direction, to apply rigidity properties of noncommutative algebras to resolve problems of independent interest to other areas of mathematics. One direction of this program is the description of the automorphism groups and invariants, such as discriminants, of noncommutative algebras via Poisson geometry and cluster algebras. Another direction targets the study of the properties of Nichols algebras and quantum groups by realizing them as cluster algebras. The geometric and combinatorial approaches will be merged in the investigation of the topology of the spectra of quantum function algebras. Noncommutative projective spaces and Calabi-Yau algebras will be studied using techniques from cluster algebras. In the reverse direction, these connections will be used to investigate homological and Poisson geometric properties of cluster algebras.