Research in Noncommutative Algebra

科学和工程中的许多模型都导致了关于交换变量方程解的数学问题。然而，从量子力学开始，大量的模型出现，导致涉及不再交换的变量的数学问题。非交换代数是研究这些结构的数学领域之一。本研究计画探讨交换与非交换设定之间的相互关系，以及透过几何学与组合学来研究非交换设定的可能方法。几何方法将称为泊松流形的结构附加到非交换对象上。组合方法研究对象的复杂内部变换，称为簇突变。这两种方法将被用来进行详细的研究的性质和对称性的非交换对象。在相反的方向上，非对易对象将显示出各种形式的刚性，这反过来将被用来解决代数，几何和组合问题的独立利益。这些研究活动将被用作研究生和本科生的培训和指导数学博士后的基础。本研究计画探讨非交换代数中的问题，以及这些结构与泊松几何、表示论、组合学、同调代数与非交换几何的关系。该计划旨在使用这些领域的技术来解决非交换代数中的问题，并在相反的方向上，应用非交换代数的刚性属性来解决其他数学领域的独立问题。这个程序的一个方向是通过泊松几何和簇代数描述非交换代数的自同构群和不变量，如判别式。另一个方向是研究Nichols代数和量子群的性质，将它们实现为簇代数。几何方法和组合方法将在量子函数代数谱的拓扑研究中得到融合。非交换射影空间和卡-丘代数将使用簇代数的技巧进行研究。在相反的方向，这些连接将被用来研究同调和泊松几何性质的集群代数。

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.