Research in Noncommutative Algebra

科学和工程中的许多模型导致了关于交换变量方程的解的数学问题。然而，从量子力学开始，出现了大量的模型，这些模型导致了涉及不再往返的变量的数学问题。非对易代数是研究这些结构的数学领域之一。本研究项目研究交换和非交换环境之间的相互关系，以及通过几何和组合学研究非交换环境的可能途径。几何方法将称为泊松流形的结构附加到非对易对象。组合方法研究物体复杂的内部转换，称为簇突变。这两种方法都将被用来详细研究非对易对象的性质和对称性。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独立感兴趣的代数、几何和组合问题。这些研究活动将被用作培训研究生和本科生以及指导数学博士后的基础。本研究项目研究非对易代数中的问题以及这些结构与泊松几何、表示论、组合学、同调代数和非对易几何的关系。该程序的目的是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性质来解决其他数学领域中独立感兴趣的问题。该程序的一个方向是通过Poisson几何和簇代数来描述非对易代数的自同构群和不变量，如判别式。另一个方向是通过将Nichols代数和量子群实现为簇代数来研究它们的性质。几何方法和组合方法将融合到量子函数代数的谱的拓扑研究中。非对易射影空间和Calabi-Yau代数将使用簇代数中的技巧来研究。在相反的方向上，这些联系将被用来研究簇代数的同调和泊松几何性质。

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.