Noncommutative Algebra

同源方法在许多领域得到越来越多的应用。最近与三角化（衍生）范畴相关的发展对交换代数、代数几何和非交换代数产生了巨大的影响。这引导我们寻找代数和几何对象的新逻辑组合结构。一些在非交换集合中发现的新理论在交换代数和代数几何中得到了意想不到的应用，而交换代数和代数几何中的许多经典理论已经通过同调载体推广到非交换集合中。该提案强调主题的同源性方面。主要研究hopfalgebra的同调积分和刚性Gorenstein环的中山阶等不变量，进一步发展非交换代数和空间上的对偶化复合体理论，并将尝试完成量子射影三维空间的分类。一般来说，非交换代数是数学和物理中的一个强大工具。当前在代数几何、数论、弦论和拓扑学等领域的研究需要对量子群、非交换Iwasawa代数和Artin-Schelter正则代数等几类非交换代数有更深入的认识。该提案包括一些突出的问题创新非交换代数，并着眼于应用到其他领域。首席研究员将与不同领域的专家进行沟通和合作。该项目对研究、教育和培养研究生和本科生具有更广泛的影响。

Homological methods are used more and more in many areas. Recentdevelopments connected with triangulated (derived) categorieshave enormous impacts to commutative algebra, algebraic geometryand noncommutative algebra. These lead us to search for newhomological-combinatorial structures of algebraic and geometricobjects. Some new theories discovered in the noncommutative settinghave unexpected applications to commutative algebra and algebraicgeometry, while many classical theories in commutative algebra andalgebraic geometry have been generalized to the noncommutativesetting via the homological vehicle. The proposal emphasizes thehomological aspects of the subject. The Principal Investigatorstudies invariants such as the homological integrals of Hopfalgebras and the Nakayama order of rigid Gorenstein rings, furtherdevelops the theory of dualizing complexes over noncommutativealgebras and spaces, and will try to complete the classificationof quantum projective three-spaces.Noncommutative algebra in general is a powerful tool in mathematicsand physics. Current research in algebraic geometry, number theory,string theory and topology require further and deeper understandingof several classes of noncommutative algebras, such as, quantumgroups, noncommutative Iwasawa algebras and Artin-Schelter regularalgebras. The proposal consists of some outstanding questions innoncommutative algebra, with an eye on applications to other areas.The Principal Investigator will communicate and collaborate withexperts in different fields. The project has broader impacts onresearch, education, and training graduate and undergraduate students.