Cluster theories and homological dimensions

聚类理论和同调维度

• 批准号: 366203-2010
• 负责人: Smith, David
• 金额: $ 1.82万
• 依托单位: Bishop's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635414
• 关键词: Cluster; theories; homological; dimensions

In scientific language, an algebra is a set of objects endowed with addition and multiplication operations satisfying some compatibility conditions. For instance, the set of all numbers form an algebra. Other kinds of algebras play an important role in many fields of sciences, for instance in physics, chemistry, engineering,biology, and of course in mathematics.My main challenge, which is the main challenge of almost every mathematician, consists in providing new tools to the scientific community for the continuation of new investigations. I wish, more specifically, to develop tools for the recognition of algebras and make explicit their main properties to allow scientists to continue the study of the objects from which these algebras came.My project consists in studying these algebras through complex algebraic structures, called modules. An algebra usually comes with infinitely many modules whose comprehension is unfortunately often quite limited. I am interested in the homological properties of the modules, and in particular in the 50 year old finitisticdimension conjecture, for which a positive answer would have huge impacts in my field. I am also interested in cluster and cluster-tilted algebras, whose recent introduction comes from the study of certain positivity problems for algebraic groups. Recently, three colleagues and I have shown that the cluster-tilted algebras are Jacobian algebras, whose study is motivated by problems coming from physics. These connections are of great interest to the mathematical sciences.

在科学语言中，代数是一组赋予满足某些兼容条件的添加和乘法操作的对象。例如，所有数字的集合形成一个代数。其他类型的代数在许多科学领域都起着重要作用，例如物理，化学，工程，生物学，当然还有数学。我的主要挑战，这是几乎每个数学家的主要挑战，是为继续进行新调查的科学界提供新工具。我希望更具体地说，能开发出识别代数的工具，并明确其主要特性，以使科学家能够继续研究这些代数来自这些代数的物体。我的项目包括通过复杂的代数结构研究这些代数，称为模块。代数通常带有无限的许多模块，不幸的是，这些模块通常是非常有限的。我对模块的同源性属性感兴趣，尤其是对50年历史的有限态度猜想，为此，积极的答案将对我的领域产生巨大影响。我也对群集和倾斜的代数感兴趣，后者最近的引入来自对代数群体对某些阳性问题的研究。最近，我和三位同事表明，倾斜的代数是雅各布式代数，他们的研究是出于物理问题的问题。这些联系引起了数学科学的极大兴趣。