Cluster theories and homological dimensions

在科学语言中，代数是一组被赋予了满足某些相容条件的加法和乘法运算的对象。例如，所有数字的集合构成一个代数。其他种类的代数在许多科学领域发挥着重要作用，例如在物理、化学、工程、生物，当然还有数学中。我的主要挑战，也是几乎所有数学家的主要挑战，在于为科学界继续进行新的研究提供新的工具。更具体地说，我希望开发出识别代数的工具，明确代数的主要性质，使科学家能够继续研究这些代数的来源。我的课题是通过称为模块的复杂代数结构来研究这些代数。代数通常有无限多个模块，不幸的是，这些模块的理解往往是非常有限的。我对模块的同调性质很感兴趣，特别是50年前的有限维猜想，对于这个猜想，一个积极的答案将对我的领域产生巨大的影响。我对簇和簇倾斜代数也很感兴趣，它们最近的引入来自于对代数群的某些正性问题的研究。最近，我和三位同事证明了簇倾斜代数是雅可比代数，雅可比代数的研究是由来自物理学的问题所激发的。这些联系引起了数学科学的极大兴趣。

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.