Cluster theories and homological dimensions

在科学语言中，代数是一组具有满足某些相容性条件的加法和乘法运算的对象。例如，所有数的集合构成一个代数。其他类型的代数在许多科学领域中发挥着重要作用，例如在物理学，化学，工程学，生物学，当然还有数学。我的主要挑战，这是几乎每一个数学家的主要挑战，包括提供新的工具，科学界继续新的调查。我希望，更具体地说，开发工具的识别代数和明确其主要性质，使科学家能够继续研究的对象，这些代数来了。我的项目包括研究这些代数通过复杂的代数结构，称为模块。一个代数通常有无穷多个模，不幸的是，这些模的理解往往相当有限。我对模的同调性质很感兴趣，特别是50年前的有限维猜想，一个肯定的答案将在我的领域产生巨大的影响。我也有兴趣在集群和集群倾斜代数，其最近的介绍来自研究某些积极的问题，代数群。最近，我和我的三个同事证明了簇倾斜代数是雅可比代数，其研究的动机是来自物理学的问题。这些联系对数学科学有着极大的兴趣。

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.