Cluster theories and homological dimensions

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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.