Mathematical Sciences: Homology and Representations Theory of Finite Dimensional Algebras and Classical Orders

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This research is concerned with the homological aspects of finite dimensional algebras. The principal investigator will examine several dimensions which measure the homological complexity of an algebra in order to obtain precise structural descriptions of higher syzygies, and to describe and exploit repetitive homological behavior. There will be a heavy emphasis on crossconnections to related areas. In particular, the principal investigator will seek replacements for the Tarsy conjecture on the global dimension of classical orders over discrete valuation domains. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.