DMS Colloquium: Saeed Nasseh

Speaker: Saeed Nasseh, Georgia Southern University

Title: Modules over finite dimensional algebras with an application to a conjecture in commutative algebra

Abstract: When \(F\) is a field and \(R\) an \(F\)-algebra, it is often important to classify the \(R\)-module structures carried by some fixed \(F\)-vector space \(V\). We will describe a classical approach to that problem, which involves some algebraic geometry and homological algebra. We will then sketch new developments of these techniques and their application to the proof of a result in commutative algebra, conjectured by Vasconcelos in 1974.

Biographical sketch

Saeed Nasseh is an associate professor in the Department of Mathematical Sciences at Georgia Southern University. Prior to that, he was a postdoctoral fellow at the University of Nebraska-Lincoln. His research areas are commutative algebra and homological algebra with some overlaps with algebraic geometry, algebraic topology, and representation theory. A major part of his recent research focuses on developing and applying differential graded (DG) homological techniques to solve problems on the vanishing of homology and cohomology over commutative rings. For instance, in a joint work with Sather-Wagstaff, using the DG methods and geometric aspects of representation theory, they provided a complete solution to a long-standing conjecture posed by Vasconcelos in 1974. In other recent (in-progress) joint works with Avramov, Iyengar, and Sather-Wagstaff again using the DG techniques, they introduce several classes of commutative noetherian local rings that satisfy another major conjecture from 1975, known as the Auslander-Reiten Conjecture (which originates from representation theory of Artin algebras). He is currently working (in separate projects with other collaborators) on developing DG methods in order to completely solve this conjecture. His past and recent results on the vanishing of (co)homology, ring homomorphisms, and homological dimensions are contained in 16 research papers to date, most of which have been published in top journals.

Faculty host: Overtoun Jenda