Speaker: Gene Abrams (University of Colorado)
Time: 9h00,Wednesday, May 16, 2018
Location: Room semina 6 Flor, Building A6, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi
Abstract: The rings studied by students in most first-year algebra courses have the "Invariant Basis Number" property: for every pair of positive integers $m$ and $n$, if the free left $R$-modules $_RR^{m}$ and $_RR^{n}$ are isomorphic, then $m=n$. For instance, the IBN property in the context of fields is simply the statement that any two bases of a vector space must have the same cardinality. Similarly, the IBN property for the ring of integers is a consequence of the Fundamental Theorem for Finitely Generated Abelian Groups.
In important work completed the early 1960's, William G. Leavitt produced a specific, universal collection of algebras which fail to have IBN. While these algebras were initially viewed as interesting but not so "main stream", these now-so-called {it Leavitt algebras} currently play a central, fundamental role in numerous lines of research in both algebra and analysis.
More generally, from any directed graph $E$ and any field $K$ one can build the {it Leavitt path algebra} $L_K(E)$. In particular, the Leavitt algebras arise in this more general context as the algebras corresponding to the graphs consisting of a single vertex. The Leavitt path algebras were first defined in 2004; as of 2018 the subject is currently experiencing a seemingly constant opening of new lines of investigation, and the significant advancement of existing lines. I'll give an overview of some of the work on Leavitt path algebras which has occurred in their fourteen years of existence, as well as mention some of the future directions and open questions in the subject.
There should be something for everyone in this presentation, including and especially algebraists, analysts, and graph theorists. We'll also present an elementary number theoretic observation which provides the foundation for one of the main results in Leavitt path algebras, a result which has had a number of important applications, including one in the theory of simple groups. The talk will be aimed at a general audience; for most of the presentation, a basic course in rings and modules will provide more-than-adequate background. |
