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Colloquia Patavina: Leavitt path algebras. Something for everyone: algebra, analysis, graph theory, number theory

Martedì 11 Giugno 2013, ore 16:00 - Aula 1A150 - Gene Abrams

ARGOMENTI: Seminari

Martedì 11 Giugno 2013 alle ore 16:00 in aula 1A150 della Torre Archimede, Gene Abrams (University of Colorado) terrà una conferenza della serie Colloquia Patavina.

La Commissione Colloquia
C. Bonotto, M.A. Garuti, M. Pavon, F. Rossi

Titolo
Leavitt path algebras
Something for everyone:
algebra, analysis, graph theory, number theory

Conferenziere
Gene Abrams (University of Colorado)

Riassunto
The rings studied by students in most first-year algebra courses turn out to have what's knownas the "Invariant Basis Number" property: for every pair of positive integers m and n, if the free left R-modules RRm and RRn are isomorphic, then m = n. For instance, the IBN property in the context of elds boils down to 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 seminal work completed the early 1960's, Bill Leavitt produced a speci c, universal collection of algebras which fail to have IBN. While it's fair to say that these algebras were initially viewed as mere pathologies, it's just as fair to say that these now-so-called 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 eld K one can build the Leavitt path algebra LK(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 fi rst de fined in 2004; as of 2013 the subject has matured well into adolescence, currently enjoying a seemingly constant opening of new lines of investigation, and the signi cant advancement of existing lines. I'll give an overview of some of the work on Leavitt path algebras which has occurred in their first 8-plus 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 recent 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.

Breve biografia
Gene Abrams is Professor of Mathematics at the University of Colorado at Colorado springs. He heard his Ph.D. in Mathematics at the University of Oregon in 1981, under the supervision of Frank Anderson. He has been at UCCS since 1983. His research interests are in noncommutative rings and their categories of modules. Since 2005 his primary focus has been on a class of rings called "Leavitt path algebras". Abrams is the author / coauthor of more than 40 research articles in refereed journals. He has given talks in numerous mathematics departments throughout the world. He was designated in 1996 as University of Colorado systemwide President's Teaching Scholar; was recipient of the 2002 Rocky Mountain Section of the Mathematical Association of America Award for Distinguished University Teaching; and was recipient (with J. Sklar) of 2010 Allendoerfer Award (for expository writing) from the MAA.

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