Research of Axel Boldt

Representation theory

I have worked on the representation theory of finite dimensional algebras. Algebras are vectorspaces in which you can multiply elements; the simplest example of an algebra is given by the set of all n-by-n matrices over a field (e.g. over the real numbers). A representation of an algebra is a homomorphism (a map that respects addition, scalar multiplication and algebra multiplication) of that algebra to a matrix algebra. The idea is to understand and classify algebras by means of their representations.

In my Diplomarbeit, written 1992 under Prof. Helmut Lenzing at the University of Paderborn, I developed a reduction formula for the characteristic polynomial of the Coxeter transformation of a path algebra and translated it into a MAPLE program. The detailed description and explanation as well as the estimation of the complexity of the program is contained in the Diplomarbeit.

Later I extended the results of the Diplomarbeit to quivers with relations and wrote an article titled "Methods to Determine Coxeter Polynomials" which has been published in the Journal for Linear Algebra and its Applications 230 (1995), p.151-164.

In August 96, I finished writing a paper with Martha Takane from UNAM, Mexico City, about Coxeterpolynomials of unicyclic quivers. It has since been published under the title "The spectral classes of unicyclic graphs" in the Journal for Pure and Applied Algebra 133 (1998), no. 1-2, p.39-49.

My Ph.D. thesis, written under the direction of Prof. Birge Huisgen Zimmermann at the Math Department of the University of California at Santa Barbara, was finished in May 1996. It deals with uniserial modules over finite dimensional algebras, especially their patterns in the Auslander-Reiten quiver. It also contains the work on Coxeter polynomials.

A joint article with Ahmad Mojiri from the University of Ottawa containing some results about Auslander-Reiten sequences involving uniserial modules was finished in February 2004. The preprint is available. It has since been published (with some modifications) as "On uniserial modules in the Auslander-Reiten quiver", Journal of Algebra, Volume 319, Issue 5, 1 March 2008, Pages 1825-1850.

Real analysis

I have also worked with Craig Calcaterra and Michael Green from Metropolitan State University, Saint Paul on some problems in analysis.

The first article deals with metric coordinate systems (a distinguished set of points of a metric space so that every other point can be uniquely identified by the set of distances to the given points) and the solvability of continuous dynamical systems specified in these coordinates. A preprint is available. The article was published as "Metric Coordinate Systems" in Communications in Mathematical Analysis, Volume 6, Number 2, Pages 79-108, February 2009.

With Craig Calcaterra, I generalized the well-known flow-box theorem from differential geometry to Lipschitz continuous vector fields on Banach spaces. A preprint is available; it has since been published as "Lipschitz Flow-box Theorem" in the Journal of Mathematical Analysis and Applications 338 (2008), p. 1108-1115.

Craig Calcaterra and I also prepared the preprint Approximating with Gaussians (2008), in which we show that linear combinations of translations of the Gaussian function exp(-x2) are dense in the space of square-integrable functions on the reals. This result entails that low-frequency trigonometric functions are also dense in a certain sense.


With Michael Janich, I wrote the preprint A Global Physician-Oriented Medical Information System (2008), a proposal for an internet-based system that would allow the world's physicians to obtain free diagnostic assistance and treatment recommendations, and would make the aggregate outcome data available to medical researchers. This proposal was submitted to Google's $10 million Project 10100. It was also submitted for publication to the journal Computers in Biology and Medicine but rejected.

I wrote the article Extending to Achieve Open Peer Review (2010), outlining a simple extension to the preprint archive. This proposed extension would allow us to establish an open peer review and open publishing process in the sciences. It was published in the Journal of Scholarly Publishing, Voluume 42, Number 2 (January 2011), pp. 238-242.

Last Change: 24-Feb-2011
Axel Boldt <>