From: "David L. Johnson"
Subject: Re: research on multilinear algebra?
Date: Mon, 20 Dec 1999 20:02:31 -0500
Newsgroups: sci.math.research
Tijl De Bie wrote:
>
> Does anyone have an idea about the future of multilinear algebra,
> I'm planning to start a doctoral research in this topic, and I would like to
> know whether there are already many research groups working on it, where are
> the best ones, what's the state of the art, what are possible application
This is, of course, a very mature field, so it tends to be very difficult to
make serious progress. There are two areas that I know about that are good
applications:
In differential geometry: There are many geometrically natural questions that
quickly boil down to multilinear algebra questions.
One particular one is a conjecture of Hopf that positive sectional curvature
implies positive Euler characteristic. This was worked on for years as a
problem in multilinear algebra, since the Gauss-Bonnet integrand which
determines the Euler characteristic is multilinear in the Riemann curvature
tensor, which is in turn quadratically related to the sectional curvature. In
the late '70's Robert Geroch provided a counter-example to that approach, but
there are still several interesting related questions.
In geometric measure theory: There is one easy method to determine when a
given submanifold M^n (or n-current) is area-minimizing, showing that it is
calibrated. In R^N, this often boils down to finding an n-form in R^N with
constant coefficients which satisfies two deceptively simple-seeming
conditions (comass-1, and achieving that bound on tangent planes to M). A
number of people have done interesting work in this area, particularly Dana
MacKenzie (sp?), Frank Morgan, and Gary Lawlor.
These applications are far from being purely algebra, but IMO it may be that
you would have to look to such areas for solvable, yet interesting, problems.
I hope this helps.
--
David L. Johnson david.johnson@lehigh.edu
Department of Mathematics http://www.lehigh.edu/~dlj0/dlj0.html
Lehigh University
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