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 14 E. Packer Avenue (610) 758-3759 Bethlehem, PA 18015-3174 "Business!" cried the Ghost, wringing its hands again. "Mankind was my business. The common welfare was my business; charity, mercy, forbearance, and benevolence, were, all, my business. The dealings of my trade were but a drop of water in the comprehensive ocean of my business!" --Dickens, "A Christmas Carol"