What are the current research interests in linear algebra? Can anyone tell me specifically?
Browsing through the titles/abstracts of recent issues of the journal Linear Algebra and Its Applications will give you a good idea of recent advances in this field. http://www.journals.elsevier.com/linear-algebra-and-its-applications/
My interests in Linear and Multilinear Algebra are in rank and canonical forms (Normal forms) of tensors. An order-k tensor is an element of the tensor product of k vector spaces. Fix a basis and an order-1 tensor is a vector, an order-2 tensor is a matrix, and order-k, k>2 tensors are essentially hypermatrices. The current research is trying to generalize concepts such as rank, eigenvalues, transformations, etc. to higher-order tensors.
Some years ago (2007, to be precise), Guenter Ziegler (a famous German mathematician) claimed in an interview that presumably, there is nothing left to discover in some areas of mathematics, like the linear algebra of finite dimensional vector spaces. Many colleagues objected - in particular, Claus Michael Ringel from Bielefeld (see http://www.math.uni-bielefeld.de/~sek/ziegler.html, which unfortunately is in German). There is a lot left to do, and what we teach students under the name "linear algebra" is far from being all that is known - for example, the results by Gantmacher and Gelfand usually never make it into classrooms. One example (out of many possible ones) of current research he cited was "Invariant subspaces of nilpotent operators" (ArXiv math.RT/0608666). His talks and papers on his homepage are a wide source of examples of others. One thing that might be misleading that part of this research does not go under the name "linear algebra" anymore, it's part of the representation theory of algebras, continues in combinatorics, algebraic Lie theory etc - but the topic is the same.
There is a lot of new research that illustrates the interplay between linear algebra and analysis. In my dissertation, for example, we proved several results on an old problem in linear algebra and operator theory, are "almost commuting" matrices perturbations of commuting matrices? Once a space of matrices is equipped with a norm, then one can consider many interesting questions. In this direction, a good research paper to start with is R. Exel and Terry Loring's paper on "Almost Commuting Unitary Matrices." The paper is only 3 pages long and is suitable for students who have studied linear algebra and real/complex analysis.
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