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Geometry and Topology in the Matrix Regularization of Membrane Theory
Just as String theory is based on the principle that a moving string
should sweep out a minimal area in space-time, Membrane theory demands
that a membrane (i.e. a surface) should sweep out a minimal volume.
In String theory, one has by now access to a large amount of
information about the classical and quantum system. In comparison,
almost nothing is known for membranes; even the classical equations of
motion are not easy to handle. So far, the only path to the
corresponding quantum system has gone through a "matrix
regularization". More specifically, functions are replaced by
sequences of matrices (of increasing dimension) in such a way that the
physically important Poisson bracket corresponds to commutators of
matrices. It turns out that this procedure is not only directly
relevant for physics, but is also of separate interest from a purely
mathematical point of view.
In this talk I will give a short introduction to this way of
regularizing by matrices, and then present some recent results on how
geometry can be encoded in matrix sequences.