Submission: 2010, Jun 8
Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions; it has since been investigated in several broader contexts, by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. The goal of this paper, which represents an expanded version of the author's ICM 2010 lecture, is to survey some of this research. I have tried to explain the underlying ideas informally through motivational remarks, examples and proof outlines (often in special cases, where the argument is more transparent), referring an interested reader to the literature for a more detailed treatment. The sections are arranged in rough chronological order, from the definition of essential dimension to open problems.
2010 Mathematics Subject Classification: 14L30, 20G10, 11E72
Keywords and Phrases: Essential dimension, linear algebraic group, Galois cohomology, cohomological invariant, quadratic form, central simple algebra, algebraic torus, canonical dimension
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