Abstract: Hilbert's solution to Waring problem in number theory shows that every positive integer is a sum of  g(k) k^{th} powers. In recent years various non-commutative analogues of this result were studied, where the aim is to present every element of a (nonabelian) finite simple group as a short product of special elements, such as powers, commutators, or values of a given (non-trivial) word  w. This is motivated by problems in profinite groups, by Ore's conjecture, etc. I will describe recent results from [LiSh], [Sh],  [LaSh1], [LaSh2], [LiOShT], as well as brand new results from [LaShT]. Connections with representation theory, probability and geometry will also be described.

References:
[LiSh]  M.W. Liebeck and A. Shalev, Diameters of finite simple groups: sharp bounds and applications, Annals of Math. 154 (2001), 383-406.
[Sh] A. Shalev, Word maps, conjugacy classes, and a non-commutative Waring-type theorem,  Annals of Math. 170 (2009), 1383-1416.
[LaSh1] M. Larsen and A. Shalev, Word maps and Waring type problems,  J. Amer. Math. Soc 22 (2009), 437-466.
[LaSh2] M. Larsen and A. Shalev, Characters of symmetric groups:  sharp bounds and applications, Inventiones Mathematicae  174 (2008), 645-687.
[LiOShT] M.W. Liebeck, E.A. O'Brien, A. Shalev and P. Tiep, The Ore Conjecture, to appear in  J. Euro. Math. Soc
[LaShT] M. Larsen, A. Shalev and P. Tiep, Waring problem for finite simple groups, in preparation.

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