In this talk, we will look at the analogues question to the classical Waring problem over objects with noncommutative structures, such as groups, Lie algebras, matrix algebras, etc. We review some work done for finite groups in recent times. For example, Shalev showed that for every finite (nonabelian) simple group of sufficiently high order every element can be expressed as values of word w of length 3. This was later improved to 2 by Larsen, Shalev and Tiep.

Larsen conjectured that a similar result should hold for matrices over finite fields. We have, for all integers k geq 1, there exists a constant C_k depending only on k, such that for all q > C_k and for all k \geq 1 every matrix in M_n(F_q) is a sum of two k-th powers. This work is done in collaboration with Krishna Kishore.