The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this talk, we address the average-case time complexity of several algorithms in group theory and show that in many cases it is linear and in some cases even constant (with respect to the length of the input). Along the way, we improve several bounds for the worst-case complexity of the word problem in groups of matrices, in particular in nilpotent groups. Most of this talk is based on joint work with Alexander Olshanskii.
Vladimir Shpilrain is a Professor of Mathematics at the City College of New York and CUNY Graduate Center. He is the (co-)author of 120+ papers and 4 monographs. His main research interests are complexity of algorithms and information security.