Nir Gadish (University of Michigan) will be speaking in the algebra seminar in two weeks. We will go out for lunch after the talk. When: June 7 May, 12-1pm Where: SMRI seminar room (Macleay Building (A12) room 301) Title: Letter-braiding invariants of words in groups. Abstract: How can we tell if a group element can be written as k-fold nested commutator? I suggest seeking computable invariants of words in groups that detect k-fold commutators. We introduce the novel theory of letter-braiding invariants - these are elementarily defined functions on words, inspired by the homotopy theory of loop-spaces and carrying deep geometric content. They give a universal finite-type invariant for arbitrary groups, extending the influential Magnus expansion of free groups that already had countless applications in low dimensional topology. As a consequence we get new combinatorial formulas for braid and link invariants, and a way to linearize automorphisms of general groups that specializes to the Johnson homomorphism of mapping class groups.