Nir Gadish, University of Pennsylvania
Event Date
2026-02-18
Event Time
03:30 pm ~ 04:30 pm
Event Location
Wachman Hall, room 407
Abstract: How can we tell if a group element can be written as a $k$-fold nested commutator? One approach is to find computable invariants of words in groups, that vanish on all $(k-1)$-fold commutators but not on $k$-fold ones. We introduce the theory of letter-braiding invariants - these are "polynomial" functions on words, inspired by the homotopy theory of loop-spaces and Koszul duality, and carrying deep geometric content. They extend the influential Magnus expansion of free groups, which already had countless applications in low dimensional topology, into a functorial invariant defined on arbitrary groups. As a consequence we get new combinatorial formulas for braid and link invariants, and a way to linearize automorphisms of general groups which specializes to the Johnson homomorphism of mapping class groups.