Letter braiding - using algebraic topology to measure words in groups

Seminar
Speaker
Nir Gadish (University of Pennsylvania)
Date
18/06/2025 - 11:30 - 10:30Add to Calendar 2025-06-18 10:30:00 2025-06-18 11:30:00 Letter braiding - using algebraic topology to measure words in groups 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 functor 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. Third floor seminar room and Zoom אוניברסיטת בר-אילן - המחלקה למתמטיקה mathoffice@math.biu.ac.il Asia/Jerusalem public
Place
Third floor seminar room and Zoom
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 functor 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.

תאריך עדכון אחרון : 10/06/2025