Lie and Poisson Brackets on Loops, their generalizations and applications

by Prof. Vladimir Chernov (Dartmouth College Mathematics)

Thursday 22 Dec 2022, 17:00 → 19:00 Europe/Kiev

https://bbb.imath.kiev.ua/b/ser-rk7-dqd-hon (ONLINE)

*Abstract:* Goldman and Turaev introduced a bialgebra structure on the space of loops on a surface. it was later generalized to the String Algebra by Chas and Sullivan.
The results of Goldman and Chas show that they can be frequently used to compute the minimal number of intersection and self intersection points of loops on the surface. 
We show that similar Andersen-Mattes-Reshetikhin Poisson Algebra always solves this problem and generalize the algebra to the Graded Poisson Algebra on the space of garlands glued out of arbitrary manifolds. This algebra can be used to define linking numbers for nonzero homologous linked submanifolds. These linking numbers can be used to study causality in spacetimes.
The talk is based on the works of the author and joint works with Cahn and with Rudyak.