From Counting Intersections of Curves on Surfaces to String Topology

In the eighties, Goldman discovered a Lie algebra structure on the vector space generated by free homotopy classes of closed curves on a surface. In the nineties, Turaev discovered that, with a minor modification, there is also a Lie cobracket on this vector space and that these two operations form a Lie bialgebra.

The Goldman-Turaev Lie bialgebra is defined by combining two familiar structures on curves: transversal intersection and the loop product. A natural question that arises is whether one can recover the minimal intersection number of curves from the Goldman-Turaev Lie bialgebra.

Our many efforts to answer this question will be the focus of this talk. These include the use of combinatorial tools, computer algorithms, hyperbolic geometry, and the accidental discovery of String Topology.

(Some of these results are joint work with Arpan Kabiraj, Fabiana Krongold, Siddartha Gadgil, and Dennis Sullivan.)

Bio
https://www.math.stonybrook.edu/~moira/