School Seminars and Colloquia

Scl, sails, and surgery

Algebra/Geometry/Topology Seminar

by Danny Calegari


Institution: Caltech
Date: Mon 10th August 2009
Time: 2:15 PM
Location: Cuming Theatre, Chemistry

Abstract: This is the third lecture in a series on scl (stable commutator length).

scl answers the question: "what is the simplest surface in a given space with prescribed boundary?" where "simplest" is interpreted in topological terms. This topological definition is complemented by several equivalent definitions - in group theory, as a measure of non-commutativity of a group; and in linear programming, as the solution of a certain linear optimization problem. On the topological side, scl is concerned with questions such as computing the genus of a knot, or finding the simplest 4-manifold that bounds a given 3-manifold. On the linear programming side, scl is measured in terms of certain functions called quasimorphisms, which arise from hyperbolic geometry (negative curvature) and symplectic geometry (causal structures). In these talks we will discuss how scl in free and surface groups is connected to such diverse phenomena as the existence of closed surface subgroups in graphs of groups, rigidity and discreteness of symplectic representations, bounding immersed curves on a surface by immersed subsurfaces, and the theory of multi-dimensional continued fractions and Klein polyhedra.

For More Information: craigw@ms.unimelb.edu.au