Stable commutator length

University Of Chicago, Chicago IL

Investigators

Abstract

In previous work, the PI developed some powerful and delicate tools for making exact calculations in 2-dimensional bounded cohomology, especially in free groups, via the stable commutator length (a relative 2-dimensional Gromov norm). Experiments and some theoretical work reveals structure in the scl spectrum of a free group analogous to the volume spectrum in hyperbolic 3-manifold topology, together with a partial explanation in terms of surgery, and connections to the theory of higher-dimensional continued fractions, enumerative combinatorics, phase locking for coupled nonlinear oscillators, and the thermodynamics of DNA. Other theoretical work reveals an abstract rigidity implicit in the polyhedral structure of the scl norm, which manifests itself in rigidity for symplectic representations of surface groups, and dynamical rigidity of Ruelle invariants and their generalizations for certain groups of symplectomorphisms. The PI will atempt to unify and explain these phenomena, and to give them a rigorous justification. For more than two thousand years, geometry has been defined and studied in 1-dimensional terms: to measure the distance between two points, find the shortest line (i.e. the smallest 1-dimensional object) that goes from one to the other. Since the 16th century, mathematicians have understood that the 1-dimensional "real" numbers are just a shadow of the 2-dimensional "complex" numbers. Recently, geometry has been "complexified" in a similar way, and mathematicians explore different aspects of the geometry of a space by probing it with 2-dimensional surfaces. Such a surface could be a soap bubble with boundary on a wire frame, or an interface between turbulent and non-turbulent flow, or an imaginary surface interpolating between strands of DNA. The PI will study the surfaces that solve "extremal" geometric problems (in many senses of the word) using tools from - and in order to solve problems that originate in - algebra, group theory, topology, and functional analysis.