Vector fields defined on closed sets in the plane have recently found several new applications in various branches of mathematics and other natural sciences. The definition of the norm of these vector fields depends on which application the fields are used for. One norm comes from looking at vector fields as tangent vectors to the Teichmuller space of the given closed set. This norm, called the Teichmuller norm, is suitable for applications in holomorphic motions and complex dynamics. It turns the space of vector fields into a space of all initial velocities of holomorphic motions of the closed set. Since the holomorphic motions of the fixed closed set are defined by an intrinsic condition on the closed set, it is natural that there is a criterion that guarantees that the vector field has bounded norm, and which is expressed purely in terms of the values of the vector field on the closed set. Such a criterion was found recently by considering the velocity of the cross ratio for every possible quadruple in the given closed set. The velocity is measured with respect to the Poincare density of the Riemann sphere punctured at three points. This leads to an alternative norm on vector fields, called the cross ratio norm. The cross ratio norm is equivalent to the Teichmuller norm, and the equivalence constant is universal and independent of the closed set or vector field. Hence, a vector field is a derivative of some holomorphic motion if, and only if, it has bounded cross ratio norm. This equivalence led to the introduction of a new canonical density on any Riemann surface. The new density is equivalent to Poincare density, and this equivalence has several important applications of the vector field theory to hyperbolic and conformal geometry. It provides a link between, on the one hand, hyperbolic geometry, Riemann surfaces, and geometric function theory, and on the other hand, Teichmuller theory, quadratic differentials and h olomorphic motions with their applications to complex dynamics.
For many years the analytical and geometrical techniques developed in dynamical systems and Teichmuller theory have been applied to a number of important nonlinear problems with broad impact on ecology, economics, and genetics. One can expect these trends to continue. This project will feature the applications of the finite earthquake theorem to various directions. This, recently stated and proved, theorem provides a way to code each finite or countable string of data into the weighted tree associated to a finite lamination and vice versa. That could turn out to be an efficient way of coding. The applications will be based on recently made program in Maple that sketches the graph of finite laminations corresponding to a given homeomorphism and lists weights associated to each leaf of those laminations. The third component of the project is based on integrating research and education.
