Infinite regions in geometry

Infinite-cluster geometry in central-force networks

We show that the infinite percolating cluster (with density P∞) of central-force networks is composed of: a fractal stress-bearing backbone (PB) and; rigid but unstressed “dangling ends” which occupy a finite volume-fraction of the lattice (PD). Near the rigidity threshold p∗, there is then a first-order transition in P∞ = PD +PB, while PB , is second-order with exponent β ′ A new mean field th...

Infinite Products in Number Theory and Geometry

We give an introduction to the theory of Borcherds products and to some number theoretic and geometric applications. In particular, we discuss how the theory can be used to study the geometry of Hilbert modular surfaces.

Poisson geometry of the infinite Toda lattice

Many important conservative systems have a non canonical Hamiltonian formulation in terms of Lie-Poisson brackets. For integrable systems, this is usually the first of two or more compatible brackets. With few notable exceptions, such as the Euler, Poisson-Vlasov, KdV, or sine-Gordon equations, for example, for infinite dimensional systems this Lie-Poisson bracket formulation is mostly formal. ...

The Geometry of Infinite-Dimensional Groups

Topological Geometrodynamics: Physics as Infinite-dimensional Geometry