In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be estimated by the dimension of a certain space of harmonic functions. Applying this to a complete manifold whose Ricci curvature is almost non-negative, we showed...

I describe two approaches to modelling the universe, the one having its origin in topos theory and differential geometry, the other in set theory. The first is synthetic differential geometry. Traditionally, there have been two methods of deriving the theorems of geometry: the analytic and the synthetic. While the analytical method is based on the introduction of numerical coordinates, and so o...

We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, Müller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application to the existence of quasiconformal mappings between manifolds, as well as an e...

Large chemical and biological systems such as fuel cells, ion channels, molecular motors, and viruses are of great importance to the scientific community and public health. Typically, these complex systems in conjunction with their aquatic environment pose a fabulous challenge to theoretical description, simulation, and prediction. In this work, we propose a differential geometry based multisca...

A new study by Terasaki et al. highlights the role of physical forces in biological form by showing that connections between stacked endoplasmic reticulum cisternae have a shape well known in classical differential geometry, the helicoid, and that this shape is a predictable consequence of membrane physics.

We introduce the C++ library Wedge, based on GiNaC, for symbolic computations in differential geometry. We show how Wedge makes it possible to use the language C++ to perform such computations, and illustrate some advantages of this approach with explicit examples. In particular, we describe a short program to determine whether a given linear exterior differential system is involutive.

String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2–dim σ–models corresponding to coset conformal field theories. Contribution to the proceedings of the Conference on Gauge Theories, Applied Supersymme...

We show that symmetries and gauge symmetries of a large class of 2-dimensional σ-models are described by a new type of a current algebra. The currents are labeled by pairs of a vector field and a 1-form on the target space of the σ-model. We compute the current-current commutator and analyse the anomaly cancellation condition, which can be interpreted geometrically in terms of Dirac structures,...