We give new algorithms for computing basis cochains for real-valued homology, cohomology, and harmonic cochains on manifold simplicial complexes. We discuss only planar, surface, and solid meshes. Our algorithms are based on a least squares formulation. Previous methods for computing homology and cohomology have relied on persistence algorithm or Smith normal form, both of which have cubic comp...

We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations using finite element or discrete exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on finding harmonic cochains cohomologous to a given cocycle. Amongst other things, this allows for localization near topol...

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz’ external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinit...

1. Classification problems in differential topology: 1/18/18 1 2. Review of the algebraic topology of manifolds: 1/23/18 4 3. Unimodular forms: 1/25/18 8 4. The intersection form and characteristic classes: 1/30/18 12 5. Tangent bundles of 4-manifolds: 2/1/18 16 6. Rokhlin’s theorem and the homotopy theorem: 2/6/18 19 7. The homotopy theorem: 2/8/18 22 8. The Hodge star: 2/13/18 22 9. Hodge the...

1. deRham Cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The Gauss-Manin Connection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Frobenius and the Hodge Filtration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4. Applications to Elliptic Curves. . . . . . . . . . . . . . . . . . . . . . . . . . ....

Geometric discretisation draws analogies between discrete objects and operations on a complex with continuum ones on a manifold. We generalise the theory to the cubic case and incorporate metric, by adding volume factors to our discrete Hodge star and then by modifying our inner product which leads to the same result.

This paper describes the recently developed mixed mimetic spectral element method for the Stokes problem in the vorticity-velocity-pressure formulation. This compatible discretization method relies on the construction of a conforming discrete Hodge decomposition, that is based on a bounded projection operator that commutes with the exterior derivative. The projection operator is the composition...

Many linear boundary value problems arising in computational physics can be formulated in the calculus of differential forms. Discrete differential forms provide a natural and canonical approach to their discretization. However, much freedom remains concerning the choice of discrete Hodge operators, that is, discrete analogues of constitutive laws. A generic discrete Hodge operator is introduce...