Lattice models of calculus and hydrodynamics
As is well-known, discretisations often behave rather differently from the continuous models which they are meant to approximate, at the very least they may contain a much more delicate structure. We will discuss two manifestations of this. While trying to construct discrete analogues of the continuum picture of the differential graded algebra of differential forms on a manifold, we find that there is not just one, but rather there are three pictures, depending upon which one of the three properties of commutativity, associativity and Leibniz is dropped. Each is a world on its own in which different tools can be applied.
We will also discuss connections with Sullivan's proposed model of lattice hydrodynamics which is `naturally' derived from physical principles in a discrete setting, rather than as an artificial discretisation of a continuous equation.
This is joint work with Dennis Sullivan and Nissim Ranade.