A major task of mathematics today is to harmonize the continuous and the
discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from
both.
E.T. Bell, ``Men of
Mathematics'' 1937
In Analysis Situs, Poincare started with a simplicial complex as his
basic discrete model. This has had broad appeal since a
finite set of points determines the simplexes and the boundary operator has a natural,
discrete definition. In over a
hundred years, this approach has not yielded a full discrete theory and convergence to the
smooth continuum has
remained elusive.
In this lecture, entirely new definitions of discrete chains are introduced in a Riemannian
manifold as idealized,
geometric analogues to differential forms. Reminiscent of the shift from simplicial to singular
homology, an intrinsic
theory of discrete calculus arises which converges to the smooth continuum.
The theory lays new
foundations for much of standard analysis in a way that captures
infinite limiting processes--- the heart of calculus--- in terms of
finite computations. Certain constructions of topology are restored to geometry at the level of
chains and cochains,
without passing to homology and cohomology. Examples include Poincare duality, intersection
of chains, and linking
number.
Smooth manifolds, metrics, fractals, vector fields, curvature, differential forms, foliations
and measures
can be discretized. Operators on forms have geometric counterparts
for chains including Lie derivative, Hodge star, codifferential and Laplace, giving them physical
meaning alongside the venerable boundary operator. Newly defined geometric products have
natural definitions such as
interior, exterior, wedge, cap, cup, slant, and convolution, with interesting commutator relations.
But what are these fluxions? The velocities of evanescent increments? And what are these same
evanescent
increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing.
May we not call them the
ghosts of departed quantities?
Bishop Berkeley, ``The Analyst'' 1734
Charles Pugh
2003-11-12