The advantage of the continuum framework is the fine symmetry and algebraic structure of differential forms. We will recall this structure and discuss its geometrical meaning. This fine structure underlies many achievements in mathematics. One beauty is the spreading out of the integer, named the Euler characteristic, over a closed manifold into an integrand defined locally by curvature ideas whose total integral is that integer. This has a finite analogue which is easy and revealing in dimension two which is the "linear in curvature" dimension. It is possible though quite difficult in the higher "nonlinear in curvature" dimensions.This spreading out of an integer generalizes to the integer or index related to the finite dimensional kernel and cokernel of certain operators defined by calculus on smooth manifolds. This is the powerful index theorem of Atiyah-Singer. The proof involves infinite dimensional analysis which is linear and manageable. In the index discussion the relevant manifolds are closed and even dimensional.
There is a generalization of the index theory to even dimensional manifolds with boundary but now the closed odd dimensional boundary must have a metric structure to explain the spreading out process. There is a new boundary term in the index theorem. The proof of this theorem uses more linear functional analysis and much of the fine algebraic structure of differential forms. In one instance for four manifolds with three dimensional closed boundary Z there is a topologically meaningful symmetric bilinear form associated to Z which is defined and non-degenerate on the space V' of exact two forms., dn and dn'. By definition this "linking form" is the integral over Z of n^dn' or the integral of n'^dn. A metric on Z allows one to construct from this bilinear form a self adjoint operator "curl" on the isomorphic space V of volume preserving vector fields. The extra term in the APS index theorem of Atiyah-Patodi-Singer is constructed from the eigenvalues of "curl" using an infinite series analogous to the classical zeta function plus analytic continuation. This is a jewel-like continuum achievement of calculus on manifolds. This is some of the good news. Some of the not-so-good news is that certain rich and important phenomena with associated nonlinear continuum math questions have resisted understanding because the required infinite dimensional nonlinear analysis is missing.
For example consider finding a path v(t) in the space V starting at some volume preserving vector field on Z that satisfies d/dt v(t)) = - (v x curl v)', where the prime means project back onto V. More generally, this equation is part of a one parameter family of equations in which to the right hand side one adds a coefficient [called viscosity] times curl curl v(t). If we assume Z is periodic three space, this family of equations is actually characterized as being the only possible equations describing continuum limits of Newtonian motions of particle systems that satisfy Galilean invariance and that the limiting particles fill space uniformly. When the viscosity coefficient is zero this equation was formulated by Euler 250 years ago and studied by Cauchy and is the Mathematician's continuum model of the motion in three space of an ideal incompressible fluid with zero friction between ideal particles. Certain Cauchy invariants underly the proof in the late 60's that this equation has solutions for some small time interval for any initial volume preserving vector field with Holder continuous first derivatives in space. Moreover recently it was even shown in this case that each particle trajectory is real analytic in time during this short time. However it is not known to this day whether smooth solutions exist for all time. The real phenomenon of turbulent motion which is very much not understood makes this a highly pursued matter. When the viscosity coefficient is non zero the added term , called the diffusion term, is known to make the problem strictly easier, yet only the same theorems are known and the question of all time existence is still open and is one of the mathematical Clay Millennium problems. It must be said, that physicists have reasonable arguments that solutions exist for all time: at the scales where observations of gases and fluids are made it is clear on physical grounds that no singularities develop. One wonders what might be the mathematical formulation that can incorporate this intuition.
Also these equations are used all over the world by engineers and other scientists to model the phenomena they study [of course one must include in such a broad claim compressible situations which have shock waves when particles come together]. This practical usage of these equations is effected by models. It is often quite successful in a wide range of situations from the circulation of blood in the heart to storms in the atmosphere of Jupiter. The models are often made by approximating the equations above by difference quotients and an adhoc product and then computing electronically. If this looks good and is consistent with data it is then used. Otherwise the model is discarded if it doesn't look right and one starts again. Since there is a physical scale in the real use of these equations and since one can observe many patterns in moving fluids like water at our finite scales one may hope that there are models at a fixed finite scale that successfully predict the behaviour relative to that approximate scale.
The plan of the lectures is too start from scratch by dividing space into many tiny cells and by examining closely the objects and operations at this finite level that parallel the corresponding elements of the continuum world. One concrete idea that emerges is that different equivalent ways of writing the continuum equations lead to difference algorithms with different properties at the finite level. We will study the question of which models are appropriate for different purposes by seeing how the fine structures alluded to above are preserved or break down at the finite level. One may think of all of the ways of writing the continuum equation as the vertices of a graph. A simple rewriting into a new form describes an edge of the graph which is labeled by the algebraic or geometric property of the continuum used in the rewriting. Also some of the writings or vertices are the physical derivations of the equations. These are the preferred models. We will describe some of these.
The simple idea is that starting from the equation defined directly from physics one may move around the edges of the graph freely in designing the model at the combinatorial finite level as long as those edge labels are also valid at the combinatorial level or at least essentially valid.
We will use the simple looking equation, divergence v(t) = 0 and d/dt[v[t]] = (v[t] x curl v[t] )' , rewritten in terms of differential forms and at the finite level in terms of cochains, to track our discussion. Since the vector product is used to write the equation above we note that under the natural isomorphism between vector fields v,v' and one forms n, n' one has v x v' corresponds to star[n ^ n'] and curl becomes star[d] on these one forms. Star is defined below. Thus the Euler equation in terms of differential one forms becomes d star[n] = 0 and d/dt n = star [n ^ star[dn]].'
Here is a beginning sample of statements:
At the continuum level: 1] many of the rewritings of the equations use the fact that the nonlinear structure coming from the wedge product of forms is commutative in the graded sense or they use associativity. Furthermore one uses exterior d is a derivation of the algebra of forms.
2] the wedge product to the top degree followed by integration is a nondegenerate pairing on the infinite dimensional spaces of forms which when written as an operator in terms of the metric on three space becomes the star operator. This star operator is an involution and encodes Poincare Duality.
At the finite level: 1]' the corresponding cup product of the objects at finite level [the cochains] can be chosen to be graded commutative or the cup product can be chosen to be associative but it cannot be chosen to be both graded commutative and associative. The corresponding coboundary operator continues to be a derivation in all cases [ this is a fundamental requirement for an admissable cup product in topology].
2]' there are recently discovered commutative cup products which define a non-degenerate pairing but one is still searching for an associative cup product that defines a non generate pairing. Poincare Duality is still present in all cases.
Finally, relative to going back from the finite to the continuum, we will discuss potential nonlinear analogues [from a recent CUNY phd thesis] of axioms of Lax-Richtmeyer that characterize in the densely defined linear equation setting when algorithms consistent with a well posed linear equation actually approximate solutions of the equation.