The Lagrange Riddle
We look at the Lagrange problem in the discrete. Before doing wo, we briefly look again at the data structure for the finite topos of delta sets. It can be given by a finite set of sets, a Dirac operator (fancy word for a matrix encoding all the incidences helping to reconstruct the face maps of the delta set) and a dimension function (from that information it is possible to get the face maps defining the delta set). The Dirac operator is a nice structure because as Connes has shown, it produces a framework which is deformable in various ways, for example into the non-commutative world. The Lagrange problem is to find an intutive and easy to formulate analog to the continuum. It is a bit more tricky as we will see that if we have a complex valued function psi on a manifold M, then { psi = 0} and (oveline{psi}=0} are in general different. They can have even completely different topological type for random functions. We just used the language of complex valued functions. To reformulate this strange behavior in a multivariable setting. Looking at the {f=0, g=0} is not the same than looking at {f=0, -g=0}. When we formulated conditions for Morse-Sard to work with vector valued functions, we had to do a choice. This is one of the rare cases, where discrete calculus differs from continuum calculus. The fabric of finite manifolds contains more treasures than continuum manifolds.
The video footage was done in Cambridge. We flew with the DJI mini from Harvard to MIT and back and used the Avata to fly above MIT and from the Salt and Pepper bridge crossing the Charles.