Quantum Multivariable Calculus Flavors

This winter, I started to revive some seeds which have been placed in the winter of 2010. It is a calculus in a general compact Riemannian manifold in which the exterior derivative is bounded. It builds on the usual exterior derivative d defined on a manifold (M,g) and builds for a real number a new derivative which now is a bounded operator. For -forms f, it is the average of over the geodesic ball of radius h. I plan to talk about this a bit more in the coming weeks (if not something more interesting should displace it. It is not that I lose interest in a subject, I often switch to other topics if they appear to be more interesting). I spent a few days this winter break to think about that old story again and it looks good. It is an opportunity also to brag about special functions like hyperelliptic functions, about partial differential equations. In any case, it is very, very classical and very explicit too. Let me just elaborate a bit about the first story mentioned in the video below. So, I hope there will be more here later on calculus on wave fronts.

Ergodic Quantum Calculus

“Quantum deformation” can mean different things. One possible is to go from a commutative algebra to a non-commutative one; this is of course triggered by the Heisenberg anti-commutation relations [q,p] = i h. This can occur already in simple single variable calculus frame works as described here: instead of the C* algebra of continuous real functions, take the crossed product with the translation s(f)(x) = f(x+1). The derivative operator Df = [s,f] gives the discrete derivative Df(x) = f(x+1)-f(x). The position operator X f = x s* and the momentum operator P=i D now satisfy [X,P]=i. This happens in the cross product C* algebra generated by s. Obviously this is a non-commutative topological space. If one starts with the von-Neumann algebra the crossed product is again a von-Neumann algebra and if $\latex \mathbb{R}$ is replaced with the compact one has a nice type I von Neumann algebra. There if one can look at random operators like the random Jacobi operator . My thesis from 1993 had been about such themes. There are lots of interesting structures to explore like the spectral type, Lyapunov exponents, isospectral Toda deformations, or Barycentric refinements, writing where the Dirac D operator can again be considered to be a random operator but over a renormalized system which is the 2:1 integral extension of the original system. That works for any automorphism of a probability space. The renormalization is rather trivial as it is a contraction both on the dynamical system level as well as on the operator level. The limiting operators are Jacobi operators over the von-Neumann Kakutani system, which is the unique fixed point of the 2:1 integral extension. The von-Neumann-Kakutani system is also known as the adding machine. It is very, very natural, at is the addition of 1 on the dyadic group of integers. Unlike an irrational rotation on the circle, there is a unique smallest translation on the dyadic group. The Koopman operator of that system has as spectrum the dual group, the Pruefer group. Lots of fun here as different parts of mathematics come together like p-adic calculus, selfsimilar similarities and continuum symmetries in the form of isospectral deformations which generalizes Abel-Jacobi maps in algebraic geometry. I also investigated a bit the cohomological aspect of the story as one can look at Df as a derivative and ask which functions are derivatives. This is very difficult even in the simplest situations like when the transformation is an irrational rotation. Diophantine properties of the rotation angle and smoothness matter as this leads to small divisor problems. As I had as a graduate student also to teach a lot, I entertained myself with always asking what happens with multi-variable topics in an ergodic setting. Take two commuting ergodic transformations T,S which define a action. One has the gradient and the curl . What is surprising (given the one dimensional non-triviality) is that in 2 and higher dimensional analogs, the cohomology is trivial. Every closed vector field [P,Q] is a gradient field. This works as long as the action is free meaning that the only group element in for which the fixed point set has measure zero, is the zero element in the group. We don not want for example that . The result that the cohomology is trivial is reviewed in this paper [PDF]. I had tried once to submit this to a journal but there seems have been nobody who appreciated it. I still think it is cute. The result has been proven earlier in a thesis of dePauw in dimensions 2 and follows from a result of Feldman-Moore in higher dimensions. One has however first to establish the equivalence of different cohomologies. The cohomology described above is and ergodic de Rham cohomology. It is equivalent to a group cohomology or a cohomology of ergodic equivalence relations. In any case, I mention this because this theory is an example of a quantum space, a space with a bounded exterior derivative. There is no limit taken. The frame work belongs to stochastic processes with n-dimensional time.