Calculus without limits ##### What is this blog about?

Quantum calculus deals with alternative flavors of calculus. It especially studies calculus on discrete or even finite sets. Examples of non-traditional calculus flavors are non-standard analysis, difference calculus or calculus on graphs or simplicial complexes. All these flavors of calculus can be seen as extensions of traditional calculus or calculus on Riemannian manifolds. The usual calculus is a special or limiting case.

##### Extending language.

One approach is to look for structures which are close to known notions of calculus. An ideal generalization of traditional calculus to the discrete does not change language. It changes the meaning of the operations, possibly after extending language. This has been realized already in the form of internal set theory IST of Ed Nelson. But this flavor of calculus is not easy to manage and teach. While intuitive, it is also more risky: it is easier to make mistakes. One can imagine historical circumstances, in which quantum calculus could been first developed and the calculus we know later been derived and have the status of generalized function theory or geometric measure theory.

##### Predicting the future

In the same way that we could in principle work with rational numbers alone, one could look exclusively at discrete geometries and look at continuum geometries as limiting cases. Computer game implementations are quite realistic but most matrix computations are kept with integers as it is faster than floating number arithmetic. Any computation on a device is by nature finite as there is only a finite amount of memory available. In some sense, already numerical analysis is already a quantized calculus. However, the theory as told in numerical analysis books is much less elegant than the actual theory (which is in the nature of things as engineering is often not elegant if it wants to be fast and efficient).

##### A simple idea

A rather general idea is that calculus links“rates of changes” with “summation” to “ predict the future”. This idea is everywhere: look at the sequence 4, 15, 40, 85, 156, 259, 400, 585 for example. How does it continue? In order to figure it out, we take derivatives 11, 25, 45, 71, 103, 141, 185, then again derivatives 14, 20, 26, 32, 38, 44 and again derivatives 6,6,6,6,6,6. We see now a pattern and can integrate the three times starting with 6,6,6,6,6,6,6 always adjusting the constant. This gives us the next term 820. We can predict the future by analyzing the past.

##### Abstract generalizations

Any geometric theory with a notion of exterior derivative on “forms” and integration of “forms” is a flavor of calculus. A nice example is Riemannian geometry, which generalizes calculus in flat space. The frame work allows then to define notions like curvature or geodesics which are so central in modern theories of gravity. These notions can be carried over to discrete spaces. But it is not only physics which motivates to look at calculus. A big “customer” of calculus ideas is computer science. Discrete versions of notions like gradient, curvature, surface etc allow a computer to “see” or to build new objects, never seen. All movies using some kind of CGI make heavy use of calculus ideas.

##### Why do we study calculus?

The power, richness and applicability of calculus are all reasons why we teach it. Calculus is a wonderful and classical construct, rich of historical connections and related with many other fields. Here is a 30 second spot. We hardly have to mention all the applications of calculus (the last 2 minutes of this 15 minute review for single variable calculus give a few). We have barely scratched the surface of what is possible when extending calculus and how it can be applied in the future.

The traditional exterior derivatives (like div,curl grad) or curvature notions based on differential forms define a traditional calculus. Integration and derivatives lead to theorems like Stokes, Gauss-Bonnet, PoincarÃ©-Hopf or Brouwer-Lefschetz. In classical calculus, the basic building blocks of space are simplices. Differential forms are functions of these simplices. In the continuum, one can not see these infinitesimal simplices. To remedy this, sheaf theoretical constructs like tensor calculus have been developed, notably by Cartan. In the discrete, when looking at graphs, the structure is simple and transparent. The theorems become easy. Here, here and here are some write ups.

##### Calculus on graphs

Calculus on graphs is probably the simplest quantum calculus with no limits: everything is finite and combinatorial. I prefer to work on graphs but one can work also with finite abstract simplicial complexes. It turns out that the category of complexes and especially graphs is quite powerful, despite its simplicity. Graphs are quite an adequate language because the Barycentric refinement of an abstract finite simplicial complex is always the Whitney complex of a finite simple graph. Most topological considerations therefore can be done on graphs. Having “trivialized calculus”, one can look at more complex constructs.

##### Interaction calculus

In interaction calculus, functions on pairs (or more generally k-tuples) of interacting (=intersecting) simplices in the graph are at the center of attention. Exterior derivatives again lead to cohomology. While in classical calculus, the derivative is df(x)=f(dx) (which is Stokes theorem as dx is the boundary chain), in second level interaction calculus, the derivative is df(x,y)=f(dx,y)+(-1)dim(x) f(x,dy). The chain complex is bigger. The cohomology groups more interesting.

##### Interaction cohomology

The analogue of Euler characteristic is the more general Wu characteristic. There are generalized versions of the just mentioned theorems. At the moment (2016), only a glimpse of the power of interaction calculus is visible. There are indications that it could be powerful: it allows to distinguish topological spaces, which traditional calculus does not: here is a case study. And here is a preprint on Interaction cohomology.

##### Open mind

Its not very helpful to look only for analogies to the continuum. Very general principles (numerical and computer science demonstrate constantly how finite machines mode things) show that the continuum can be emulated very well in the discrete. We also have to look out for new things. A surprise for example is that there is a Laplacian for discrete geometries which is always invertible: discovered in February 2016 and proven in the fall 2016, it leads to invariants and potential theory different from the usual Hodge Laplacian. Unlike the Hodge Laplacian which is the square of the Dirac Laplacian, this new Laplacian has a quantized and completely finite potential theory, no super symmetry. The total energy of the geometry is the Euler characteristic.

##### Project page

For more, see the project page on my personal web page. The entries here are more like drafts, sometimes expository, sometimes research logs which I leave as it is for me also interesting to see later, where and how I was stuck. So, it can happen that initially in an entry, the story has not yet been clear but at the end been understood.