Here is the code to compute a basis elements of the cohomology groups of an arbitrary simplicial complex. It takes 6 lines in mathematica without any outside libraries. This is so simple, it compares in complexity with computations in basic planimetric computations in a triangle (implementing one of the Euclid constructions on a computer). We just compute the Dirac operator … ….
When replacing the circle group with the dyadic group of integers, the Riemann zeta function becomes an explicit entire function for which all roots are on the imaginary axes. This is the Dyadic Riemann Hypothesis.
A finite abstract simplicial complex has a natural connection Laplacian which is unimodular. The energy of the complex is the sum of the Green function entries. We see that the energy is also the number of positive eigenvalues minus the number of negative eigenvalues. One can therefore hear the Euler characteristic. Does the spectrum determine the complex?
One of the attempts to quantize space without losing too much symmetry is ergodic theory. Much of my thesis belongs to this program. It is a flavor of quantum calculus, as “no limits” are involved. The story is closely related to Jacob Feldman, one of my heroes of my graduate and postdoc time. I write this blog entry after having … ….
In the context of quantum calculus one is interested in discrete structures like graphs or finite abstract simplicial complexes studied primarily in combinatorics or combinatorial topology. Are they geometry? Are they calculus? What is geometry? In MathE320 I try to use the following definition: Geometry is the science of shape, size and symmetry. The symmetry statement is borrows from Klein’s … ….
The mathematics of evolving fields with two complex components is known already in Jones calculus.
This blog entry delivers an other example of an elliptic complex which can be used in discrete Atiyah-Singer or Atiyah-Bott type setups as examples. We had seen that when deforming an elliptic complex with an integrable Lax deformation, we get complex elliptic complexes. We had wondered in that blog entry whether a complex can lead to quaternion-valued fields. The discussion … ….
The strong ring is a category of geometric objects G which are disjoint unions of products of
simplicial complexes. Each has a Dirac operator D and a connection operator L. Both are related in
various ways to topology.
Implementing the Dirac operator D for products of simplicial complexes without going to the Barycentric refined simplicial complex has numerical advantages. If G is a finite abstract simplicial complex with n elements and H is a finite abstract simplicial complex with m elements, then is a strong ring element with n*m elements. Its Barycentric refinement is the Whitney complex of … ….
In the book ‘This Idea Must Die: Scientific Theories That Are Blocking Progress’, there are two entries which caught my eye because they both belong to interests of mine: geometry and calculus. The two articles are provided below. [I believe it is “fair use” as a reprint of these two articles helps not only to promote the book but also … ….