More on random manifolds - Quantum Calculus

Update: July 26, 2026: A write-up.

If $f_G(t)=1+f_0 t + f_1 t^2 + \cdots + f_q t^{q+1}$ is the simplex generating function of a simplicial complex G, then define $K_G(t) = \int_0^t f_G(s) \; ds$ , the anti-derivative. Gauss-Bonnet is then the elegant formula $f_G = \sum_{v \in V} K_{S(v)}$ . It combines all possible Gauss-Bonnet formulas for all valuations. [The simplest of them is the Euler Handshake formula, which can be seen as Gauss-Bonnet the valuation $X(G) = f_1(G)$, the number of edges. Then $K_{S(v)} = deg(v)/2$ is the vertex degree.] We have also seen other theorems generalize to functional versions. In the video, I show how to get a functional version of the index formula. This then produces a more general random sub-manifold theorem:

Theorem: $E[f_H(t)] = 2-2K_G(t) + f_G(t)$ giving the expected generating function for a random level set H in G.

As in the scalar version (for t=-1), the probability space on $\Omega = \{ -1,1\}^V$ has to be chosen correctly. It is the Bayes measure, the average over all Binomial measures on $\Omega$ .

In the example chosen in the presentation, I use a 3 dimensional torus, obtained using a Stanley-Reisner product (Barycentric refinement of the Cartesian product). It is quite a large graph already. For the picture, I just took a random submanifold. The graph is show to the right (cut open so that it has the shape of a cube). To the left, I show the two dimensional analog. It is the product of two path graphs.

I will post some code I wrote last week which allows to experiment and also generate (for small simplicial complexes at least) to compute all submanifolds and verify the above formula.

It is important that all we do here is finite and exact. It is not a limit theorem. It is a perfectly exact identity for polynomials.

Author: oliverknill