An** abstract delta set **(G,D,R) is a finite set G with n elements, a selfadjoint **Dirac matrix** with and a **dimension vector** defining a partition and Hilbert spaces called the spaces of **k-forms**. The **exterior derivative** maps to . The **Hodge Laplacian** is a block diagonal matrix defining the Hodge blocks and are the** Betti numbers** quantifying how large the **k’th cohomology** is. The kernel of is the vector space **k-harmonic forms**. The **Euler characteristic ** of G is the **total energy of space**, where is the **energy of a simplex** . If is the cardinality of , then is called the **Euler-Poincare formula.** It follows directly from the **McKean-Singer symmetry** stating that the non-zero eigenvalues of on agree with the non-zero eigenvalues of on implying for positive k so that the super trace of the **heat operator** is time independent and agrees with . The **super trace** of an operator A on is defined as complementing the **trace** . The **Euler-Poincare formula** just restates that the super trace of the heat operator at is the same than the super trace of the heat operator at as the limiting operator is the projection onto** harmonic forms**, the kernel of L which is . The McKean-Singer symmetry more generally implies that the super trace of is independent of for any **entire function** satisfying . An example is the **heat operator** which has the property that satisfies the **heat equation** . An other example is the **wave operator** or its real part , which has the property that solves the **wave equation** . If we look at as a **curvature** because it adds up to Euler characteristic, we can also look at as a **time dependent curvature**. We can still look at the entries as energies attached to the simplex . These deformed curvature forms still satisfy Gauss-Bonnet and more.

Call an element g in **locally injective** if whenever or is contained in some . Such a locally injective k-form defines a **total order** on the **unit ball ** of any simplex . The simplex therefore can send its energy to the on which is maximal. If all simplices have moved their shipping, we end up with the energy concentrated on . Since all packages have been integers, the value on each is an integer meaning it is a **divisior**. No energy was lost so that we have the **Poincare-Hopf theorem** . Given a **probability measure** on the set of locally injective functions, we get the **index expectation curvature** as **expectation** . By Fubini, we have the Gauss-Bonnet formula . In the case when is the **product probability space** (take IID random variables on with values in and assuming the uniform Lebesgue probability measure on , we end up with the **k-form curvature**. It can be seen as the situation, where the energy on the simplex is distributed equally to each of the k-simplices which either contain x or are contained in x. An explicit formula for that is , where means strict inclusion and the **degree** is the number of with $latx y \subset z$. In the case of d-manifolds G, spaces where each unit sphere is a (d-1)-sphere (defined recursively), we can simplify formulas. In that case, the 0-curvatures as well as the d-curvatures are all constant zero provided the measure is symmetric with respect to the involution . These are **Dehn-Sommerville** manifestations for k=0. To summarize what we just did is in a rather general combinatorial frame work to show that the **analytic index** is the **topological index** . To the analytic index, one uses that D also defines a map from to . Seen as such, we have a **Atiyah-Singer statement**. (I wrote once something about discrete Aityah-Singer in 2017).

What we did in a few lines gives precise definitions and results in a combinatorial linear algebra frame work which takes a considerable amount of time to establish in the continuum where we are blind about the k-dimensional part of space. We need quite a bit of functional analysis and algebra and topology and differential geometry in order just to make sense of the expressions that are involved. We need for example to understand the spectral property of the self-adjoint Laplacian. A common frame work in the continuum is that is a smooth Riemannian d-manifold. A very special case of Atiyah-Singer is the Gauss-Bonnet-Chern theorem for even dimensional compact Riemannian manifolds. It takes a considerable amount of effort to set up the mathematics: we need tensor calculus just to define notions like the Riemann curvature tensor, we need elliptic regularity to establish that the spectral behavior of the Laplacian is nice. In order to define a square root of the Laplacian we can still use the algebra of differential forms which as a vector space hosts the **Clifford algebra** structure, the frame work used to define for example the Dirac matrices. Note that in the discrete setup we do not require at all that the d is the traditional exterior derivative. We can use it for higher characteristics like **quadratic characteristic** in which are pairs of intersecting simplices in a simplicial complex for which the dimensions add up to k. The total energy is then** Wu characteristic **, which is the **Ising type energy** . Everything is the same. We have then **k-form curvatures **adding up to Wu characteristic. We can also **deform the Dirac matrix** using the** isospectral flow** which does not affect the Laplacian but defines new exterior derivatives (as well as some **“dark energy” ** as develops a diagonal part . The new can be used to define **distance** via the **Connes distance formula** (which only needs the Dirac operator and obviously works in the discrete and also with deformed Dirac matrices. If the shrink then space expands. One can also see this from the solution where u'(0) is in the ortho-complement of the kernel of D. If D gets smaller, the propagation speed appears to get smaller, which can be read as if space expands). Also the **nonlinear wave operator** defining can be used to define a curvature motion. Just take . It works because the super trace of the nonlinear wave operator is still constant by McKean Singer. Apropos Connes: the delta set (G,D,R) of course immediately also gives a **spectral triple** (A,H,D), where is the algebra of functions on and , which is **summable** because we are in finite dimensions. Interestingly, the free evolution of the Dirac matrix in its isospectral class always produces an expansion of space with an **inflationary start**. This is not a “Deus ex machina frame work” but a mathematical result which holds for any positive dimensional finite geometry (as well as in the continuum when deforming the Dirac operator on a Riemannian manifold.) There is some Mathematica code illustrating this on this Wolfram community blog. (And Here (.nb file) is the mathematica notebook to this community blog entry. )

In order to see the relation between the discrete and the continuum, the **integral geometric** formalism helps. Take an even dimensional compact **Riemannian d-manifold** M. By **Nash’s embedding theorem**, we can assume that M is isometrically embedded in a larger dimensional Euclidean space E. There is a natural probability space of **Morse function** on E: take the set of linear functions with n in the standard sphere in E. There is a natural probability measure on S coming from the rotationally symmetric **Haar measure **on the compact **Lie group** of all rotations SO(E) of E. Now, almost all such linear functions induce Morse functions on the manifold M. The traditional continuum Poincare-Hopf theorem on M tells that , where is the divisor on M giving non-zero integer index at finitely many points. In the Morse case, the support of are the critical points of the Morse function and the index takes values in at critical points, depending on whether the **Morse index** (the number of negative eigenvalues of the Hessian ) is odd or even. The index expectation must now be the Gauss-Bonnet-Chern integrand. It is a curvature that is locally invariant under rotations of the frame bundle defined near a point. It was Hermann Weyl who once showed that there is only one curvature which satisfies this property. The Gauss-Bonnet-Chern integrand had already been known by Heinz Hopf. The theorem slowly developed from smooth convex hypersurfaces in E to hypersurfaces in general, then larger codimension until Chern freed it from embeddings in an ambient space. The Nash-Embedding theorem also establishes the result. But how do we relate the discrete from the continuum? We do not have a metric given on G yet but if we take G to be a **fine triangulation** of the manifold M, then each point and simplex in G has spacial coordinates attached in the ambient Euclidean space E we have placed M into. This means that G is geometrically realized as a **d-dimensional polyhedron** in E or what one sometimes calls a **concrete simplicial complex** (in comparison **an abstract simplicial complex** is freed from any Euclidean parts). The probability measure of Morse functions now produces discrete curvatures on G. These curvatures (at least when averaged over some region) converge now to the Gauss-Bonnet-Chern curvature. A possibility to do this is to assume that the vertices of G are an dense set in the compact manifold M. We can now average the curvature and get close to which converges to for .