# News about this project

**Interaction cohomology [PDF]**is a case study: like Stiefel-Whitney classes, interaction cohomology is able to distinguish the cylinder from the Möbius strip. The cohomology also admits the Lefschetz fixed point theorem. More on the miniblog.

**Wu Characteristic**Handout for Mathtable.

^{T}produce invariants as described on the same miniblog. local copy.

_{m}of a finite simple graph show a remarkable universality: the graph spectra converge to a distribution which only depends on the maximal dimension of a complete subgraph. For graphs without triangles, the distribution is related to the smooth equilibrium measure of the Julia set of the quadratic map z

^{2}-2. In higher dimension, the universal distributions are unidentified, but appears to be non-smooth with discrete or singular continuous components. Local copy.

_{7}example, which is 6 dimensional for us). We also believe that chromatic number 5 is maximal for surfaces (attained only for nonorientable surfaces like the projective plane (an example found by Jenny)). [Dec. 2014/Jan. 2015 updates there are examples due to Fisk showing that the chromatic number 5 can occur for tori. It really seems to matter that the complement of a torus in a 3 sphere is not simply connected. There is evidence that the chromatic number of any surface is 3,4 or 5: any 2D surface S can be placed into a closed 4D unit ball B, so that the complement of S intersected with int(B) is simply connected. For orientable surfaces we can place S even into the 3-dimensional boundary of B. By coloring int(B)-S (the problem being to make the interior 5 colorable by subdivision or collaps), we could color S.]

_{n}, which are entire functions. We prove that the roots converge to the axes Re(s)=1. This is equivalent that the roots of the Laplace zeta function of the circular graphs converge to the axes Re(s)=1/2. We also derive discrete Basel problem values like zeta(2)=(n

^{2}-1)/12 or zeta(4) = (n

^{2}-1)(n

^{2}+11)/45 which lead in the limit to the classical Basel values zeta(2) = pi

^{2}/6 or zeta(4)=pi

^{4}/90 for the circle. [Updates: Dec 18: The Kubert connection with Milnor's results. Dec 24: the local maximum of the imaginary part in Figure 3 is at the height of the first root of the Riemann zeta function.

_{p},z

^{2}+a,z

^{2}+b,z

^{2}+c) with prime p. The computer is since still looking for more. [Update January 22, 2014: Some slides]

July 18: We found that Cheboratev and Shamis have proven the forest theorem already. We are of course disappointed but also reassured. The paper is now upgraded to count colored trees. The linear algebra results are much stronger and give this too. The update will appear also on the ArXiv. update blog.

^{*}on a graph or manifold. This is the first writeup on this system. Its rough on the edges, chatty and repetitive and maybe even has a forbidding style, but details to most computations should be there. ArXiv. Source code to experiment with the system will be posted later.

^{T}G in terms of products of minors of F and G, where F,G are arbitrary matrices of the same size. The proof is done using the exterior algebra. An update of June 10, 2013 includes Mathematica code. July 6: added that the main result implies an identity for usual determinants: for any two matrices F,G of the same shape det(1+F

^{T}G) = sum_P det(F_P) det(G_P), where P runs over all possible minors, with 1 for the empty minor. See also the [ update log with Mathematica code to copy paste. ] August 6: article.

^{*}on any compact Riemannian manifold or finite simple graph. It also deforms the exterior derivative d but the Laplacian L=D

^{2}stays the same as does cohomology. Classical wave or heat evolution on the geometry are not affected neither. Besides the deformed D(t) = d(t) + d(t)

^{*}+ b(t) the new exterior derivative defines a new Dirac operator C(t) = d(t) + d(t)

^{*}which in the spirit of noncommutative geometry defines a new geometry on the manifold or graph. We prove that the geometry always expands, with a fast inflationary start - as in cosmology. The McKean-Singer supersymmetry relation still holds: the nonlinear unitary evolution U(t) - which naturally replaces the Dirac wave evolution - has the property that str(U(t))= chi(G) at all times. However, supersymmetry is not visible. At t=0, a fermion f and its partner Df are orthogonal at t=0. Already after a short time, the super partner D(t) f is so close to the fermionic subspace that it must be taken as a fermion. Supersymmetry is not broken, but invisible. This holds we take symmetries of quantum mechanics serious. An other feature of the system is that if we do not constrain the evolution to the real, a complex structure evolves. It is absent at t=0 and asymptotically for large t, but it is important in the early part of the evolution. We illustrate in the simplest case like the circle or the two point graph but have computer code which evolves any graph.

^{2}is a block matrix, where each block is the Laplacian on p-forms. The McKean-Singer formula telling that str(exp(-t L) is the Euler characteristic for all t reflects a symmetry. It has combinatorial consequences for counting paths in the simplex space. It also helped to construct graphs which are Dirac isospectral. The matrix is also valuable for doing computations in geometry. Already Poincare has used it in 1900. Today, one can with a dozen lines of computer algebra system code produce the cohomology groups for any graph. The Dirac operator also allows to to see the graph theoretical Gauss-Bonnet-Chern theorem as an example of a discrete index theorem.

**algebraic invariant**(cup) with a

**topological invariant**(cup) and an

**analytic invariant**(cri). [Update Nov 13, 2012: The original cat was renamed topological category tcat(G) since it is - similarly than the geometric category gcat(G) - not yet a homotopy invariant. (While the Fox graph is an example with gcat(G)=3, tcat(G)=cat(G)=2, the dunce hat G is homotopic to a point and satisfies cat(G)=1 but tcat(G)=2 because it is not contractible).