January 8, 2023 Finite Topology for Finite Geometries [ArXiv]. Using
topology on simplicial complexes might appear to be a small matter, but it actually changes the perspective and notions
considerably. It leads to new theorems which are going to come in the next weeks.
April 24, 2020: The
Energy article is up: O. knill, The energy of a simplicial complex, Linear Algebra and its Applications 600 (2020) 96-129
contains results obtained from 2016-2018. The main proof was presented here
The paper is behind a paywall, but there is a share link from Elsevier,
for a few days.
August 18, 2019:
Energized Simplicial Complexes
with local updated version.
Simplicial complexes can be equipped with divisor like functions. The unimodularity and energy
theorem generalizes to this. Some slides
July 21, 2019:
The counting matrix of a simplicial complex,
(local [PDF]).
About a Bosonic sibling of the connection Laplacian. The matrix is in SL(n,Z) and
(very exciting) leads to a zeta function which always
satisfies a functional equation.
[June 18, 2017] The paper On the arithmetic of graphs (ArXiv)
and [PDF] looks at rings of graphs or simplicial complexes.
It deals with various rings of network with main focus on the Zykov Ring which is dual and isomorphic
to the Sabidussi ring. The main result is that the Kuenneth formula holds for the strong Sabidussi product.
(There had been some blogging about this: Jan 15, 2017Jan 27, 2017,
June 10,2017 and
Jun 9, 2017.)
Some Updates.
[June 4, 2017] Some slides about the Hardy-Littlewood prime race:
[May 29, 2017] The paper On a Dehn-Sommerville functional for simplicial complexes establishes a connection between the f-vector rsp. the generating function of a simplicial complex with the
trace of the Green function operator. It also relates the trace of the "hydrogen operator" L-L^{-1} with
the sum of the Euler characteristic of the unit spheres. There had been some
blogging about this already. Local copy.
[April-May, 2017] Various pages were added on the quantum calculus blog: this includes also something about the mass gap and quantum plane which prompted me to speak some text to older unpublished slides. In order not to have them rot on my harddrive, they have been thrown to youtube now:
[March 19, 2017] The paper Helmholtz free energy for finite abstract simplicial complexes [ArXiv]
uses a new Gauss-Bonnet formula to prove that the internal energy of a complex, the total sum over all Green function entries
is Euler characteristic. Taking the energy analogy seriously, we add entropy S(p) to the internal energy U(p) of a probability
measure p and minimize free energy F=U-T S. We observe catastrophes, discontinuous changes of the free energy in dependence of T.
local copy.
Apropos Entropy: clip from Movie "arrival".
[January 14, 2016] While finishing writing down the proof that Green functions form a topological invariant: Something about arithmetic in networks. The join operation plays an important role in the proof but it leads to an interesting quest: is there a fundamental theorem of graph arithmetic?
[December 1,2016] About Euler and Fredholm illustrating
the unimodularity theorem using prime and GCD graphs introduced here.
[November 24, 2016] A discussion written over thanksgiving about the unimodularity theorem status on Quantum calculus. The result is now more general and the proof more transparent based on a generalized multiplicative Poincaré-Hopf lemma assuring that the Fredholm characteristic is the product of Euler-Poincare-indices φ(G) = ∏_{x} ω(x), similarly as Euler characteristic is the sum χ(G) = ∑_{x} ω(x). The Fredholm determinant is a multiplicative analogue of Euler characteristic.
[October 18, 2016] Handout [PDF] for a Mathtable talk on
Bowen Lanford Zeta functions of graphs. The proof comes soon.
[August 28, 2016] Some updates of the
Morse theoretical genesis of numbers.
[August 21, 2016]
Primes, Graphs and Cohomology(local copy [PDF]): counting is a Morse theoretical
process. It also provides a prototype of graphs for which all cohomology groups can be computed
and where Morse cohomology is equivalent to simplicial cohomology.
Some updates (miniblog).
[August 21, 2016]
Particles and Primes: primes in the two complete associative
division algebras C and H show some affinities with Leptons and Hadrons.
[May 26, 2016] Keiji Miura shared a movie showing an application of
Poincaré-Hopf for touch screen devices. Pretty cool.
[March 18, 2016] 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.
[January 23, 2016],
Some Slides about Wu characteristic.
[January 17, 2016] Gauss-Bonnet for multi-linear valuations [ArXiv]
develops multi-linear valuations on graphs. An example of a quadratic valuation was constructed by Wu 1959.
We prove that the Wu characteristic is multiplicative,
invariant under Barycentric refinements and that for d-graphs (discrete d-manifolds),
the formula w(G) = X(G) -X(dG) holds, where dG is the boundary. After developing Gauss-Bonnet
and Poincare-Hopf theorems for multilinear valuations, we prove the existence of multi-linear
Dehn-Sommerville invariants, settling a conjecture of Gruenbaum from 1970.
Local version [PDF]. And here is a miniblog.
[October 4, 2015] Barycentric characteristic numbers.
We outline a proof that for d-graphs, the k-th Barycentric characteristic number is zero if
k+d is even. This is a two page, math only writeup.
October 6: The document has now references included.
[September 20, 2015]
A complete rewrite
of the universality result using the Barycentric operator A for which the eigenvectors of
A^{T} produce invariants as described on the same miniblog.
local copy.
[September 7, 2015] Some Chladni figures (nodal surfaces) in the case of
Barycentric refinements of the triangle G in the case m=1,2
and m=3, a
rounded discrete square, or
larger and
even larger square.
A cylinder.
More updates in the miniblog.
[August 31, 2015] For a lecture of the course MathE320 in the extension school, a
worksheet
on Barycentric refinement.
[August 23, 2015]
A Sard theorem for Graph Theory: ,
with a miniblog for updates.
How to we define level surfaces, discrete algebraic sets in graph theory? How large
is the set of critical values? What is the second derivative test or how does one do
Lagrange extremization in a network? Things go pretty much in the same way as in the continuum, but there
are some twists. Local copy.
[August 9, 2015] The graph spectrum of barycentric refinements:
See a miniblog with code and updates.
The graph spectra of Barycentric refinements G_{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.
[June 21, 2015]
The Jordan-Brouwer theorem for graphs. The
theme is well suited to test definitions and geometric
notions. We prove a general Jordan-Brouer-Schoenflies separation theorem for knots of codimension one.
The inductive definition of spheres (as we found out during this research put forward already
by Alexander Evako) works very well. The proof would not have been possible without the tool of the
graph product found earlier.
(Local copy).
[May 27, 2015]
Kuenneth formula in graph theory.
Slides [PDF], 2014.
Having stumbled over new product for finite graphs, we introduce
de Rham cohomology for general finite simple graphs graphs, show a discrete de Rham theorem,
prove the Kuenneth formula and Eilenberg-Zilber theorem and prove that the dimension is super additive
dim(G x H) >= dim(G) + dim(H) like Hausdorff dimension in the continuum. Products GxH
have explicitly computable chromatic numbers, and if
G,H are geometric, then G x H is geometric and even Eulerian. The product allows to define
joins, new notions of homotopy, discrete manifolds or fibre bundles.
(local copy). Updates.
[Feb 1, 2015] Some diary notes on the miniblog.
Theoretical obstacles seem gone. The problem is to implement the procedure and see it work.
[Jan 11,2015]
"Graphs with Eulerian Unit spheres"
is written in the context of coloring problems
but addresses the fundamental question "what are lines and spheres" in graph theory. We define
d-spheres inductively as homotopy spheres for which each unit sphere is a (d-1) sphere.
In order to define lines in a graph, we need a unique geodesic flow. Because such a flow requires
a fixed point free involution on each unit sphere, we restrict to the subclass of Eulerian
graphs. Such graphs with Eulerian unit spheres are the topic of this paper. Eulerian spheres are
very exciting since if we could extend a general 2-sphere to an Eulerian 3-sphere, it
would prove the 4-color theorem. The paper also gives a short independent classification
of all Platonic solids in d-dimensions, which only uses Gauss-Bonnet-Chern:
these are d-spheres for which all unit spheres are (d-1)-dimensional Platonic solids.
(local copy)
[Dec 21,2014] Coloring graphs using topology.
A geometric approach to graph coloring. We hope it could be a path towards
``seeing why the four color theorem is true". The idea is to embed the graph in a higher
dimensional graph and made 4 colorable by cutting it up. It works in examples but not yet
systematically. (local copy, containing updates),
mini blog, some illustration.
[Oct 12,2014] We looked at various variational problems and especially
Characteristic Length and Clustering [ArXiv].
(local copy) and [update log]
We see more indications that Euler characteristic is the most interesting functional and see
correlations between dimension and length-cluster coefficient. A tiny mathematical lemma proven expresses clustering coefficient with
a relative characteristic length allowing to look at clustering and length-cluster coefficient in general metric spaces.
[Oct 5, 2014] Curvature from Graph Colorings and (Local copy).
This is an extension of the
Index expectation theorem but with a much smaller probability space:
the set of colorings. It uses the remark that the discrete Poincaré-Hopf theorem holds also for
locally injective functions aka colorings. Averaging over all colorings gives curvature.
The topic mixes
chromatic graph theory, integral geometry and is motivated by results known in
differential geometry (like the Fary-Milnor theorem of 1950 which writes total curvature of a knot
as an index expectation) and is elementary.
[July, 2014:] A summer HCRP project with Jenny Nitishinskaya on graph coloring problems
seen from a differential geometric and topological point of view.
A summary report [PDF].
It has been turbulent as we were in uncharted territory.
We conjecture that 4 colors suffice, as for any orientable surface like sphere or torus.
(note that we look for graphs where every unit sphere
is a cyclic graph, disqualifying the K_{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.]
[June 17, 2014:] Update on Binet paperLocal copy. Substantially enhanced, including PseudoPaffians
and Chebotarev-Shamis. [See the mini update blog]
[Mar 23, 2014:] "If Archimedes would have known functions ..."
contains a
Pecha-Kucha talk, a short summary of calculus on finite simple graph,
a collection of calculus problems and some historical remarks.
ArXiv and
local copy [PDF] with
updates.
[February 8, 2014:] "Classical mathematical structures within topological graph theory",
ArXiv, and Local copy [PDF] consists of some preparation notes for a talk at the
a AMS session in January.
See also the slides.
[January 12, 2014:]
A notion of graph homeomorphism.,
(local [PDF])
We find a notion of homeomorphism between finite simple graphs which preserves
basic properties like connectivity, dimension, cohomology and homotopy type and which for triangle free
graphs includes the standard notion of homeomorphism of graphs. The notion is inspired by pointless
topology and Cech constructions. The fact that homeomorphisms with
non-zero Lefschetz numbers have fixed open invariant sets, can be seen as a Kakutani fixed point
theorem for finite simple graphs.
[December 15, 2013:]
The zeta function of circular graphs [ARXIV]
(local [PDF].
The Riemann zeta function is the Dirac zeta function of the circle. We study the roots of the zeta function
of the circular graphs C_{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.
[December 1, 2013:] On quadratic orbital networks [ARXIV],
and local [PDF].
Some remarks in the case of quadratic orbital networks. Was written after finding a disconnected quadratic
network (Z_{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 13, 2013] Counting rooted forests in a network.
We prove that the number of rooted spanning forests in a finite simple graph is det(1+L) where L is the combinatorial
Laplacian of the graph. Compare that with the tree theorem of Kirchhoff which tells that
the pseudo determinant Det(L) is the number of rooted spanning trees in a finite simple graph.
The result can also be interpreted as a voting count: assume that in a social network everybody can vote one of the
friends as "president". If the network forbids any cyclic nominations to prevent groups from tempering with the vote,
then the number of possible voting patterns is det(L+1).
See ArXiv.
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.
[July 13, 2013] The Euler characteristic of an even-dimensional graph. We argue that Euler
characteristic is an interesting functional on four dimensional geometric graphs
because Euler curvature as an average of two dimensional
curvatures of random two dimensional geometric subgraphs. Since Euler curvature is conceptionally close to scalar
curvature, which integrates to the Hilbert action, the Euler characteristic should be an interesting analogue.
ArXiv.
[June 24, 2013] A Isospectral deformation of the Dirac operator: More details about the
integrable system which deforms D=d+d^{*} 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.
[June 9, 2013] Some expanded notes [PDF] from
a talk given on June 5 at an ILAS meeting.
ArXiv and
Slides [PDF].
The talk covered on some linear algebra related to the Dirac operator D of a graph and to
demonstrate how natural this object is. The language of graphs is also a natural frame work
in which one can see essential ideas of multi-variable calculus in arbitrary dimensions. Stokes theorem
on graphs was covered in this talk in even less than
6 minutes 40 seconds.
[May 31, 2013] A Cauchy-Binet theorem for Pseudo-Determinants [PDF],
ArXiv, Jun 1, 2013.
This paper generalizes the classical Cauchy-Binet theorem
for pseudo determinants and more: it gives an expression for the coefficients of the characteristic
polynomial of the matrix F^{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.
[May 31, 2013] An integrable evolution equation in geometry ,
[ArXiv, Jun 1, 2013].
A bit more back to the roots when working on integrable systems in grad school. It introduces a Noether symmetry by doing an isospectral deformation of the
Dirac operator D=d+d^{*} 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.
[January 6, 2013] The The McKean-Singer Formula in Graph Theory [PDF]
[ArXiv].
This paper deals with the Dirac operator D on general finite simple graphs G. It is a matrix
associated with G and contains geometric information. The square L=D^{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.
[November 4, 2012] The Lusternik-Schnirelmann theorem for graphs
[ArXiv].
With Frank Josellis,
we prove cup(G) ≤ tcat(G) ≤ crit(G) for a general finite simple graph G where cup(G) is the cup length,
tcat(G) is the minimal number of in G contractible subgraphs covering G and crit(G) is the minimal number of
critical points an injective function can have on G.
This implies cup(G) ≤ cat(G) ≤ cri(G) for a general finite simple graph G,
where cat(G) is the minimum over all tcat(H) with H homotopic to G and cri(G) is the minimal
crit(H) for an graph H homotopic to G.
The Lusternik-Schnirelmann theorem links an 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).
[June 4, 2012:] A fixed point theorem for graphs
[ArXiv, June 4, 2012] proves a general Lefschetz formula for
graph endomorphisms, leading to fixed point results like a discrete Brouwer theorem which generalizes the
edge theorem of Nowakowski-Rival. Unlike in the continuum, we have to look at simplices as the basic "points".
With the right notion of "degree" of a simplex with respect to T, the proof is pretty close to Hopfs proof in the classical case,
which essentially boils down to "circular graphs have Euler characteristic 0" and "fixed points have Euler characteristic 1" and
"every attractor of an endomorphism is either a circular graph or fixed point".
The paper also gives a formula for the zeta function of T which involves the signature and dimension of prime simplex orbits.
[May 1, 2012:] A continuation shows that curvature K(x) is zero for odd dimensional geometric graphs.
This is proven by showing that the symmetric index j(f,x) = [i(f,x) + i(-f,x)]/2 is constant zero for odd dimensional geometric graphs,
a result which holds for odd dimensional Riemannian manifolds.
In the discrete, we need to define level surfaces B(f,x) = { f=c } in unit spheres S(x). We show that each B(f,x)
is a polytop which can be completed to become geometric. For general simple graphs, the symmetric index j(f,x) satisfies
j(f,x) = [2-chi(S(x))-chi(B(x))]/2 (a formula which also holds in the manifold case). For odd dimensional graphs in particular,
j(f,x) = -chi(B(f,x))/2 which is zero by Poincaré-Hopf and induction. Curvature K(x) as the expectation E[j(f,x)]
over a probability space of scalar functions f is therefore zero too.
[Feb 20, 2012:] Index expectation (ArXiv
brings in some probability theory. It will be used to show that
curvature for odd dimensional geometric graphs is constant zero:
if we integrate over all Morse functions on a graph and average the indices, we get curvature. Since each individual index
function adds up to Euler characteristic, simply taking expectation over all fields gives Gauss-Bonnet.
While this does not simplify the proof of Gauss-Bonnet in the discrete, it most likely will simplify Gauss-Bonnet-Chern
for Riemannian manifolds.
[Jan 29, 2012:] An expository paper [PDF] which might be extended more in the future.
It deals with Gauss-Bonnet, Poincare-Hopf and Green Stokes in a graph theoretical setting.
[Jan 4, 2012:] A discrete analogue of Poincaré Hopf (ArXiv )
for a general simple graphs G. Computer experimentation were essential to try different approaches, starting with small dimensions and guided by the continuum
to find the index which works for random graphs. Discretisation would have been difficult because
the index is classically defined as the degree of a sphere map (needing algebraic topology to be understood properly)
and the analogue of spheres in graph theory can
be pretty arbitrary graphs. Even with a computer, it needed months of experimentation.
Morse theory is relief also in the continuum.
[Dec 19, 2011:] A paper (ArXiv)
on the dimension and Euler characteristic of random graphs provides
explicit formulas for the expectation of inductive dimension dim(G) or Euler characteristic X(G), which are considered
random variables over Erdoes-Renyi probability spaces. Most results were found first experimentally using brute force
computations before proving them. The paper belongs to the mathematics of complex networks.
Dimension and Euler characteristic mixes in some geometry.
[Update Jan 2018: this paper of 2013 mentions the expectation of Euler
characteristic on page 16. Kahle obviously was unaware of my 2011 paper. I still think (2018) that the formula
appeared first here.]
[Nov 21 2011:] A paper on higher dimensional Gauss-Bonnet which fits
the occasion of Chern's birthday of October 26, 1911,
The result was obtained in the summer of 2009 but illustrating it with examples took time. One can define curvature K(x) which
depends only on the unit sphere of a vertex x in a graph G=(V,E) such that the sum of K(x) over V is Euler characteristic
X(G). To see this implemented in Mathematica visit the code page.
[Jul 6, 2010] This project started in spring 2009. The subject is simple topology or discrete differential geometry initiated in this paper.
The goal is to understand graphs on a geometric level and investigate
discrete analogues of structures which are known in differential geometry.
This website is up since July 2010.