Papers and updates
- [March 19, 2017] The paper Helmholtz free energy for finite abstract simplicial complexes [ARXIV]
is posted. [Local PDF]
- [February 12, 2017] The paper
"Sphere Topology and Invariants" is posted.
- [December 24, 2016]
On Fredholm determinants in topology
gives the proof of the unimodularity theorem as announced in October. [Local PDF]
- [October 18, 2016] Handout [PDF] for a Mathtable talk on
Bowen Lanford Zeta functions of graphs.
- [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.
- [August 21, 2016]
Particles and Primes: primes in the two complete associative
division algebras C and H show some affinities with Leptons and Hadrons.
- [June 19, 2016] Something about primes,
Goldbach in Division algebras [ArXiv],
local copy [PDF]
A larger report on experiments:
local copy [PDF].
- [March 18, 2016]Interaction cohomology
Interaction cohomology [PDF]
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 theorem.
- [March 8, 2016] Wu Characteristic Handout [PDF] for a Mathtable talk.
- 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.
And here is a miniblog.
- [October 4, 2015 Barycentric characteristic numbers.
We give 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, a detailed writeup will need some time.
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
AT produce invariants for which Euler characteristic is an example
Updates are on the same miniblog.
- [August 23, 2015]
A Sard theorem for Graph Theory:
(See a miniblog with updates).
Sard works surprisingly well in a discrete setup. The topic also allows to
do a bit of quantum multivariable calculus like looking at level surfaces,
or doing Lagrange extremization. local copy.
- [August 9, 2015]
The graph spectrum of barycentric refinements [ArXiv]:
We look at the graph spectral
distribution of barycentric refinements Gm of a finite simple graph.
The spectrum converges 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 z2 -2. In higher
dimensions d>1, the universal distributions are still unidentified but appears do not look smooth.
Mathematica code and
- [June 21, 2015]
The Jordan-Brouwer theorem for graphs [ArXiv]
We prove a general Jordan-Brouer-Schoenflies separation theorem for knots of codimension one.
The proof is close to Jordan's, Brouwer's or Alexander's papers.
- [May 27, 2015]
Kuenneth formula in graph theory.
While procrastinating a bit on programming the geometric coloring algorithm,
we found a new product for finite graphs. Its pretty exciting as it allows to define
de Rham cohomology for general finite simple graphs graphs (only a good product allows
even to talk about de Rham cohomology for graphs), show a discrete de Rham theorem,
prove the Kuenneth formula rsp. 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, homotopy, discrete manifolds or fibre bundles. (local copy).
- [Jan 11,2015] "Graphs with Eulerian Unit spheres"
addresses questions like "what are lines and spheres" in graph theory. We define
d-spheres inductively as homotopy spheres for which every unit sphere is a (d-1) sphere.
To define lines in a graph, we need a unique geodesic flow. Because such a flow defines
a fixed point free involution on each unit sphere, we must restrict to a subclass of Eulerian
graphs. Such graphs with Eulerian unit spheres are the topic of this. Eulerian spheres are
exciting since if we could extend a general 0sphere to an Eulerian 3 sphere, it
would prove the four color theorem. The paper also gives a short independent classification
of all Platonic solids in d-dimensions these are d-spheres for which all unit spheres are
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 kept updated) and
- Oct 12, 2014: We look at various variational problems and especially
Characteristic Length and Clustering [ArXiv].
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: the short note Curvature from Graph Colorings (local copy)
extends the index expectation theorem but with graph colorings
a much smaller probability space.
- Jun 17, 2014: The Binet paper
is finally updated, actually substantially enhanced. We use now more natural
definitions and include Pseudo Pfaffians and also mention Chebotarev-Shamis,
since it is so nice. [See the mini update blog]
Local copy of the revision.
- Mar 23, 2014: "If Archimedes would have known functions ..."
local copy [PDF] with
- Feb 8, 2014: "Classical mathematical structures within topological graph theory",
Local copy kept up to date [PDF], notes for
a AMS session in January.
- Jan 12, 2014:
A notion of graph homeomorphism [ARXIV] and
- Dec 15, 2013:The zeta function of circular graphs [ARXIV] and local [PDF]. [Updates]
- Dec 1, 2013: On quadratic orbital networks [ARXIV],
and local [PDF].
- Nov 25, 2013: Natural orbital networks [ARXIV], local file [PDF].
- Nov 17, 2013: Dynamically generated networks [ArXiv] local file [PDF].
See the project page.
- Counting rooted forests in a network.
The number of rooted spanning forests in a finite simple graph is det(1+L) where L is the combinatorial Laplacian.
ArXiv. [update blog]
- The Euler characteristic of an even-dimensional graph. We argue that Euler
characteristic is an interesting functional because Euler curvature as an average of two dimensional
curvatures of random two dimensional geometric subgraphs.
- Isospectral deformations of the Dirac operator (ArXiv). More details about the
integrable dynamical system in geometry.
- The Dirac operator of a graph [PDF], consists of notes to the talk
It is also on the [ArXiv]. The Slides [PDF],
- Cauchy-Binet for Pseudo-Determinants [PDF], [updates], ArXiv, Jun 1, 2013
- An integrable evolution equation in geometryArXiv, Jun 1, 2013
- The McKean-Singer Formula in Graph Theory [PDF], ArXiv, Jan 8, 2013
- The Lusternik-Schnirelmann theorem for graphs [PDF], ArXiv, Nov 4 (updated Nov 13), 2012 and updates.
- A Brouwer fixed point theorem for graph endomorphisms [PDF], ArXiv, June 4, 2012 and updates.
Fixed Point Theory and Applications.2013, 2013:85. DOI: 10.1186/1687-1812-2013-85.
- An index formula for simple graphs [PDF], ArXiv May 2012 and updates.
- On index expectation and curvature for networks [PDF], ArXiv Feb 2012 and updates.
- The theorems of Green-Stokes,Gauss-Bonnet and Poincare-Hopf in Graph Theory[PDF]. ArXiv Jan 2012, and updates.
- A graph theoretical Poincare-Hopf Theorem [PDF]. ArXiv Jan 2012, updates
- On the Dimension and Euler characteristic of random graphs [PDF]. ArXiv Dec 2011 and Updates
- Chern-Gauss-Bonnet theorem for graphs [PDF], On ArXiv Nov 2011 and Updates
- A discrete Gauss-Bonnet type theorem [PDF].On ArXiv [Sep 2010].Elemente der Mathematik, 67,1, pp1-44, 2012
See also the Code page.