Papers and updates

January 24, 2024, Density of wave fronts (motivates related discrete models)
September 13, 2024, Colorful Rings of Partitions.
September 2, 2024, Gauss Bonnet for Form Curvatures.
August 24, 2024, On Symmetries of Finite Geometries.
June 24, 2024 Quadratic fusion inequality local PDF.
May 29, 2024 Morse and Lusternik-Schnirelmann for graphs, (local copy),
May 22, 2024 Eigenvalue bounds revised (code and more references)
Jan 14, 2024 Manifolds from Partitions [ArXiv], Local PDF (updated).
Jan 14, 2024 Manifolds from Partitions [ArXiv], Local PDF.
Dec 21, 2023 Discrete algebraic sets in discrete manifolds [PDF], Local PDF.
Nov 5, 2023 Arboricity of manifolds [ArXiv], Local PDF.
Oct 22, 2023 Arboricity of manifolds [PDF], Local PDF.
Sept. 4, 2023 Three tree theorem [PDF], Local PDF.
July 23, 2023 Cohomology of measurable sets, local PDF
June 18, 2023 Weierstrass and the pendulum, see also local version [PDF]
May 21, 2023 Cohomology of Open Sets , [PDF]
April 3, 2023 Spectral monotonicity [ArXiv],[PDF]
February 6, 2023 Higher Characteristics [ArXiv][PDF]
January 12, 2023, Sphere formula [ArXiv], [PDF]
January 8, 2023: Finite Topology for Finite Geometries [ArXiv] local [PDF].
May 22, 2022: Eigenvalue bounds of the Kirchhoff Laplacian [Arxiv] and local [PDF] and The Tree-Forest Ratio [Arxiv]. local [PDF].
January 23, 2022: Analytic Torsion [Arxiv] local version [PDF].
August 10, 2021: Shannon Capacity, Chess, DNA and Umbrellas (ArXiv) and local copy [PDF].
July 17, 2021: Product formula for curvature: The curvature of graph products [PDF].
June 27, 2021: Coloring Discrete Manifolds [ArXiv]. Local copy [PDF].
June 17, 2021: A few remarks on graph arithmetic[ArXiv]. Local Copy [PDF].
January 18, 2021: Graph Complements of circular graphs. Local copy[PDF], Mathematica CODE [TXT].
December 13, 2020: Complexes, Graphs, Homotopy and Shannon Capacity [Arxiv] and local version [PDF].
October 18, 2020: Green functions of Energized complexes. Local version [PDF].
August 23, 2020: Division algebra valued energized simplicial complexes, local version [PDF]
June 28, 2020: a bit more reflection on Positive curvature and physics local version [PDF].
June 19, 2020: On a theorem of Grove and Searle [PDF] and Local version [PDF].
May 24-May 31, 2020: A Dehn type quantity for Riemannian manifolds (V2) and a Local copy [PDF]. There is an entry Chopping up Riemannian manifolds in the quantum calculus blog and a mini blog roll on this.
April 24, 2020: The Energy article is available for a few days from this share link from Elsevier.
January 19, 2020: In Index expectation curvature for manifolds [ArXIV] local version [PDF]
January 5, 2020: Integral geometric Hopf conjectures [ArXiv], local [PDF]
December 23, 2019: Constant index expectation curvature for graphs or Riemannian manifolds [PDF], local [PDF]
December 1, 2019: More on Poincare-Hopf and Gauss-Bonnet [ArXiv]

updated local version [PDF]

November 10, 2019: Poincaré-Hopf for vector fields on graphs [ArXiv] and local [PDF]
October 6, 2019: A simple sphere theorem for graphs [PDF] (The Mickey mouse theorem). local version [PDF].
August 18, 2019: Energized Simplicial Complexes. Simplicial complexes can be equipped with divisor like functions. The unimodularity and energy theorem generalizes to this.
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.
July 7, 2019: The energy of a simplicial complex local copy [PDF] A new write up about older results related to the energy of a complex.
June 15, 2019: Parametrized Poincare-Hopf theorem, (local copy [PDF])
May 30, 2019: On Numbers and Graphs , and local copy [PDF]
May 12, 2019: Dehn-Sommerville from Gauss-Bonnet, and updated local copy [PDF].
May 5, 2019: The average simplex cardinality of a finite abstract simplicial complex. Local copy [PDF].
March 24, 2019: A Reeb sphere theorem in graph theory [ArXiv]. ( local version [PDF].
November 25, 2018: Cartan's Magic Formula for Simplicial Complexes [PDF], Local [PDF]
[August 21, 2018]Eulerian edge refinements, geodesics, billiards and sphere coloring
[July 22, 2018]Some theorems in math
[June 18, 2018] Combinatorial manifolds are Hamiltonian. (Updated on a local copy [PDF])
[April 22, 2018] The amazing world of simplicial complexes (local copy [PDF] for AMS meeting on April 22, 2018.
[March 18, 2018] The Cohomology for Wu Characteristics, (local copy)
[March 4, 2018] The Hydrogen identity for Laplacian (local copy).
[February 4, 2018] Listening to the cohomology of graphs [ArXiv], with Local PDF.
[January 14, 2018] The paper Elementary Dyadic Riemann zeta function is on the (ArXiv), with Local PDF.
[November 26, 2017] Paper One can hear the Euler characteristic of a simplicial complex [ArXiv]. Local PDF. There had been some blogging about this on quantum calculus.
[August 20, 2017] A follow up of the strong ring: "Atiyah-Singer and Atiyah-Bott for simplicial complexes". with local copy [PDF]
[August 5, 2017] Paper The strong ring of simplicial complexes introduces a ring of geometric objects in which one can compute quantities like cohomologies faster. local copy.
[June 18, 2017] Paper: On the arithmetic of graphs ArXIV, and local copy PDF) and Updates. (There had been some blogging about this: Jan 15, 2017) Jan 27, 2017, June 10,2017 and Jun 9, 2017.)
[May 29, 2017] The paper "On a Dehn-Sommerville functional for simplicial complexes" is posted. There had been some blogging about this.
[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. [Local PDF]
[December 24, 2016] ArXIV version"> 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: [ArXiv] 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 A^T 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 G_m 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. local copy, Mathematica code and Miniblog.
[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. (local copy).
[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). Updates.
[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 updates and some illustration.
Oct 12, 2014: We look at various variational problems and especially Characteristic Length and Clustering [ArXiv]. (local copy). 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 ..." ArXiv"ArXiv and local copy [PDF] with updates.
Feb 8, 2014: "Classical mathematical structures within topological graph theory", ArXiv, and Local copy kept up to date [PDF], notes for a AMS session in January. and slides. updates.
Jan 12, 2014: A notion of graph homeomorphism [ARXIV] and local [PDF]. [Updates]
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. ArXiv.
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], [updates]
Cauchy-Binet for Pseudo-Determinants [PDF], [updates], ArXiv, Jun 1, 2013
An integrable evolution equation in geometry ArXiv, 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

Papers and updates

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