## ArXiv Stuff

All ArXiv- Mandelbulb, Mandelbrot, Mandelring and Hopfbrot
- Cohomology of Open Sets
- Spectral Monotonicity of the Hodge Laplacian
- Characteristic Topological invariants
- The sphere formula
- Finite topologies for finite geometries
- On Graphs, Groups and Geometry
- The Babylonian Graph
- The Tree-Forest Ratio
- Eigenvalue bounds of the Kirchhoff Laplacian
- Analytic torsion for graphs
- The curvature of graph products
- Coloring discrete manifolds
- Remarks about the Arithmetic of Graphs
- Graph complements of circular graphs
- Complexes, Graphs, Homotopy, Products and Shannon Capacity
- reen Functions of Energized complexes
- Division algebra valued energized simplicial complexes
- Positive Curvature and Bosons
- On a theorem of Grove and Searle
- A Dehn type quantity for Riemannian manifolds
- On index expectation curvature for manifolds
- Integral geometric Hopf conjectures
- Constant index expectation curvature for graphs or Riemannian manifolds
- More on Poincare-Hopf and Gauss-Bonnet
- Poincare Hopf for vector fields on graphs
- A simple sphere theorem for graphs
- Energized simplicial complexes
- The counting matrix of a simplicial complex
- The energy of a simplicial complex
- A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs
- More on Numbers and Graphs
- Dehn-Sommerville from Gauss-Bonnet
- Average Simplex cardinality
- Reeb Sphere Theorem
- Cartan's Magic Formula for Simplicial Complexes
- Eulerian Edge Refinements, Geodesics, Billiards and Sphere colorings
- Some Fundamental Theorems in Mathematics
- Combinatorial Manifolds are Hamiltonian
- The amazing world of simplicial complexes
- The Cohomology for Wu Characteristic
- Hydrogen Identity for Laplacians
- Listening to the cohomology of graphs
- An Elementary Dyadic Riemann Hypothesis
- One can hear the Euler characteristic of a simplicial complex
- On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes
- THe strong ring of simplicial complexes
- On the arithmetic of graphs
- On a Dehn-Sommerville functional for simplicial complexes
- Helmholtz free energy for finite abstract simplicial complexes
- Sphere geometry and invariants
- Fredholm determinants in topology
- On particles and Primes
- On Primes, Graphs and Cohomology
- Some experiments in number theory
- Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes
- Gauss-Bonnet for Multi-linear valuations
- Universality for Barycentric subdivision
- Sard Theorem for Graph theory
- Graph spectrum of Barycentric refinements
- Jordan-Brower theorem for graphs
- The Kuenneth formula for graphs
- Graphs with Eulerian Unit spheres
- Coloring graphs using topology
- Characteristic Length and Clustering
- Curvature from Graph Colorings
- If Archimedes would have known functions
- Classical mathematical structures within topological graph theory
- A notion of graph homeomorphism
- The zeta function of circular graphs [ARXIV]
- Natural orbital networks [ARXIV]
- On quadratic orbital networks [ARXIV]
- Dynamically generated networks [ARXIV]
- Counting rooted trees [ARXIV]
- Illustrating Mathematics using 3D printers
- Isospectral Deformations of the Dirac operator
- The Dirac operator of a graph
- Cauchy-Binet for Pseudo determinants
- An integrable evolution equation in geometry
- Thinking like Archimedes with a 3D printer, with Elizabeth Slavkovsky.
- The McKean-Singer Formula in Graph Theory [PDF], Jan 6, 2013
- The Lusternik-Schnirelmann theorem for graphs, ArXiv, November 4, 2012.
- Selfsimilarity in the Birkhoff sum of the cotangent function [2012] , Project page.
- Brower fixed point for graph endomorphisms, [ArXiv] 2012
- An index formula for simple graphs [Arxiv] 2012
- On index expectation and curvature for networks [ArXiv] 2012
- A graph theoretical Poincare-Hopf Theorem [ArXiv] (2012)
- On the Dimension and Euler characteristic of random graphs [ArXiv] (2011)
- A graph theoretical Gauss-Bonnet-Chern Theorem [ArXiv] (2011). See project page
- With J. Ramirez: A structure from motion inequality [ArXiv] (2007),project page
- With J. Ramirez: Omnidirectional structure from motion problem [ArXiv] (2007) ,project page
- With J. Ramirez: On Ullman's Theorem in Computer Vision [ArXiv] (2007) ,project page
- A multivariable Chinese Reminder Theorem [PDF] (2005), update (2012), ArXiv, project page
- Fluctuation bounds for subharmonic functions [PDF], (2004),MP-ARC, Jan19, 2000 or on research age.
- With J. Carlson, A. Chi and M. Lezama: An artificial intelligence experiment in college math education [PDF] (2003)
- On Hausdorffs moment problem in higher dimensions [PDF] , (2000)
- An existence theorem for Vlasov gas dynamics in regions with moving boundaries, MP-ARC 00-38, Jan24, (2000)
- Random Schroedinger operators arising from lattice gauge fields [PDF] (2000)
- On the cohomology of discrete abelian group actions, MP-ARC, (2000)
- Inversion of the two dimensional Radon transformation by diagonalisation (1997)

**Failed projects:**

- The Problem of Positive Kolmogorov-Sinai entropy for the Standard map
This was an attempt to prove that the Standard map T(x,y) = (2x-y+c sin(x),y) on T
^{2}has metric entropy bounded below by log(c/2). The idea was to push Herman's subharmonic estimates to a real analytic situation (but not complex analytic frame work) using multi-linear algebra. The approach is described a bit more in this math table talk of 2004[PDF]. It also lead to This paper from 2000 which I find one of my best papers I have been writing as it introduces a novel homogenisation approach to estimate fluctuations of subharmonic functions (which is an extremely classical area of mathematics). All these papers were written during a stressful time with severe time constraints and unmovable deadlines (postdoc times have fixed timelines and after that time is over, the game is over). It is tempting in such situations to make a hail Mary pass using a difficult unsolved problem. - A deterministic displacement theorem [PDF] This was work on an important open problem in Hamiltonian dynamics. The goal had been to show that the Hamiltonian n-body problem has almost everywhere solutions (also non-collision singularities have measure zero). This is a much too underappreciated problem in Celestial mechanics. The most natural approach to the problem is to analyze non-collsion singularities where particles escape to infinity in finite time. But the construction of examples of such initial conditions was insane. An toerh approach therefore is to try to take a continuum measure of masses and to see this measure evolve using an integro-partial differential equation, then show that this measure describes the average of solutions of the real problem. I had hoped that the simplest mean field model, the Vlasov system would do the job. Evolve a Poisson cloud of particles using the n-body dynamics and show that the density moves according to Vlasov. "The statement in the displacement theorem announcement is incorrect. One would either have to look at a Vlasov-BBGKY hierarchy or then make additional randomness assumption. Without that, correlations develop which would then have to be evolved using higher order correlations etc. (This was pointed out to me by H. Spohn sometime in 1998). The announceement been an attempt to solve the open problem that the Hamiltonian Newtonian n-body problem has a solutions for almost all initial conditions. Unfortunately, the BBGKY stuff is technically complicated. When introducing randomness, one enters an other class of dynamical systems which Boltzman type equations. The work (which was done in 1996-1997 while I was in Arizona and Texas) was an opportunity to learn about Poisson processes as well as Vlasov dynamics. Section 5.4 in the probability book profited from this research. Tackling a BBGKY expansion similar to a Taylor expansion turned out to too technical for me. The result proven in that electronic announcement is that the first derivative for correlations is zero. This is correct. The conclusion that the correlations remain zero for later time is false. What actually happens is that higher derivatives are no more zero in general. Analyze higher derivatives leads to higher order Vlasov equations, a version of the so called BBGKY hierarchy. But this is then no more a Poisson process. The picture of evolving Poisson processes is tempting, as it interprets a probability measure as an average of finite point processes. My quest had been to see the Vlasov equation as an integral equation which describes the mean of a probability space of n-body problems. The existence of solutions of the Vlasov equation then would lead to almost everywhere existence of solutions of the n-body problems. The almost everywhere existence of solutions to the n-body problem is a famous open problem and part of Simon's problems. See this list of open problems. The problem of singular potential in the Newtonian problem is not the difficulty as there are existence theorems for Vlasov dynamics with the -1/|x| potential. Vlasov appears still a promising approach for that existence theorem (analyzing non-collision singularities looks in comparison extremely hard; even establishing existence of particles moving to infinity in finite time was a tour de force by mathematicians like Saari or Xia). There is hope that some mean field theory can help to estimate averages of multi-particle n-body processes and establish global existence for almost all initial conditions (one of the celebrated Barry Simon's open problems in mathematical physics).