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Oliver Knill

Publications and source records attributed to Oliver Knill.

At least 37 records · Page 2Linked to original sources
arXiv2024

Morse and Lusternik-Schnirelmann for graphs

Both Morse theory and Lusternik-Schnirelmann theory link algebra, topology and analysis in a geometric setting. The two theories can be formulated in finite geometries like graph theory or within finite abstract simplicial complexes. We work here mostly in graph theory and review the Morse inequalities b(k)-b(k-1) + ... + b(0) less of equal than c(k)-c(k-1) + ... + c(0) for the Betti numbers b(k) and the minimal number c(k) of Morse critical points of index k and the Lusternik-Schnirelmann inequalities cup+1 less or equal than cat less or equal than cri, between the algebraic cup length cup, the topological category cat and the analytic number cri counting the minimal number of critical points of a function.

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arXiv2024

Eigenvalue bounds of the Kirchhoff Laplacian

We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prove that in general the weak Brouwer-Haemers lower bound d(k) + (n-k) holds for all eigenvalues l(k) of the Kirchhoff matrix of a quiver.

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arXiv2024

Manifolds from Partitions

If f maps a discrete d-manifold G onto a (k+1)-partite complex P then H(G,f,P),the set of simplices x in G such that f(x) contains at least one facet in P defines a (d-k)-manifold.

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arXiv2023

Discrete Algebraic sets in Discrete Manifolds

A discrete d-manifold is a finite simple graph G=(V,E) where all unit spheres are (d-1)-spheres. A d-sphere is a d-manifold for which one can remove a vertex to make it contractible. A graph is contractible if one can remove a vertex with contractible unit sphere to get a contractible graph. We prove a discrete Morse-Sard theorem: if G=(V,E) is a d-manifold and f:V to R^k an arbitrary map, then for any c not in f(V), a level set { f = c } is always a (d-k)-manifold or empty. While a priori open sets in the simplicial complex of G, they are sub-manifolds in the Barycentric refinement of G. Level sets are orientable if G is orientable. Any complex-valued function psi on a discrete 4-manifold M defines so level surfaces {psi=c} which are except for c in f(V) always 2-manifolds or empty.

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arXiv2023

Arboricity and Acyclic Chromatic Number

A theorem of Hakimi, Mitchem and Schmeichel from 1996 states that the edge arboricity arb(G) of a graph is bounded above by the acyclic chromatic number acy(G). We can improve this HMS inequality by 1, if acy(G) is even. We review also results about acyclic chromatic numbers in the context of a Grünbaum conjecture from 1973.

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arXiv2023

On the arboricity of manifolds

The arboricity of a discrete 2-sphere is always 3. The arboricity of any other discrete 2-dimensional surface is always 4. For d-manifolds of dimension larger than 2, the arboricity can be arbitrary large and must be larger than d.

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arXiv2023

The Three Tree Theorem

We prove that every 2-sphere graph different from a prism can be vertex 4-colored in such a way that all Kempe chains are forests. This implies the following three tree theorem: the arboricity of a discrete 2-sphere is 3. Moreover, the three trees can be chosen so that each hits every triangle. A consequence is a result of an exercise in the book of Bondy and Murty based on work of A. Frank, A. Gyarfas and C. Nash-Williams: the arboricity of a planar graph is less or equal than 3.

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arXiv2023

On the cohomology of measurable sets

If T is an ergodic automorphism of a Lebesgue probability space (X,A,m), the set of coboundries B = db =T(b)+b with symmetric difference + form a subgroup of the set of cocycles A. Using tools from descriptive set theory, Greg Hjorth showed in 1995 that the first cohomology group H=A/B is uncountable. This can surprise, given that in the case of a finite ergodic probability space, H has only 2 elements. Hjorth's proof used descriptive set theory in the complete metric space (A,d(a,b)=m(a+b)), leading to the statement that B is meager in A. We use a spectral genericity result of Barry Simon to establish the same. It leads to the statement noted first by Karl Petersen in 1973 that for a generic a in A, the induced system T_a is weakly mixing, which is slightly stronger than a result of Nate Friedman and Donald Ornstein about density of weakly mixing in the space of all induced systems T_a coming from an ergodic automorphism T.

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arXiv2023

Mandelbulb, Mandelbrot, Mandelring and Hopfbrot

A topological ring R, an escape set B in R and a family of maps z^d+c defines the degree d Mandelstuff as the set of parameters for which the closure of the orbit of 0 does not intersect R. If B is the complement of a ball of radius 2 in C or H it is the complex or quaternionic Mandelbrot set; in a vector space with polar decomposition x=|x| U(t) like R=R^m, the map z^d+c is defined as the map z=|z| U(t) to z^d=|z|^d U(d t). We review some Jacobi Mandelstuff of quadratic maps on almost periodic Jacobi matrices which have the spectrum on Julia sets. In a Banach algebra R, one can define A^d=|A|^d U^d with A=|A| U written as the product of a self-adjoint |A| and unitary element U. In R^4, the Hopf parametrization leads to the Hopfbrot, which has White-Nylander Mandelbulbs in R=R^3 as traces and the standard Mandelbrot sets in C as codimension 2 traces. It is an open problem of White whether Mandelbulbs in higher dimensions are connected. The document contains an appendix with a proof of the Douady-Hubbard theorem.

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arXiv2023

Weierstrass elliptic functions for the pendulum

The mathematical pendulum is traditionally solved using a Jacobi elliptic functions. We solve it here using the Weierstrass elliptic function. Every initial condition of the pendulum produces an elliptic curve and a point which by the dynamics of the pendulum is translated linearly on the torus.

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arXiv2023

Cohomology of open sets

If G is a finite abstract simplicial complex and K is a subcomplex of G and U=G-K is the open complement of K in G, the Betti vectors of K and U and G satisfy the inequality b(G) less or equal b(K)+b(U).

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arXiv2023

Spectral monotonicity of the Hodge Laplacian

If K,G are finite abstract simplicial complexes and K is a subcomplex of G then the eigenvalues of the Hodge Laplacian of K are smaller or equal than the eigenvalues of the Hodge Laplacian of G, provided the eigenvalue lists are padded left.

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arXiv2023

Characteristic Topological Invariants

The higher characteristics w_m(G) for a finite abstract simplicial complex G are topological invariants that satisfy k-point Green function identities and can be computed in terms of Euler characteristic in the case of closed manifolds, where we give a new proof of w_m(G)=w_1(G). Also the sphere formula generalizes: for any simplicial complex, the total higher characteristics of unit spheres at even dimensional simplices is equal to the total higher characteristic of unit spheres at odd dimensional simplices.

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arXiv2023

The Sphere Formula

The sphere formula states that in an arbitrary finite abstract simplicial complex, the sum of the Euler characteristic of unit spheres centered at even-dimensional simplices is equal to the sum of the Euler characteristic of unit spheres centered at odd-dimensional simplices. It follows that if a geometry has constant unit sphere Euler characteristic, like a manifold, then all its unit spheres have zero Euler characteristic or the space itself has zero Euler characteristic. Especially, odd-dimensional manifolds have zero Euler characteristic, a fact usually verified either in algebraic topology using Poincaré duality together with Riemann-Hurwitz then deriving it from the existence of a Morse function, using that the Morse indices of the function and its negative add up to zero in odd dimensions. Gauss Bonnet also shows that odd-dimensional Dehn-Sommerville spaces have zero Euler characteristic because they have constant zero curvature. Zero curvature phenomenons can be understood integral geometrically as index expectation or as Dehn-Sommerville relations.

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arXiv2023

Finite topologies for finite geometries

Without leaving finite mathematics and using finite topological spaces only, we give a definition of homeomorphisms of finite abstract simplicial complexes or finite graphs. Besides exploring the definition in various contexts, we add some remarks like that the general Lefschetz formula works for any continuous map on any finite topological space. We also noted that any higher order Wu characteristic as well as their cohomology are topological invariants which are not homotopy invariants. Energy theorems allow to express these topological invariants in terms of interaction energies of local open sets.

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arXiv2022

The Babylonian Graph

The Babylonian graph B has the positive integers as vertices and connects two if they define a Pythagorean triple. Triangular subgraphs correspond to Euler bricks. What are the properties of this graph? Are there tetrahedral subgraphs corresponding to Euler tesseracts? Is there only one infinite connected component? Are there two Euler bricks in the graph that are disconnected? Do the number of edges or triangles in the subgraph generated by the first n vertices grow like of the order n W(n), where n is the product log? We prove here some first results like the threshold where B(n) becomes non-planar. In an appendix, we include handout from a talk on Euler cuboids given in the year 2009.

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arXiv2022

The Tree-Forest Ratio

The number of rooted spanning forests divided by the number of spanning rooted trees in a graph G with Kirchhoff matrix K is the spectral quantity tau(G)= det(1+K)/det(K) of G by the matrix tree and matrix forest theorems. We prove that that under Barycentric refinements, the tree index T(G)=log(det(K))/n and forest index F(G)=log(det(1+K))/n and so the tree-forest index i=F-G=log(tau(G))/n converge to numbers that only depend on the size of the maximal clique in the graph. In the 1-dimensional case, all numbers are known: T(G)=0, F(G)=i(G) =2 log(phi), where phi is the golden ratio. The convergent proof uses the Barycentral limit theorem assuring the Kirchhoff spectrum converges weakly to a measure dk on the positive real axis that only depends on dimension of G. Trees and forests indices are potential values i = U(-1)-U(0) for the subharmonic function U(z)=int_R log|x-z| dk(x) defined by the Riesz measure dk=Delta U which only depends on the dimension of G. The potential U(z) is defined for all z away from the support of dk and finite at z=0. Convergence follows from the tail estimate k[x,infty] < C exp(-a x) where the decay rate a only depends on the maximal dimension. With the normalized zeta function zeta(s) = (1/n) sum_k lambda_k^-s, we have for all finite graphs of maximal dimension larger than 1 the identity i(G) = sum_t (-1)^(s+1) zeta(s)/s. The limiting zeta function zeta(s) = int_R x^(-s) dk(x) is analytic in s for s<0. The Hurwitz spectral zeta function zeta_z(s)=U_s(z) = int_R (x-z)^(-s) dk(x) complements U(z) = int_R log(x-z) dk(x) and is analytic for z in C - R^+ and for fixed z in C-R^+ is an entire function in s in C.

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