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Oliver Knill

Publications and source records attributed to Oliver Knill.

At least 73 records · Page 4Linked to original sources
arXiv2019

Energized simplicial complexes

For a simplicial complex with n sets, let W^-(x) be the set of sets in G contained in x and W^+(x) the set of sets in G containing x. An integer-valued function h on G defines for every A subset G an energy E[A]=sum_x in A h(x). The function energizes the geometry similarly as divisors do in the continuum, where the Riemann-Roch quantity chi(G)+deg(D) plays the role of the energy. Define the n times n matrices L=L^--(x,y)=E[W^-(x) cap W^-(y)] and L^++(x,y) = E[W^+(x) cap W^+(y)]. With the notation S(x,y)=1_n omega(x) =delta(x,y) (-1)dim(x) and str(A)=tr(SA) define g=S L^++ S. The results are: det(L)=det(g) = prod_x in G h(x) and E[G] = sum_x,y g(x,y) and E[G]=str(g). The number of positive eigenvalues of g is equal to the number of positive energy values of h. In special cases, more is true: A) If h(x) in -1, 1}, the matrices L=L^--,L^++ are unimodular and L^-1 = g, even if G is a set of sets. B) In the constant energy h(x)=1 case, L and g are isospectral, positive definite matrices in SL(n,Z). For any set of sets G we get so isospectral multi-graphs defined by adjacency matrices L^++ or L^-- which have identical spectral or Ihara zeta function. The positive definiteness holds for positive divisors in general. C) In the topological case h(x)=omega(x), the energy E[G]=str(L) = str(g) = sum_x,y g(x,y)=chi(G) is the Euler characteristic of G and phi(G)=prod_x omega(x), a product identity which holds for arbitrary set of sets. D) For h(x)=t^|x| with some parameter t we have E[H]=1-f_H(t) with f_H(t)=1+f_0 t + cdots + f_d t^d+1 for the f-vector of H and L(x,y) = (1-f_W^-(x) cap W^-(y)(t)) and g(x,y)=omega(x) omega(y) (1-f_W^+(x) cap W^+(y)(t)). Now, the inverse of g is g^-1(x,y) = 1-f_W^-(x) cap W^-(y)(t)/t^dim(x cap y) and E[G] = 1-f_G(t)=sum_x,y g(x,y).

math.CO↗
arXiv2019

The counting matrix of a simplicial complex

For a finite abstract simplicial complex G with n sets, define the n x n matrix K(x,y) which is the number of subsimplices in the intersection of x and y. We call it the counting matrix of G. Similarly as the connection matrix L which is L(x,y)=1 if x and y intersect and 0 else, the counting matrix K is unimodular. Actually, K is always in SL(n,Z). The inverse of K has the Green function entries K^(-1)(x,y)=w(x) w(y) |W^+(x) intersected W^+y|, where W^+(x) is the star of x, the sets in G which contain x and w(x)=(-1)^dim(x). The matrix K is always positive definite. The spectra of K and K^(-1) always agree so that the matrix Q=K-K^(-1) has the spectral symmetry spec(Q)=-spec(Q) and the zeta function z(s) summing l(k)^(-s) with eigenvalues l(k) of K satisfies the functional equation z(a+ib)=z(-a+ib). The energy theorem in this case tells that the sum of the matrix elements of K^(-1)(x,y) is equal to the number sets in G. In comparison, we had in the connection matrix case the identity that the sum of the matrix elements of L^(-1) is the Euler characteristic of G.

math.CO↗
arXiv2019

The energy of a simplicial complex

A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. This matrix is always unimodular so that the inverse g of L has integer entries g(x,y). In analogy to Laplacians on Euclidean spaces, these Green function entries define a potential energy between two simplices x,y. We prove that the total energy summing all matrix elements g(x,y) is equal to the Euler characteristic X(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to X(G).

math.CO↗
arXiv2019

A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs

Given a locally injective real function g on the vertex set V of a finite simple graph G=(V,E), we prove the Poincare-Hopf formula f_G(t) = 1+t sum_{x in V} f_{S_g(x)}(t), where S_g(x) = { y in S(x), g(y) less than g(x) } and f_G(t)=1+f_0 t + ... + f_{d} t^{d+1} is the f-function encoding the f-vector of a graph G, where f_k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t=-1, the parametric Poincare-Hopf formula reduces to the classical Poincare-Hopf result X(G)=sum_x i_g(x), with integer indices i_g(x)=1-X(S_g(x)) and Euler characteristic X. In the new Poincare-Hopf formula, the indices are integer polynomials and the curvatures K_x(t) expressed as index expectations K_x(t) = E[i_x(t)] are polynomials with rational coefficients. Integrating the Poincare-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f_G(t) = 1+sum_{x} F_{S(x)}(t), where F_G is the anti-derivative of f_G. A similar computation is done for the generating function f_{G,H}(t,s) = sum_{k,l} f_{k,l}(G,H) s^k t^l of the f-intersection matrix f_{k,l}(G,H) counting the number of intersections of k-simplices in G with l-simplices in H. Also here, the computation is reduced to 4 n^2 computations for graphs of half the size: f_{G,H}(t,s) = sum_{v,w} f_{B_g(v),B_g(w)}(t,s) - f_{B_g(v),S_g(w)}(t,s) - f_{S_g(v),B_g(w)}(t,s) + f_{S_g(v),S_g(w)}(t,s), where B_g(v)= S_g(v)+{v} is the unit ball of v.

math.CO↗
arXiv2019

More on Numbers and Graphs

In this note we revisit a "ring of graphs" Q in which the set of finite simple graphs N extend the role of the natural numbers and the signed graphs Z extend the role of the integers. We point out the existence of a norm which allows to complete Q to a real or complex Banach algebra R or C.

math.CO↗
arXiv2019

Dehn-Sommerville from Gauss-Bonnet

We give a zero curvature proof of Dehn-Sommerville for finite simple graphs. It uses a parametrized Gauss-Bonnet formula telling that the curvature of the valuation G to f_G(t)=1+f0 t + ... + fd t^(d+1) defined by the f-vector of G is the anti-derivative F of f evaluated on the unit sphere S(x). Gauss Bonnet is then parametrized, f_G(t) = sum_x F_(S(x))(t), and holds for all simplicial complexes G. The Gauss-Bonnet formula chi(G)=sum_x K(x) for Euler characteristic chi(G) is the special case t=-1. Dehn-Sommerville is equivalent to the reflection symmetry f_G(t)+(-1)^d f_G(-1-t)=0 which is equivalent to the same symmetry for F. Gauss-Bonnet therefore relates Dehn-Sommerville for G with Dehn-Sommerville for the unit spheres S(x), where it is a zero curvature condition. A class X_d of complexes for which Dehn-Sommerville holds is defined inductively by requiring chi(G)=1+(-1)^d and S(x) in X_(d-1) for all x. It starts with X_(-1)={}. Examples are simplicial spheres, including homology spheres, any odd-dimensional discrete manifold, any even-dimensional discrete manifold with chi(G)=2. It also contains non-orientable ones for which Poincar'e-duality fails or stranger spaces like spaces where the unit spheres allow for two disjoint copies of manifolds with chi(G)=1. Dehn-Sommerville is present in the Barycentric limit. It is a symmetry for the Perron-Frobenius eigenvector of the Barycentric refinement operator A. The even eigenfunctions of A^T, the Barycentric Dehn-Sommerville functionals, vanish on X like 22 f1 - 33 f2 + 40 f3 - 45f4=0 for all 4-manifolds.

math.CO↗
arXiv2019

The average simplex cardinality of a finite abstract simplicial complex

We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of a finite abstract simplicial complex G. The functional is a homomorphism from the monoid of simplicial complexes to the rationals: the formula Dim^+(G + H) = Dim^+(G) + Dim^+(H) holds for the join + similarly as for the augmented inductive dimension dim^+(G) = dim(G)+1 where dim is the inductive dimension dim(G) = 1+ sum_x dim(S(x))/|G| with unit sphere S(x) (a recent theorem of Betre and Salinger). In terms of the generating function f(t) = 1+v_0 t + v_1 t^2 + ... +v_d t^(d+1) defined by the f-vector (v_0,v_1, \dots) of G for which f(-1) is the genus 1-X(G) with Euler characteristic X and f(1)=|G|+1 is the augmented number of simplices, the average cardinality is the logarithmic derivative Dim^+(f) = f'(1)/f(1) of f at 1. Beside introducing the average cardinality and establishing its compatibility with arithmetic, we prove two results: 1) the inequality dim^+(G)/2 <= Dim^+(G) with equality for complete complexes. 2) the limit C_d of Dim^+(G_n) for n to infinity is the same for any initial complex G_0 of maximal dimension d and the constant c_d is explicitly given in terms of the Perron-Frobenius eigenfunction of the universal Barycentric refinement operator and is for positive d always a rational number in the open interval ((d+1)/2,d+1).

math.CO↗
arXiv2019

A Reeb sphere theorem in graph theory

We prove a Reeb sphere theorem for finite simple graphs. The result bridges two different definitions of spheres in graph theory. We also reformulate Morse conditions in terms of the center manifolds, the level surface graphs {f=f(x)} in the unit sphere S(x). In the Morse case these graphs are either spheres, the empty graph or the product of two spheres.

math.CO↗
arXiv2018

Cartan's Magic Formula for Simplicial Complexes

Cartan's magic formula L_X = i_X d + d i_X = (d+i_X)^2=D_X^2 relates the exterior derivative d, an interior derivative i_X and its Lie derivative L_X. We use this formula to define a finite dimensional vector space of vector fields X on a finite abstract simplicial complex G. This space has a Lie algebra structure satisfying L_[X,Y] = L_X L_Y - L_Y L_X as in the continuum. Any such vector field X defines a coordinate change on the finite dimensional vector space l^2(G) which play the role of translations along the vector field. If i_X^2=0, the relation L_X=D_X^2 with D_X=i_X+d mirrors the Hodge factorization L=D^2, where D=d+d^* we can see f_t = - L_X f defining the flow of X as the analogue of the heat equation f_t = - L f and view the Newton type equations f'' = -L_X f as the analogue of the wave equation f'' = -L f. Similarly as the wave equation is solved by u(t)=exp(i Dt) u(0) with complex valued u(t)=f(t)-i D^-1 f_t(t), also any second order differential equation f'' = -L_X f is solved by u(t) = exp(i D_X t) u(0) in l^2(G,C}). If X is supported on odd forms, the factorization property L_X = D_X^2 extends to the Lie algebra and i_[X,Y] remains an inner derivative. If the kernel of L_X on p-forms has dimension b_p(X), then the general Euler-Poincare formula holds for every parameter field X. Extreme cases are i_X=d^*, where b_k are the usual Betti numbers and X=0, where b_k=f_k(G) are the components of the f-vector of the simplicial complex G. We also note that the McKean-Singer super-symmetry extends from L to Lie derivatives. It also holds for L_X on Riemannian manifolds. the non-zero spectrum of L_X on even forms is the same than the non-zero spectrum of L_X on odd forms. We also can deform with D_X' = [B_X,D_X] of D_X=d+i_X + b_X, B_X=d_X-d_X^*+i b_X the exterior derivative d governed by the vector field X.

math.CA↗
arXiv2018

Eulerian edge refinements, geodesics, billiards and sphere coloring

A finite simple graph is called a 2-graph if all of its unit spheres S(x) are cyclic graphs of length 4 or larger. A 2-graph G is Eulerian if all vertex degrees of G are even. An edge refinement of a graph splits an edge (a,b) to two edges (a,c),(c,b) and connects the newly added vertex c to the intersection of S(a) with S(b). Theorem I assures that every 2-graph can be rendered Eulerian by successive edge refinements. The construction is explicit using geodesic cutting. After the refinement, we have an Eulerian 2-graph that carries a natural geodesic flow. We construct some ergodic ones. A 2-graph with boundary is finite simple graph for which every unit sphere is either a path graph of length n larger than 1 or a cyclic graph of length larger than 3. 2-balls are special 2-graphs are simply connected with a circular boundary. Theorem II tells that every 2-ball can be edge refined using interior edges to become Eulerian if and only if its boundary has length divisible by 3. A billiard map is defined already if all interior vertices have even degree. We will construct some ergodic billiards in 2-balls, where the geodesics bouncing off at the boundary symmetrically and which visit every interior edge exactly once. A consequence of Theorem II is that an Eulerian billiard which is ergodic must have a boundary length that is divisible by 3. We also construct other 2-graphs like tori with ergodic geodesic flows. This clashes with experience in the continuum, where tori have periodic points minimizing the length in homology classes of paths. Ergodic Eulerian 2-graphs or billiards are exciting because they satisfy a Hopf-Rynov result: there exists a geodesic connection between any two vertices. We get so unique canonical metric associated to any ergodic Eulerian graph. It is non-local in the sense that two adjacent vertices can have large distance.

cs.DM↗
arXiv2018

Combinatorial manifolds are Hamiltonian

Extending a theorem of Whitney of 1931 we prove that all connected d-graphs are Hamiltonian for positive d. A d-graph is a type of combinatorial manifold which is inductively defined as a finite simple graph for which every unit sphere is a (d-1)-sphere. A d-sphere is d-graph such that removing one vertex renders the graph contractible. A graph is contractible if there exists a vertex for which the unit sphere and the graph without that vertex are both contractible. These inductive definitions are primed with the assumptions that the empty graph 0 is the (-1)-sphere and that the one-point graph 1 is the smallest contractible graph. The proof is constructive and shows that unlike for general graphs, the complexity of the construction of Hamiltonian cycles in d-graphs is polynomial in the number of vertices of the graph.

cs.DM↗
arXiv2018

The Cohomology for Wu Characteristics

While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu characteristics w_k(G) = sum_(x_1,x_2,...,x_k) w(x_1)...w(x_k) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More general is the k-intersection number w_k(G_1,...G_k), where x_i in G_i. We define interaction cohomology H^p(G_1,...,G_k) compatible with w_k and invariant under Barycentric subdivison. It allows to distinguish spaces which simplicial cohomology can not: it can identify algebraically the Moebius strip and the cylinder for example. The cohomology satisfies the Kuenneth formula: the Poincare polynomials p_k(t) are ring homomorphisms from the strong ring to the ring of polynomials in t. The Dirac operator D=d+d^* defines the block diagonal Hodge Laplacian L=D^2 which leads to the generalized Hodge correspondence b_p(G)=dim(H^p_k(G)) = dim(ker(L_p)) and Euler-Poincare w_k(G)=sum_p (-1)^p dim(H^p_k(G)) for Wu characteristic. Also, like for traditional simplicial cohomology, isospectral Lax deformation D' = [B(D),D], with B(t)=d(t)-d^*(t)-ib(t), D(t)=d(t)+d(t)^* + b(t) can deform the exterior derivative d. The Brouwer-Lefschetz fixed point theorem generalizes to all Wu characteristics: given an endomorphism T of G, the super trace of its induced map on k'th cohomology defines a Lefschetz number L_k(T). The Brouwer index i_T,k(x_1,...,x_k) = product_j=1^k w(x_j) sign(T|x_j) attached to simplex tuple which is invariant under T leads to the formula L_k(T) = sum_T(x)=x i_T,k(x). For T=Id, the Lefschetz number L_k(Id) is equal to the k'th Wu characteristic w_k(G) of the graph G and the Lefschetz formula reduces to the Euler-Poincare formula for Wu characteristic.

math.CO↗
arXiv2018

The hydrogen identity for Laplacians

For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, where H=(d+d^*)^2 is the sign-less Hodge Laplacian defined by sign-less incidence matrix d and where L is the connection Laplacian. Any spectral information about L directly leads to estimates for the Hodge Laplacian H=(d+d^*)^2 and allows to estimate the spectrum of the Kirchhoff Laplacian H_0=d^* d. The hydrogen identity implies that the random walk u(n) = L^n u with integer n solves the one-dimensional Jacobi equation Delta u=H^2 with (Delta u)(n)=u(n+2)-2 u(n)+u(n-2). Every solution is represented by such a reversible path integral. Over a finite field, we get a reversible cellular automaton. By taking products of complexes such processes can be defined over any lattice Z^r. Since L^2 and L^(-2) are isospectral, by a theorem of Kirby, the matrix L^2 is always similar to a symplectic matrix if the graph has an even number of simplices. The hydrogen relation is robust: any Schrödinger operator K close to H with the same support can still can be written as $K=L-L^{-1}$ where both L(x,y) and L^-1(x,y) are zero if x and y do not intersect.

math.SP↗
arXiv2018

Listening to the cohomology of graphs

We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite simple Barycentric refined graph and the spectrum of the connection Laplacian L of G determine each other: we prove that L-L^(-1) is similar to the Hodge Laplacian H of G which is in one dimensions the direct sum of the Kirchhoff Laplacian H0 and its 1-form analog H1. The spectrum of a single choice of H0,H1 or H alone determines the Betti numbers b0,b1 of G as well as the spectrum of the other matrices. It follows that b0 is the number of eigenvalues 1 of L and that b1 is the number of eigenvalues -1 of L. For a general abstract finite simplicial complex G, we express the matrix entries g(x,y) = w(x) w(y) X( St(x) cap St(y) ) of the inverse of L using stars St(x)= { z in G | x subset of z } of x and w(x)=(-1)^dim(x) and Euler characteristic X. One can see W+(x)=St(x) and W-(x)={ z in G | z subset x } as stable and unstable manifolds of a simplex x in G and g(x,y) =w(x) w(y) X(W+(x) cap W+(y)) as heteroclinic intersection numbers or curvatures and the identity L g=1 as a collection of Gauss-Bonnet formulas. The homoclinic energy w(x)=X(W+(x) cap W-(x)) by definition adds up to X(G). The matrix M(x,y)=w(x) w(y) X(W-(x) cap W-(y)) is similar to L(x,y)=X(W-(x) cap W-(y)). The sum of the matrix entries of M is the definition of Wu characteristic. For dimension 2 and higher we don't know yet how to recover the Betti numbers from the eigenvalues of the matrix H or from L. So far, it can only be obtained from a collection of block matrices, via the Hodge relations b_k = dim(H_k). A natural conjecture is that for a Barycentric refinement of a complex G, the spectrum of L determines the Betti vector. We know this now in one dimensions.

cs.DM↗
arXiv2018

An Elementary Dyadic Riemann Hypothesis

The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As a consequence of the spectral formula chi(G)=sum_x (-1)^dim(x) = p(G)-n(G), where p(G) is the number of positive eigenvalues and n(G) is the number of negative eigenvalues of L, both the Euler characteristic chi(G)=zeta(0)-2 i zeta'(0)/pi as well as determinant det(L)=e^zeta'(0)/pi can be written in terms of zeta. (II) As a consequence of the generalized Cauchy-Binet formula for the coefficients of the characteristic polynomials of a product of matrices we show that for every one-dimensional simplicial complex G, the functional equation zeta(s)=zeta(-s) holds, where zeta(s) is the Zeta function of the positive definite squared connection operator L^2 of G. Equivalently, the spectrum sigma of the integer matrix L^2 for a 1-dimensional complex always satisfies the symmetry sigma = 1/sigma and the characteristic polynomial of L^2 is palindromic. The functional equation extends to products of one-dimensional complexes. (III) Explicit expressions for the spectrum of circular connection Laplacian lead to an explicit entire zeta function in the Barycentric limit. The situation is simpler than in the Hodge Laplacian H=D^2 case where no functional equation was available. In the connection Laplacian case, the limiting zeta function is a generalized hypergeometric function which for an integer s is given by an elliptic integral over the real elliptic curve w^2=(1+z)(1-z)(z^2-4z-1), which has the analytic involutive symmetry (z,w) to (1/z,w/z^2).

math.CO↗
arXiv2017

One can hear the Euler characteristic of a simplicial complex

We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite abstract simplicial complex G matches the number b of even dimensional simplices in G and that the number n of negative eigenvalues matches the number f of odd-dimensional simplices in G. The Euler characteristic X(G) of G therefore can be spectrally described as X(G)=p-n. This is in contrast to the more classical Hodge Laplacian H which acts on the same Hilbert space, where X(G) is not yet known to be accessible from the spectrum of H. Given an ordering of G coming from a build-up as a CW complex, every simplex x in G is now associated to a unique eigenvector of L and the correspondence is computable. The Euler characteristic is now not only the potential energy summing over all g(x,y) with g=L^{-1} but also agrees with a logarithmic energy tr(log(i L)) 2/(i pi) of the spectrum of L. We also give here examples of L-isospectral but non-isomorphic abstract finite simplicial complexes. One example shows that we can not hear the cohomology of the complex.

math.CO↗
arXiv2017

On Atiyah-Singer and Atiyah-Bott for finite abstract simplicial complexes

A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an analytic index dim(ker(D)) -dim(ker(D^*)) of a differential complex D:E -> F. In the discrete, a complex D can be called elliptic if a McKean-Singer spectral symmetry applies as this implies str(exp(-t D^2)) is t-independent. In that case, the analytic index of D is the sum of (-1)^k b_k(D), where b_k(D) is the k'th Betti number, which by Hodge is the nullity of the (k+1)'th block of the Hodge operator L=D^2. It can also be written as a topological index summing K(v) over the set of zero-dimensional simplices in G and where K is an Euler type curvature defined by G and D. This can be interpreted as a Atiyah-Singer type correspondence between analytic and topological index. Examples are the de Rham differential complex for the Euler characteristic X(G) or the connection differential complex for Wu characteristic w_k(G). Given an endomorphism T of an elliptic complex, the Lefschetz number X(T,G,D) is defined as the super trace of T acting on cohomology defined by E. It is equal to the sum i(v) over V which are contained in fixed simplices of T, and i is a Brouwer type index. This Atiyah-Bott result generalizes the Brouwer-Lefschetz fixed point theorem for an endomorphism of the simplicial complex G. In both the static and dynamic setting, the proof is done by heat deforming the Koopman operator U(T) to get the cohomological picture str(exp(-t D^2) U(T)) in the limit t to infinity and then use Hodge, and then by applying a discrete gradient flow to the simplex data defining the valuation to push str(U(T)) to V, getting curvature K(v) or the Brouwer type index i(v).

math.GN↗