Oliver Knill | Harvard University (original) (raw)

Papers by Oliver Knill

arXiv: Differential Geometry, 2020

We look at the functional Y(M) = int_M K(x) dV(x) for compact Riemannian 2d-manifolds M, where K(... more We look at the functional Y(M) = int_M K(x) dV(x) for compact Riemannian 2d-manifolds M, where K(x) = C sum_p prod_{k=1}^d K_(p(2k-1),p(2k)) with 1/C=d!(4pi)^d involves the sectional curvatures K_(ij)(x) of an orthonormal frame in the tangent space T_xM and sums over all permutations p of {1,...,2d}. Like the Gauss-Bonnet-Chern integrand which integrates up to Euler characteristic X(M), the curvature K is coordinate independent. Like X(M), also Y(M) is metric independent so that delta(M)=Y(M)-X(M) is a topological invariant of the manifold. We prove delta(M) >= 0. In the surface case d=1, the curvature K is the Gauss-curvature and Y(M)=X(M). We also have Y(M)=X(M) for spheres or for 2d-manifold which is a product of 2-manifolds or then for the 6-manifold SO(4), where Y(M)=0 implying that SO(4) as well as its universal cover Spin(4) can not carry a metric of positive curvature. For the simply-connected non-negative curvature 8-manifold SU(3) we have Y(M)>0 and X(M)=0. We define...

arXiv (Cornell University), Jun 21, 2020

A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with ci... more A theorem of Grove and Searle directly establishes that positive curvature 2d manifolds M with circular symmetry group of dimension 2d ≤ 8 have positive Euler characteristic χ(M): the fixed point set N consists of even dimensional positive curvature manifolds and has the Euler characteristic χ(N) = χ(M). It is not empty by Berger. If N has a co-dimension 2 component, Grove-Searle forces M to be in {RP 2d , S 2d , CP d }. By Frankel, there can be not two codimension 2 cases. In the remaining cases, Gauss-Bonnet-Chern forces all to have positive Euler characteristic. This simple proof does not quite reach the record 2d ≤ 10 which uses methods of Wilking but it motivates to analyze the structure of fixed point components N and in particular to look at positive curvature manifolds which admit a U (1) or SU (2) symmetry with connected or almost connected fixed point set N. They have amazing geodesic properties: the fixed point manifold N agrees with the caustic of each of its points and the geodesic flow is integrable. In full generality, the Lefschetz fixed point property χ(N) = χ(M) and Frankel's dimension theorem dim(M) < dim(N k) + dim(N l) for two different connectivity components of N produce already heavy constraints in building up M from smaller components. It is possible that S 2d , RP 2d , CP d , HP d , OP 2 , W 6 , E 6 , W 12 , W 24 are actually a complete list of even-dimensional positive curvature manifolds admitting a continuum symmetry. Aside from the projective spaces, the Euler characteristic of the known cases is always 1, 2 or 6, where the jump from 2 to 6 happened with the Wallach or Eschenburg manifolds W 6 , E 6 which have four fixed point components N = S 2 + S 2 + S 0 , the only known case which are not of the Grove-Searle form N = N 1 or N = N 1 + {p} with connected N 1 .

arXiv (Cornell University), Nov 26, 2017

We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite... more 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 χ(G) of G therefore can be spectrally described as χ(G) = p − n. This is in contrast to the more classical Hodge Laplacian H which acts on the same Hilbert space, where χ(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 ∈ 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 x∈G y∈G g(x, y) with g = L −1 but also agrees with a logarithmic energy tr(log(iL))2/(iπ) of the spectrum of L. We also give here examples of isospectral but non-isomorphic abstract finite simplicial complexes. One example shows that we can not hear the cohomology of the complex.

arXiv (Cornell University), Jan 19, 2020

Index expectation curvature K(x) = E[i f (x)] on a compact Riemannian 2dmanifold M is an expectat... more Index expectation curvature K(x) = E[i f (x)] on a compact Riemannian 2dmanifold M is an expectation of Poincaré-Hopf indices i f (x) and so satisfies the Gauss-Bonnet relation M K(x) dV (x) = χ(M). Unlike the Gauss-Bonnet-Chern integrand, these curvatures are in general non-local. We show that for small 2d-manifolds M with boundary embedded in a parallelizable 2d-manifold N of definite sectional curvature sign e, an index expectation K(x) with definite sign e d exists. The function K(x) is constructed as a product k K k (x) of sectional index expectation curvature averages E[i k (x)] of a probability space of Morse functions f for which i f (x) = i k (x), where the i k are independent and so uncorrelated.

arXiv (Cornell University), Mar 4, 2018

For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identit... more For any 1-dimensional simplicial complex G defined by a finite simple graph, the hydrogen identity |H| = L−L −1 holds, where |H| = (|d| + |d| *) 2 is the sign-less Hodge Laplacian defined by the sign-less incidence matrix |d| and where L is the connection Laplacian. Having linked the Laplacian spectral radius ρ of G with the spectral radius of the adjacency matrix its connection graph G allows for every k to estimate ρ ≤ r k − 1/r k , where r k = 1 + (P (k)) 1/k and P (k) = max x P (k, x), where P (k, x) is the number of paths of length k starting at a vertex x in G. The limit r k − 1/r k for k → ∞ is the spectral radius ρ of |H| which by Wielandt is an upper bound for the spectral radius ρ of H = (d + d *) 2 , with equality if G is bipartite. We can relate so the growth rate of the random walks in the line graph G L of G with the one in the connection graph G of G. The hydrogen identity implies that the random walk ψ(n) = L n ψ on the connection graph G with integer n solves the 1-dimensional Jacobi equation ∆ψ = |H| 2 ψ with ∆u(n) = u(n + 2) − 2u(n) + u(n − 2) and assures that every solution is represented by such a reversible path integral. The hydrogen identity also holds over any finite field F. There, the dynamics L n ψ with n ∈ Z is a reversible cellular automaton with alphabet F G. By taking products of simplicial complexes, such processes can be defined over any lattice Z r. Since L 2 and L −2 are isospectral, by a theorem of Kirby, L 2 is always similar to a symplectic matrix if the graph has an even number of simplices. By the implicit function theorem, the hydrogen relation is robust in the following sense: any matrix K with the same support than |H| can still be written as K = L − L −1 with a connection Laplacian satisfying L(x, y) = L −1 (x, y) = 0 if x ∩ y = ∅.

arXiv (Cornell University), Aug 9, 2015

Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the... more Given a finite simple graph G, let G1 be its barycentric refinement: it is the graph in which the vertices are the complete subgraphs of G and in which two such subgraphs are connected, if one is contained into the other. If λ0 = 0 ≤ λ1 ≤ λ2 ≤ • • • ≤ λn are the eigenvalues of the Laplacian of G, define the spectral function F (x) = λ [nx] on the interval [0, 1], where [r] is the floor function giving the largest integer smaller or equal than r. The graph G1 is known to be homotopic to G with Euler characteristic χ(G1) = χ(G) and dim(G1) ≥ dim(G). Let Gm be the sequence of barycentric refinements of G = G0. We prove that for any finite simple graph G, the spectral functions FG m of successive refinements converge for m → ∞ uniformly on compact subsets of (0, 1) and exponentially fast to a universal limiting eigenvalue distribution function F d which only depends on the clique number respectively the dimension d of the largest complete subgraph of G and not on the starting graph G. In the case d = 1, where we deal with graphs without triangles, the limiting distribution is the smooth function F (x) = 4 sin 2 (πx/2). This is related to the Julia set of the quadratic map T (z) = 4z − z 2 which has the one dimensional Julia set [0, 4] and F satisfies T (F (k/n)) = F (2k/n) as the Laplacians satisfy such a renormalization recursion. The spectral density in the d = 1 case is then the arc-sin distribution which is the equilibrium measure on the Julia set. In higher dimensions, where the limiting function F still remains unidentified, F appears to have a discrete or singular component. We don't know whether there is an analogue renormalization in d ≥ 2. The limiting distribution has relations with the limiting vertex degree distribution and so in 2 dimensions with the graph curvature distribution of the refinements Gm.

arXiv (Cornell University), Jun 15, 2019

Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E)... more Given a locally injective real function g on the vertex set V of a finite simple graph G = (V, E), we prove the Poincaré-Hopf formula f G (t) = 1 + t x∈V f Sg(x) (t), where S g (x) = {y ∈ S(x), g(y) < 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 Poincaré-Hopf formula reduces to the classical Poincaré-Hopf result [5] χ(G) = x i g (x), with integer indices i g (x) = 1−χ(S g (x)) and Euler characteristic χ. In the new Poincaré-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 over Q. Integrating the Poincaré-Hopf formula over probability spaces of functions g gives Gauss-Bonnet formulas like f G (t) = 1+ x F S(x) (t), where F G (t) is the anti-derivative of f [4, 14]. A similar computation holds for the generating function f G,H (t, s) = 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 4n 2 computations for graphs of half the size: f G,H (t, s) = v,w f Bg(v),Bg(w) (t, s)− f Bg(v),Sg(w) (t, s) − f Sg(v),Bg(w) (t, s) + f Sg(v),Sg(w) (t, s), where B g (v) = S g (v) + {v} is the unit ball of v.

arXiv (Cornell University), Jul 7, 2019

A finite abstract simplicial complex G defines a matrix L, where L(x, y) = 1 if two simplicies x,... more 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 = L −1 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 E(G) = x,y g(x, y) is equal to the Euler characteristic χ(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to χ(G).

arXiv (Cornell University), Dec 1, 2019

We illustrate connections between differential geometry on finite simple graphs G = (V, E) and Ri... more We illustrate connections between differential geometry on finite simple graphs G = (V, E) and Riemannian manifolds (M, g). The link is that curvature can be defined integral geometrically as an expectation in a probability space of Poincaré-Hopf indices of coloring or Morse functions. Regge calculus with an isometric Nash embedding links then the Gauss-Bonnet-Chern integrand of a Riemannian manifold with the graph curvature. There is also a direct nonstandard approach [18]: if V is a finite set containing all standard points of M and E contains pairs which are closer than some positive number. One gets so finite simple graphs (V, E) which leads to the standard curvature. The probabilistic approach is an umbrella framework which covers discrete spaces, piecewise linear spaces, manifolds or varieties.

arXiv (Cornell University), Oct 18, 2020

G to a ring K with conjugation x * defines χ(A) = x∈A h(x) and ω(G) = x,y∈G,x∩y =∅ h(x) * h(y). D... more G to a ring K with conjugation x * defines χ(A) = x∈A h(x) and ω(G) = x,y∈G,x∩y =∅ h(x) * h(y). Define L(x, y) = χ(W − (x) ∩ W − (y)) and g(x, y) = ω(x)ω(y)χ(W + (x) ∩ W + (y)), where W − (x) = {z | z ⊂ x}, W + (x) = {z | x ⊂ z} and ω(x) = (−1) dim(x) with dim(x) = |x| − 1. 1.2. The following relation [8] only requires the addition in K Theorem 1. χ(G) = x,y∈G g(x, y) 1.3. The next new quadratic energy relation links simplex interaction with multiplication in K. Define |h| 2 = h * h = N(h) in K. Theorem 2. ω(G) = x,y∈G ω(x)ω(y)|g(x, y)| 2. 1.4. The next determinant identity holds if h maps G to a division algebra K and det is the Dieudonné determinant [1]. The geometry G can here be a finite set of sets and does not need the simplical complex axiom stating that G is closed under the operation of taking non-empty finite subsets. Theorem 3. det(L) = det(g) = x∈G h(x). 1.5. If h : G → K takes values in the units U(K) of K, like i.e. Z 2 , U(1), SU(2), S 7 of the division algebras R, C, H, O, the unitary group U(H)∩K of an operator C *-algebra K ⊂ B(H) for some Hilbert space H or the units in a ring K = O K of integers of a number field K, and if G is a simplicial complex, then: Theorem 4. If h(x) * h(x) = 1 for all x ∈ G, then g * = L −1 .

arXiv (Cornell University), May 30, 2019

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

arXiv (Cornell University), Aug 20, 2017

A linear or multi-linear valuation on a finite abstract simplicial complex can be expressed as an... more 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(e −tD 2) is t-independent. In that case, the analytic index of D is χ(G, D) = k (−1) k b k (D), where b k 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 v∈V K(v), where V is 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 χ(G) or the connection differential complex for Wu characteristic ω k (G). Given an endomorphism T of an elliptic complex, the Lefschetz number χ(T, G, D) is defined as the super trace of T acting on cohomology defined by D and G. It is equal to the sum v∈V i(v), where V is the set of zerodimensional simplices 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(e −tD 2 U (T)) in the limit t → ∞ 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 the zero dimensional set V , getting curvature K(v) or the Brouwer type index i(v).

arXiv (Cornell University), Aug 21, 2016

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomo... more The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtration is related to the Mertens function, the Poincaré-Hopf indices at critical points correspond to the values of the Moebius function. The Morse inequalities link number theoretical quantities like the prime counting functions relevant for the distribution of primes with cohomological properties of the graphs. The just given picture is a special case of a discrete Morse cohomology equivalent to simplicial cohomology. The special example considered here is a case where the graph is the Barycentric refinement of a finite simple graph.

arXiv (Cornell University), Jul 19, 2021

We show that the curvature K G * H (x, y) at a point (x, y) in the strong product G * H of two fi... more We show that the curvature K G * H (x, y) at a point (x, y) in the strong product G * H of two finite simple graphs is equal to the product K G (x)K H (y) of the curvatures.

arXiv (Cornell University), Jun 18, 2021

The arithmetic of N ⊂ Z ⊂ Q ⊂ R can be extended to a graph arithmetic N ⊂ Z ⊂ Q ⊂ R, where N is t... more The arithmetic of N ⊂ Z ⊂ Q ⊂ R can be extended to a graph arithmetic N ⊂ Z ⊂ Q ⊂ R, where N is the semi-ring of finite simple graphs and where Z, Q are integral domains culminating in a Banach algebra R. An extension of Q with a single network completes to the Wiener algebra A(T). We illustrate the compatibility with topology and spectral theory. Multiplicative linear functionals like Euler characteristic, the Poincaré polynomial, zeta functions can be extended naturally. These functionals can also help with number theoretical questions. The story of primes is a bit different as the new integers Z are not a unique factorization domain, because there are many additive primes and because a simple sieving argument shows that most graphs in Z are multiplicative primes unlike in Z, where most are not.

arXiv (Cornell University), Jan 18, 2021

In this report, we study graph complements Gn of cyclic graphs Cn or graph complements G + n of p... more In this report, we study graph complements Gn of cyclic graphs Cn or graph complements G + n of path graphs. Gn are circulant, vertex-transitive, claw-free, strongly regular, Hamiltonian graphs with a Zn symmetry and Shannon capacity 2. Also the Wiener and Harary index are known. The explicitly known adjacency matrix spectrum leads to explicit spectral zeta function and tree or forest quantities. The forest-tree ratio of Gn converge to e in the limit when n goes to infinity. The graphs Gn are all Cayley graphs and so Platonic in the sense that they all have isomorphic unit spheres G + n−3. The graphs G 3d+3 are homotop to wedge sums of two d-spheres and G 3d+2 , G 3d+4 are homotop to d-spheres, G + 3d+1 are contractible, G + 3d+2 , G + 3d+3 are homotop to d-spheres. Since disjoint unions are dual to Zykov joins, graph complements of all 1-dimensional discrete manifolds G are homotop to either a point, a sphere or a wedge sums of spheres. If the length of every connected component of a 1-manifold is not divisible by 3, the graph complement of G must be a sphere. In general, the graph complement of a forest is either contractible or a sphere. It also follows that all induced strict subgraphs of Gn are either contractible or homotop to spheres. The f-vectors Gn or G + n satisfy a hyper Pascal triangle relation, the total number of simplices are hyper Fibonacci numbers. The simplex generating functions are Jacobsthal polynomials, generating functions of k-king configurations on a circular chess board. While the Euler curvature of circle complements Gn is constant by symmetry, the discrete Gauss-Bonnet curvature of path complements G + n can be expressed explicitly from the generating functions. There is now a non-trivial 6-periodic Gauss-Bonnet curvature universality in the complement of Barycentric limits. The Brouwer-Lefschetz fixed point theorem produces a 12-periodicity of the Lefschetz numbers of all graph automorphisms of Gn. There is also a 12-periodicity of Wu characteristic. This corresponds to 4-periodicity in dimension as n → n + 3 is homotop to a suspension. These are all manifestations of stable homotopy features, but purely combinatorial.

arXiv (Cornell University), Jun 17, 2018

Extending a theorem of Whitney of 1931 we prove that all connected d-graphs are Hamiltonian for d... more Extending a theorem of Whitney of 1931 we prove that all connected d-graphs are Hamiltonian for d ≥ 1. 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 its 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.

arXiv (Cornell University), Mar 20, 2017

arXiv (Cornell University), Jan 6, 2020

The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has... more The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic χ(M) > 0 and that in the case of negative curvature χ(M)(−1) d > 0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S 2 ×S 2. By formulating curvature integral geometrically, these questions can be explored for finite simple graphs, where it leads to linear programming problems. In this more expository document we aim to explore also a bit of the history of the Hopf conjecture and mention some strategies of attacks which have been tried. We illustrate the new integral theoretic µ curvature concept by proving that for every positive curvature manifold M there exists a µ-curvature K satisfying Gauss-Bonnet-Chern χ(M) = M K dV such that K is positive on an open set U of volume arbitrary close to the volume of M .

arXiv (Cornell University), Jul 22, 2019

For a finite abstract simplicial complex G with n sets, define the n×n matrix K(x, y) = |W − (x) ... more For a finite abstract simplicial complex G with n sets, define the n×n matrix K(x, y) = |W − (x) ∩ W − (y)| which is the number of subsimplices in x ∩ 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) = ω(x)ω(y)|W + (x) ∩ W + y|, where W + (x) is the star of x, the sets in G which contain 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 σ(Q) = −σ(Q) and the zeta function ζ(s) = n k=1 λ −s k defined by the eigenvalues λ k of K satisfies the functional equation ζ(a+ib) = ζ(−a+ib). The energy theorem in this case tells that the total potential energy is x,y K −1 (x, y) = |G| = x 1 is the number sets in G. In comparison, we had in the connection matrix case the identity x,y L −1 (x, y) = χ(G) = x ω(x). K(x, y) = |W − (x) ∩ W − (y)| = 2 |x∩y| − 1 .