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Oliver Knill

Publications and source records attributed to Oliver Knill.

At least 19 recordsLinked to original sources
arXiv2026

On Natural Groups

A group (G,*) is natural if its group operation is uniquely determined by a metric structure (G,d) on G in the following sense: every group structure (G,.) for which all right translations are isometries of (G,d) must be isomorphic to (G,*). Examples included all connected Lie groups or all groups generated by involutions (Sutherland). The orientation rigidity theorem of Leemann and de la Salle allows to upgrade the non-abelian structure theorem: every non-abelian group of cardinality not larger than the continuum that is not generalized dicyclic is natural. A consequence is that all non-metabelian Lie groups, all simple group of cardinality not larger then the continuum, all homeomorphism-, diffeomorphism -or symplectomorphism groups of manifolds, or automorphism groups probability spaces or non-abelian crystallographic groups are natural. Also the abelian structure theorem is extended: affine rigidity of Jarosz, together with Mazur-Ulam's theorem implies that the additive group of every real or complex vector spaces is natural and that all connected abelian Banach Lie groups are natural. A theorem of Babai implies that every Boolean group is natural for every cardinality. Undecided is whether C_p^k is natural for cardinals k>c and odd prime p, and whether the additive group of any field F is natural, or whether the cardinality assumption in the non-abelian structure theorem is needed. A major question is whether G^2=G implies that G is natural. This is already open for abelian groups: does 2G=G imply that G is natural?

math.GR↗
arXiv2026

Elements of finite geometry I

This is a snapshot of a first part on a possibly much longer text on finite geometries, meaning graphs or finite abstract simplicial complexes. In in this first batch we review 12 subjects: Gauss-Bonnet, Poincare-Hopf, Index expectation, Euler's gem, Euler-Poincare,Unimodularity, Brouwer-Lefschetz, Sphere formula, Level sets, Index formula, Quadratic cohomology and Higher characteristic.

math.HO↗
arXiv2026

Euler Characteristics of Random Manifolds

We prove that the expectation of the Euler characteristic X(H) of random level surface H in a given simplicial complex G is E[X(H)] =2-2K(G)-X(G), where K(G)=1-f_0/2+f_1/3- ... is the curvature functional of G and X(G)=f_0-f_1+f_2-... is the Euler characteristics. More generally, the expectation of the f-vector of a submanifold is explicitly linked to the f-vector of the host manifold.

math.CO↗
arXiv2026

Remarks about the Moebius-Kantor graph

The Moebius-Kantor graph MK=G(8,3) is a Cayley graph of three non-abelian groups, the Pauli group P(1), the semi-dihedral group SD(16), as well as the dihedral group D(16) of order 16. In topological graph theory, it illustrates the Heawood number 7 of the torus and leads to the Tucker group Aut(MK), the unique group of genus 2. We compute the Lefschetz numbers to illustrate the Brouwer-Lefschetz fixed point theorem. MK is also the dual of the 2-skeleton complex of the 3-sphere G. The graph represents one of flat Clifford tori of a Hopf fibration in the 3-sphere G=K(2,2,2,2) reflecting that Coxeter saw that MK is a subgraph of the tesseract G*. It carries a metric d so that (MK,d) has only one algebraic group structure (P(1),*) that preserves the metric. It makes the Pauli group natural, similarly as the Moebius ladder M(16) makes the dihedral group D(16) natural, forcing the algebraic structure from the metric structure.

math.GT↗
arXiv2026

Calculus on Wave Fronts

We define a deformation of the exterior derivative that is a bounded operator and preserves the symmetries of the geometry. It satisfies a modified wave equation that honors the strong Huygens principle in all dimensions.

math.AP↗
arXiv2026

Remarks about Connection and Dirac matrices

The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the eigenvalues for subcomplexes. This gives general upper bounds of the eigenvalues both for L and D in terms of inclusion or intersection degrees. We conjecture that L always dominates both D and the inverse of L in a weak Loewner sense. In a second part we look at dynamical systems (G,T), where T is a simplicial map on G. Both L and D generalize to dynamical versions of L and D. The modified L is still unimodular with an explicit Green function inverse and modified Dirac part still comes from an exterior derivative d. We also review the Lefschetz fixed point theorem for a simplicial map T on a simplicial complex G which implies the Brouwer fixed point theorem: any simplicial map on a contractible finite abstract simplicial complex G has a fixed simplex.

math.CO↗
arXiv2026

Density of wave fronts

We prove that wave fronts on a flat torus become dense. As a corollary, wave fronts become dense for a square billiard or for the geodesic flow on the flat Klein bottle or the cube surface.

math.DG↗
arXiv2025

Dehn Sommerville Manifolds

Dehn-Sommerville manifolds are a class of finite abstract simplicial complexes that generalize discrete manifolds. Despite a simpler definition in comparison to manifolds, they still share most properties of manifolds. They especially satisfy all Dehn-Sommerville symmetries telling that half of the f-vector entries are redundant. They also share other properties with q-manifolds: for every Dehn-Sommerville q-manifold G and any function g: V(G) to A={0, ..., k} with positive k, the set of x such that g(x) contains A is a Dehn-Sommerville (q-k)-manifold if not empty. We also see that for Dehn-Sommerville q-manifolds, all higher characteristics w_m(G) agree with Euler characteristic that the chromatic number is bounded above by 2q+2 and that odd-dimensional Dehn-Sommerville manifolds are flat and form a monoid under the join operation. In general, Dehn-Sommerville manifolds are invariant under edge refinement, Barycentric refinement and Cartesian products.

math.CO↗
arXiv2025

Remarks on the Brouwer Conjecture

The Brouwer conjecture (BC) in spectral graph theory claims that the sum of the largest k Kirchhoff eigenvalues of a graph are bounded above by the number m of edges plus k(k+1)/2. We show that (BC) holds for all graphs with n vertices if n is larger or equal than 4 times the square of the maximal vertex degree. We also note that (BC) for graphs implies (BC) for quivers.

math.CO↗
arXiv2025

Interacting Geodesics on Discrete Manifolds

We define an evolution of multiple particles on a discrete manifold $G$. Each particle alone moves on geodesics and particles can interact if they are on the same facet. They move deterministically and reversibly on the frame bundle $P$ of the abstract simplicial complex $G$. Particles are signed and each is represented by a totally ordered maximal simplex $p \in P$ in $G$. The motion of divisors on $P$ also defines a time dependent reversible deformation of space.

math.DS↗
arXiv2025

Geodesics for Discrete manifolds

The geodesic flow on a finite discrete q-manifold with or without boundary is defined as as a permutation of its ordered q-simplices. This allows to define geodesic sheets and a notion of sectional curvature.

math.CO↗
arXiv2025

Soft Barycentric Refinement

The soft Barycentric refinement preserves manifolds with or without boundary. In every dimension larger than one, there is a universal spectral central limiting measure that has affinities with the Barycentric limiting measure one dimension lower. Ricci type quantities like the length of the dual sphere of co-dimension-2 simplex stay invariant under soft refinements. We prove that the dual graphs of any manifold can be colored with 3 colors, which is in the 2-dimensional case a special case of the Groetzsch theorem. It follows that the vertices of a soft Barycentric refined q-manifold G' can be colored by q+1 or q+2 colors.

math.CO↗
arXiv2024

Colorful Rings of Partition

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play together when proving lim p(n)^(1/n)=1 and introduce finite or infinite rings of partitions.

math.HO↗
arXiv2024

On Symmetries of Finite Geometries

The isospectral set of the Dirac matrix D=d+d* consists of orthogonal Q for which Q* D Q is an equivalent Dirac matrix. It can serve as the symmetry of a finite geometry G. The symmetry is a subset of the orthogonal group or unitary group and isospectral Lax deformations produce commuting flows d/dt D=[B(g(D)),D] on this symmetry space. In this note, we remark that like in the Toda case, D_t=Q_t* D_0 Q_t with exp(-t g(D))=Q_t R_t solves the Lax system.

nlin.SI↗
arXiv2024

Gauss-Bonnet for Form Curvatures

We look at curvatures that are supported on k-dimensional parts of a simplicial complex G. These curvature all satisfy the Gauss-Bonnet theorem, provided that the k-dimensional simplices cover $G$. Each of these curvatures can be written as an expectation of Poincare-Hopf indices. Linear or non-linear wave dynamics with discrete or continuous time allow to deform these curvatures while keeping the Gauss-Bonnet property.

math.CO↗
arXiv2024

Fusion inequality for quadratic cohomology

Classical simplicial cohomology on a simplicial complex G deals with functions on simplices x in G. Quadratic cohomology deals with functions on pairs of simplices (x,y) in G x G that intersect. If K,U is a closed-open pair in G, we prove here a quadratic version of the linear fusion inequality. Additional to the quadratic cohomology of G there are five additional interaction cohomology groups. Their Betti numbers are computed from functions on pairs (x,y) of simplices that intersect. Define the Betti vector b(X) computed from pairs (x,y) in X x X with x intersected y in X a and b(X,Y) with pairs in X xY with x intersected y in K. We prove the fusion inequality b(G) <= b(K)+b(U)+b(K,U)+b(U,K)+b(U,U) for cohomology groups linking all five possible interaction cases. Counting shows f(G) = f(K)+f(U) + f(K,U)+f(U,K)+f(U,U) for the f-vectors. Super counting gives Euler-Poincare sum_k (-1)^k f_k(X)=\sum_k (-1)^k b_k(X) and sum_k (-1)^k f_k(X,Y)=sum_k (-1)^k b_k(X,Y) for X,Y in {U,K}. As in the linear case, also the proof of the quadratic fusion inequality follows from the fact that the spectra of all the involved Laplacians L(X),L(X,Y) are bounded above by the spectrum of the quadratic Hodge Laplacian L(G) of G.

math.CO↗
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.

math.CO↗