Oliver Knill

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Mathematician at Harvard University since 2000, Research Fellowship at University of Texas at Austin from 1997-2000

Remarks about the Arithmetic of Graphs

The arithmetic of N, Z, Q, R can be extended to a graph arithmetic where...
06/18/2021 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Graph complements of circular graphs

Graph complements G(n) of cyclic graphs are circulant, vertex-transitive...
01/18/2021 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Complexes, Graphs, Homotopy, Products and Shannon Capacity

A finite abstract simplicial complex G defines the Barycentric refinemen...
12/14/2020 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Green functions of Energized complexes

If h is a ring-valued function on a simplicial complex G we can define t...
10/19/2020 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Constant index expectation curvature for graphs or Riemannian manifolds

An integral geometric curvature is defined as the index expectation K(x)...
12/24/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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More on Poincare-Hopf and Gauss-Bonnet

We illustrate connections between differential geometry on finite simple...
12/02/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Poincare Hopf for vector fields on graphs

We generalize the Poincare-Hopf theorem sum_v i(v) = X(G) to vector fiel...
11/11/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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A simple sphere theorem for graphs

A finite simple graph G is declared to have positive curvature if every ...
10/07/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Energized simplicial complexes

For a simplicial complex with n sets, let W^-(x) be the set of sets in G...
08/19/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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The energy of a simplicial complex

A finite abstract simplicial complex G defines a matrix L, where L(x,y)=...
07/07/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs

Given a locally injective real function g on the vertex set V of a finit...
06/15/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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More on Numbers and Graphs

In this note we revisit a "ring of graphs" Q in which the set of finite ...
05/31/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Dehn-Sommerville from Gauss-Bonnet

We give a zero curvature proof of Dehn-Sommerville for finite simple gra...
05/13/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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The average simplex cardinality of a finite abstract simplicial complex

We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of...
05/06/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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A Reeb sphere theorem in graph theory

We prove a Reeb sphere theorem for finite simple graphs. The result brid...
03/25/2019 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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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...
11/26/2018 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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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...
08/22/2018 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Combinatorial manifolds are Hamiltonian

Extending a theorem of Whitney of 1931 we prove that all connected d-gra...
06/17/2018 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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The Cohomology for Wu Characteristics

While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu ch...
03/19/2018 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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The hydrogen identity for Laplacians

For any finite simple graph G, the hydrogen identity H=L-L^(-1) holds, w...
03/05/2018 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Listening to the cohomology of graphs

We prove that the spectrum of the Kirchhoff Laplacian H0 of a finite sim...
02/05/2018 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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An Elementary Dyadic Riemann Hypothesis

The connection zeta function of a finite abstract simplicial complex G i...
01/15/2018 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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One can hear the Euler characteristic of a simplicial complex

We prove that that the number p of positive eigenvalues of the connectio...
11/27/2017 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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

A linear or multi-linear valuation on a finite abstract simplicial compl...
08/21/2017 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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The strong ring of simplicial complexes

We define a ring R of geometric objects G generated by finite abstract s...
08/05/2017 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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On a Dehn-Sommerville functional for simplicial complexes

Assume G is a finite abstract simplicial complex with f-vector (v0,v1, ....
05/30/2017 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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A structure from motion inequality

We state an elementary inequality for the structure from motion problem ...
08/18/2007 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Space and camera path reconstruction for omni-directional vision

In this paper, we address the inverse problem of reconstructing a scene ...
08/17/2007 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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On Ullman's theorem in computer vision

Both in the plane and in space, we invert the nonlinear Ullman transform...
08/17/2007 ∙ by Oliver Knill, et al. ∙ 0 ∙ share

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Oliver Knill

Featured Co-authors

Jose Ramirez-Herran

Remarks about the Arithmetic of Graphs

Graph complements of circular graphs

Complexes, Graphs, Homotopy, Products and Shannon Capacity

Green functions of Energized complexes

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 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

The average simplex cardinality of a finite abstract simplicial complex

A Reeb sphere theorem in graph theory

Cartan's Magic Formula for Simplicial Complexes

Eulerian edge refinements, geodesics, billiards and sphere coloring

Combinatorial manifolds are Hamiltonian

The Cohomology for Wu Characteristics

The 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 a Dehn-Sommerville functional for simplicial complexes

A structure from motion inequality

Space and camera path reconstruction for omni-directional vision

On Ullman's theorem in computer vision