Research profile

Warner A. Miller

Works on discrete & graph Ricci, geometrization and numerics & visualization.

NP-RC-875092

9
topics: 3
co-authors

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Discrete & graph Ricci Geometrization Numerics & visualization Surgery Rotational symmetry & warped products (Non)collapsing 3-manifolds Ricci flow

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Paul M. Alsing
1 joint worklatest 2017
Shing-Tung Yau
1 joint worklatest 2017
Rory Conboye
1 joint worklatest 2016

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Papers

Their work in this field since 2016, with the older papers the site recommends. Titles open each paper's page, with its abstract, PDF and HTML versions; Connections shows the papers closest to it. The arXiv listing has the complete record.

2017

math.DGSep 21, 2017arXiv:1709.08494

A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery

Paul M. Alsing, Warner A. Miller, Shing-Tung Yau

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure – a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.

Ricci flow Singularities Surgery (Non)collapsing Geometrization 3-manifolds Rotational symmetry & warped products Discrete & graph Ricci Numerics & visualization

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2016

math.DGMar 10, 2016arXiv:1603.03113

Piecewise Flat Curvature and Ricci Flow in Three Dimensions

Rory Conboye, Warner A. Miller

Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each type of curvature, with the integrals based on the parallel transport of vectors around hinges. Computations for triangulations of a diverse set of manifolds show these piecewise flat curvatures to converge to their smooth values. The Ricci curvature also gives a piecewise flat Ricci flow as a fractional rate of change of edge-lengths, again converging to the smooth Ricci flow for the manifolds tested.

Ricci flow Discrete & graph Ricci

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Nearby

Also on these topics

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Shuliang BaiShared topics: Numerics & visualization, Discrete & graph Ricci, Ricci flow
Ke FengShared topics: Geometrization, Discrete & graph Ricci, 3-manifolds
Huabin GeShared topics: Geometrization, Surgery, Discrete & graph Ricci
Wenshuai JiangShared topics: Geometrization, Surgery, Discrete & graph Ricci
Grigori PerelmanShared topics: Geometrization, Surgery, Discrete & graph Ricci

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NP-RC-875092 is Warner A. Miller's permanent number in the Ricci flow community on Neckpinch. It stays with them through any change of name or post. More about these numbers.

A window on one part of a person's work: papers here go back to 2016 and cover Ricci flow and its neighbouring flows, with the older work the site recommends, so this page is no measure of anyone's whole output. Being listed is not an endorsement by or of the person. If two people share this name, their work may appear together here. How it is put together.