About

Crichton Ogle is a professor in the Department of Mathematics at The Ohio State University. He holds a PhD from Brandeis University, obtained in 1984. His areas of expertise include algebraic topology and K-theory. His research focuses on algebraic topology, with specific emphasis on applications within algebra. Ogle is involved in the academic community through his role at Ohio State, contributing to the department's teaching and research activities. His contact information includes an email at ogle.1@osu.edu, a phone number (614) 292-8609, and an office located at MW 410, 231 W 18th Ave, Columbus, OH 43210.

Research topics

• Mathematics
• Pure mathematics
• Combinatorics
• Discrete mathematics
• Physics

Selected publications

• Pade Approximants for Geodesy

  arXiv (Cornell University) · 2026-05-05

  preprintOpen accessSenior author

  In this note we analyze the use of Padé approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs are, in essence, expansions in 1/r, i.e., expansions about the point at infinity. Their domain of convergence is generically the exterior of the Brillouin sphere. However, for synthetic models with analytic topography and density the region of convergence may be larger, with the deviation decreasing as the structural complexity of the planet increases.

  PublisherDOI

• Pade Approximants for Geodesy

  ArXiv.org · 2026-05-05

  articleOpen accessSenior author

  In this note we analyze the use of Padé approximants for downward continuation beyond the radius of convergence of spherical harmonic expansions (SHEs), and for identifying the complex singularities of the gravitational potential. SHEs are, in essence, expansions in 1/r, i.e., expansions about the point at infinity. Their domain of convergence is generically the exterior of the Brillouin sphere. However, for synthetic models with analytic topography and density the region of convergence may be larger, with the deviation decreasing as the structural complexity of the planet increases.

  PublisherOA PDF

• Divergence beneath the Brillouin sphere and the phenomenology of prediction error in spherical harmonic series approximations of the gravitational field

  Reports on Progress in Physics · 2024-06-20 · 9 citations

  articleOpen access

  Abstract The Brillouin sphere is defined as the smallest sphere, centered at the origin of the geocentric coordinate system, that incorporates all the condensed matter composing the planet. The Brillouin sphere touches the Earth at a single point, and the radial line that begins at the origin and passes through that point is called the singular radial line. For about 60 years there has been a persistent anxiety about whether or not a spherical harmonic (SH) expansion of the external gravitational potential, V , will converge beneath the Brillouin sphere. Recently, it was proven that the probability of such convergence is zero. One of these proofs provided an asymptotic relation, called Costin’s formula, for the upper bound, E N , on the absolute value of the prediction error, e N , of a SH series model, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>V</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>λ</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> , truncated at some maximum degree, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mo movablelimits="true">max</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math> . When the SH series is restricted to (or projected onto) a particular radial line, it reduces to a Taylor series (TS) in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mn>1</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:math> . Costin’s formula is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mi>N</mml:mi></mml:msub><mml:mo>≃</mml:mo><mml:mi>B</mml:mi><mml:msup><mml:mi>N</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>b</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>R</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math> , where R is the radius of the Brillouin sphere. This formula depends on two positive parameters: b , which controls the decay of error amplitude as a function of N when r is fixed, and a scale factor B . We show here that Costin’s formula derives from a similar asymptotic relation for the upper bound, A n on the absolute value of the TS coefficients, a n , for the same radial line. This formula, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>≃</mml:mo><mml:mi>K</mml:mi><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> , depends on degree, n , and two positive parameters, k and K , that are analogous to b and B . We use synthetic planets, for which we can compute the potential, V , and also the radial component of gravitational acceleration, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>r</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mi>∂</mml:mi><mml:mi>V</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>∂</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math> , to hundreds of significant digits, to validate both of these asymptotic formulas. Let superscript V refer to asymptotic parameters associated with the coefficients and prediction errors for gravitational potential, and superscript g to the coefficients and predictions errors associated with g r . For polyhedral planets of uniform density we show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>7</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:mn>5</mml:mn><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:math> almost everywhere. We show that the frequency of oscillation (around zero) of the TS coefficients and the series prediction errors, for a given radial line, is controlled by the geocentric angle, α , between that radial line and the singular radial line. We also derive useful identities connecting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>K</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>B</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mi>K</mml:mi><mml:mi>g</mml:mi></mml:msup></mml:mrow></mml:math> , and B g . These identities are expressed in terms of quotients of the various scale factors. The only other quantities involved in these identities are α and R . The phenomenology of ‘series divergence’ and prediction error (when r &lt; R ) can be described as a function of the truncation degree, N , or the depth, d , beneath the Brillouin sphere. For a fixed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>r</mml:mi><mml:mtext>⩽</mml:mtext><mml:mi>R</mml:mi></mml:mrow></mml:math> , as N increases from very low values, the upper error bound E N shrinks until it reaches its minimum (best) value when N reaches some particular or optimum value, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> . When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>&gt;</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> , prediction error grows as N continues to increase. Eventually, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>N</mml:mi><mml:mo>≫</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:mi>opt</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> , prediction errors increase exponentially with rising N . If we fix the value of N and allow <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mi>R</mml:mi><mml:mrow><mml:mo>/</mml:mo></mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:math> to vary, then we find that prediction error in free space beneath the Brillouin sphere increases exponentially with depth, d , beneath the Brillouin sphere. Because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:msup><mml:mi>b</mml:mi><mml:mi>g</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mi>V</mml:mi></mml:msup><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> everywhere, divergence driven prediction error intensifies more rapidly for g r than for V , both in terms of its dependence on N and d . If we fix both N and d , and focus on the ‘lateral’ variations in prediction error, we observe that divergence and prediction error tend to increase (as does B ) as we approach high-amplitude topography.

  PublisherOA PDFDOI

• The uniform homotopy category

  Journal of Pure and Applied Algebra · 2023-05-22 · 1 citations

  articleSenior author
  PublisherDOI

• On the Domain of Convergence of Spherical Harmonic Expansions

  Communications in Mathematical Physics · 2022-01-01 · 11 citations

  article
  PublisherDOI

• Non-convergence of the spherical harmonic expansion of gravitational potential below the Brillouin sphere: The continuous case

  Journal of Mathematical Physics · 2021-10-01 · 2 citations

  articleOpen access1st authorCorresponding

  For a singleton planet, P, with gravitational potential, V, we show that for each ɛ &amp;gt; 0, there exists a planet P′ with gravitational potential V′, with (P′, V′) “ɛ-close” to (P, V) (in an appropriate C0-sense), for which the spherical harmonic expansion of V′ does not extend more than a distance ɛ below the Brillouin sphere of P′.

  PublisherOA PDFDOI

• The Uniform Homotopy Category

  arXiv (Cornell University) · 2021-09-17

  preprintOpen accessSenior author

  This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the respective Lipschitz and uniform settings. Cubical sets and uniform spaces admit the additional compatible structures of categories of (co)fibrant objects. A categorical equivalence between classical homotopy categories of cubical sets and spaces lifts to a full and faithful embedding from an associated Lipschitz homotopy category of cubical sets into an associated uniform homotopy category of uniform spaces. Bounded cubical cohomology generalizes to a representable theory on the Lipschitz homotopy category. Bounded singular cohomology on path-connected spaces generalizes to a representable theory on the uniform homotopy category. Along the way, this paper develops a cubical analogue of Kan's Ex^infinity functor and proves a cubical approximation theorem for uniform maps.

  PublisherOA PDFDOI

• Invertibility in Category Representations

  arXiv (Cornell University) · 2020-10-21 · 1 citations

  preprintOpen accessSenior author

  Inverse categories are categories in which every morphism x has a unique pseudo-inverse y in the sense that xyx=x and yxy=y. Persistence modules from topological data analysis and similarly decomposable category representations factor through inverse categories. This paper gives a numerical condition, decidable when the indexing category is finite, characterizing when a representation of a small category factors through an inverse category.

  PublisherOA PDFDOI

• On the chromatic localization of the homotopy completion tower for\n $\\mathcal{O}$-algebras

  arXiv (Cornell University) · 2020-12-18 · 1 citations

  preprintOpen access1st authorCorresponding

  The completion tower of a nonunital commutative ring is a classical\nconstruction in commutative algebra. In the setting of structured ring spectra\nas modeled by algebras over a spectral operad, the analogous construction is\nthe homotopy completion tower. The purpose of this brief note is to show that\nlocalization with respect to the Johnson-Wilson spectrum $E(n)$ commutes with\nthe terms of this tower.\n

  PublisherOA PDFDOI

• On the chromatic localization of the homotopy completion tower for $\mathcal{O}$-algebras

  arXiv (Cornell University) · 2020-12-18

  preprintOpen access1st authorCorresponding

  The completion tower of a nonunital commutative ring is a classical construction in commutative algebra. In the setting of structured ring spectra as modeled by algebras over a spectral operad, the analogous construction is the homotopy completion tower. The purpose of this brief note is to show that localization with respect to the Johnson-Wilson spectrum $E(n)$ commutes with the terms of this tower.

  PublisherOA PDFDOI

Frequent coauthors

• Ronghui Ji

  Indiana University – Purdue University Indianapolis

  10 shared

• Bobby Ramsey

  The Ohio State University

  8 shared

• Sanjeevi Krishnan

  Sree Balaji Dental College and Hospital

  3 shared

• Andrew Salch

  3 shared

• Jim Fowler

  3 shared

• Ovidiu Costin

  The Ohio State University

  3 shared

• Nikolas Schonsheck

  2 shared

• Rodica D. Costin

  The Ohio State University

  2 shared

Awards & honors

• Graduate Teaching Awards