Clayton Shonkwiler » Research

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Background

I passed my orals in Differential Geometry (major) and Logic & Finite Model Theory (minor) in April, 2006. My committee consisted of Herman Gluck (chair), Chris Croke and Scott Weinstein. You can read my orals syllabus and the actual questions I was asked. My advisor is Herman Gluck and I have a CV.

My work is mostly focused on topics near the intersection of topology and differential geometry. For most of the last couple years I have worked on topics in geometric link theory, especially generalizations of the Gauss linking integral. I also dabble a bit in low-dimensional topology and contact geometry. Of late I have become interested in some peculiarities of the Hodge Decomposition Theorem for manifolds with boundary.

Papers

Some papers I’ve written:

• “Some examples of Poincaré Duality angles”. Preprint, 2008.
• “Triple linking numbers, Hopf invariants and integral formulas for three-component links”, with Dennis DeTurck, Herman Gluck, Rafal Komendarczyk, Paul Melvin and David Shea Vela-Vick. In preparation.
• “Higher-dimensional linking integrals”, with David Shea Vela-Vick. Submitted, 2008. arXiv:0801.4022 [math.GT]

Talks

These are talks I’ve given on various subjects.

• The triple linking number is an ambiguous Hopf invariant — Geometry–Topology Reading Seminar, University of Pennsylvania, Apr. 15, 2008. See also the version with overlays.
• What is a Poincaré Duality angle? — Graduate Student Geometry–Topology Seminar, University of Pennsylvania, Apr. 2, 2008.
• Higher-dimensional linking integrals — 2008 Graduate Student Topology Conference, University of Illinois, March 29, 2008. See also the version with overlays.
• Higher-dimensional linking integrals — Philadelphia Area Contact/Topology Seminar, Bryn Mawr College, Feb. 14, 2008.
• The classification of links up to link-homotopy (4 parts) — Philadelphia Area Contact/Topology Seminar, Bryn Mawr College, Nov. 8‒Dec. 13, 2007.
• Link complements and the classification of links up to link-homotopy — Graduate Student Geometry–Topology Seminar, University of Pennsylvania, Oct. 31, 2007.
• Geometric linking integrals in Sⁿ × R^m — Pizza Seminar, University of Pennsylvania, Oct. 12, 2007.
• Introduction to Minimal Surfaces — Pre-Colloquium Talk, University of Pennsylvania, Oct. 18, 2006. See also the dynamic slide version and the versions without embedded video: dynamic and printable.
• The Four Vertex Theorem and its Converse — Pizza Seminar, University of Pennsylvania, Oct. 6, 2006. See also the dynamic slide version.
• The Gauss Linking Integral in S³ and H³ — Graduate Student Geometry–Topology Seminar, University of Pennsylvania, Sept. 27, 2006
• Four Isoperimetric Properties of Homogeneous Spherical Membranes — Graduate Student Geometry–Topology Seminar, University of Pennsylvania, Dec. 7, 2005
• Pictures and Syzygies: An exploration of pictures, cellular models and free resolutions — Senior Talk, The University of the South, April 2003
• Picture groups for links — REU final presentation, Louisiana State University, August 2002

Notes

Here are a few notes I’ve written up in the course of my work. The quality of writing isn’t particularly high, but they give some idea of what I’m up to these days.

• Linking integrals via invariant forms on USⁿ — Some examples of computing Lk(K, L) − Lk(K, −L) for submanifolds K and L of odd spheres by way of invariant forms on the unit tangent bundle.
• Convolution linking integrals via invariant forms on USⁿ and UHⁿ — An extension of the previous note which proves a general convolution formula for Lk(K, L) − Lk(K, −L) on all odd spheres. At the end I include a conjecture for a similar formula for Lk(K, L) in hyperbolic n-space for odd n.
• Linking integrals on even spheres via invariant forms on USⁿ — A proof that essentially the same convolution formula for linking as above holds in even spheres. This gives rise to a general formula for Lk(K, L) + (−1)ⁿ Lk(K, −L) for closed, connected, oriented submanifolds of Sⁿ for all n.
• Linking integrals on Sⁿ — The above three notes, packaged into a single document and slightly re-written in a more publication-friendly style.
• An integral formula for μ₁₂₃ — An argument in the style of Polyak & Viro that yields a (nasty) integral formula for the Milnor invariant of a 3-component link. Note: there are good reasons for thinking there is a problem with this argument, so use with caution.
• A tangle picture of an arbitrary 3-component link — A quick proof that, up to link homotopy, any 3-component link is of a given standard form.
• Principal angles in terms of inner products — A technique for determining the principal angles between two k-planes using only the inner products between basis vectors for the k-planes.
• Some examples of Poincaré Duality angles — Computations of Poincaré Duality angles in some example Riemannian manifolds with boundary.
• cp2negcurv
• s2xs2-diagonal