Research

The common theme of my research is an interest in the interplay between geometry, topology, differential forms, and vector fields. One thread of my work develops new invariants for Riemannian manifolds with boundary which seem to measure how close the manifold is to being closed and develops connections to the problem of Electrical Impedance Tomography. Another thread is motivated by Arnol′d’s search for higher order helicities of fluid flows and has led to, among other things, a simple new integral formula for Milnor's triple linking number and the definition of “iterated helicities”. Yet another thread is devoted to understanding what Legendrian contact homology can tell us about the geometry of a Legendrian knot.

For a (much) more detailed description of my research interests, please see my research statement.

Publications

(Note: the first appearance of each collaborator’s name is linked to his/her webpage.)

“Legendrian contact homology and nondestabilizability”
Joint with David Shea Vela-Vick.
J. Symplectic Geom. 9 (2011), no. 1, 33–44
arXiv: 0910.3914 [math.GT]
“Higher-dimensional linking integrals”
Joint with David Shea Vela-Vick.
Proc. Amer. Math. Soc. 139 (2011), no. 4, 1511–1519
MR: 2748445; doi: 10.1090/S0002-9939-2010-10603-2; arXiv: 0801.4022 [math.GT]
“Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links”
Joint with Dennis DeTurck, Herman Gluck, Rafal Komendarczyk, Paul Melvin and David Shea Vela-Vick.
Mat. Contemp. 34 (2008) 251–283
MR: 2588614; arXiv: 0901.1612 [math.GT]

Preprints

In Preparation

“Homotopy periods of link maps and μ-invariants of Borromean links”
Joint with Frederick R. Cohen and Rafal Komendarczyk.
“Homotopy invariants of links and Koschorke’s conjecture”
Joint with Frederick R. Cohen and Rafal Komendarczyk.
“Rulings and augmentations for bordered Legendrian knots”
Joint with Joshua Sabloff and David Shea Vela-Vick.

Talks

Invited Talks

Higher Helicities, Geometric Linking Integrals, and Koschorke’s Conjecture — Knots & Applications workshop on Entanglement and Linking, Centro di Ricerca Matematica Ennio de Giorgi, Pisa, Italy, May 18, 2011.
The search for higher helicities — Southeast Geometry Conference, May 8, 2011.
The complete Dirichlet-to-Neumann map for differential forms — Geometry and Topology Seminar, Tulane University, Apr. 14, 2011.
The search for higher helicities — AMS Special Session on Knots, Links, 3-Manifolds, and Physics, Joint Mathematics Meetings, Jan. 8, 2011.
The complete Dirichlet-to-Neumann map for differential forms — Geometry–Topology Seminar, University of Pennsylvania, Dec. 9, 2010.
The search for higher helicities — VIGRE Colloquium, University of Georgia, Apr. 6, 2010.
Poincaré duality angles on Riemannian manifolds with boundary — Geometry and Topology Seminar, Tulane University, Mar. 9, 2010.
Poincaré duality angles on Riemannian manifolds with boundary — Geometry Seminar, University of Rochester, Mar. 4, 2010.
Legendrian contact homology and nondestabilizability — Geometry–Topology Seminar, University of Pennsylvania, Dec. 10, 2009.
Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links — Geometry and Topology Seminar, Caltech, Oct. 16, 2009.
Poincaré duality angles on Riemannian manifolds with boundary — Geometry/Topology Seminar, Duke University, Sept. 15, 2009.
Linking Integrals in Hyperspheres — Bi-Co Math Colloquium, Bryn Mawr College, Apr. 13, 2009. Essentially the same slides as the Sewanee talk mentioned below.
Poincaré duality angles for Riemannian manifolds with boundary — Geometry–Topology Seminar, Temple University, Dec. 2, 2008.
Linking Integrals in Hyperspheres — Sewanee Homecoming Lecture, The University of the South, Oct. 24, 2008. You will need a recent version of Adobe Reader to view the movies embedded in the linked PDF.
Higher-dimensional linking integrals — Philadelphia Area Contact/Topology Seminar, Bryn Mawr College, Feb. 14, 2008.

Contributed Talks

Poincaré duality angles on Riemannian manifolds with boundary — Lehigh University Geometry and Topology Conference, June 5, 2009.
Higher-dimensional linking integrals — 2008 Graduate Student Topology Conference, March 29, 2008. See also the version with overlays.

Other Talks

Poincaré duality angles on Riemannian manifolds with boundary — Ph.D. thesis defense, University of Pennsylvania, Apr. 13, 2009.
Recovering cup products from boundary data — Geometry–Topology Reading Seminar, University of Pennsylvania, Feb. 24, 2009.
Invariant differential forms in a cohomogeneity one manifold — Graduate Student Bridge Seminar, University of Pennsylvania, Feb. 18, 2009.
Poincaré duality angles for Riemannian manifolds with boundary — Graduate Student Geometry–Topology Seminar, University of Pennsylvania, Feb. 18, 2009.
The Dirichlet-to-Neumann map for differential forms — Graduate Student Geometry–Topology Seminar, University of Pennsylvania, Oct. 1, 2008.
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.
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

Graduate School

I received my Ph.D. from the University of Pennsylvania in May, 2009; my advisors were Dennis DeTurck and Herman Gluck and my thesis was on Poincaré duality angles on Riemannian manifolds with boundary. Along the way, 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.

Notes

Some old research notes that will probably never be published.

Linking integrals on Sⁿ — A general convolution formula for Lk(K, L) + (−1)ⁿ Lk(K, −L), where K and L are closed, connected, oriented submanifolds of Sⁿ. This formula is arrived at using invariant forms on the unit tangent bundle in the odd case, then extended to the even case using a geometric trick. This result was superseded by the paper “Higher-dimensional linking integrals”, but may be of some independent interest.
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. This result was superseded by the much nicer integral formula for μ₁₂₃ given in the paper “Pontryagin invariants and integral formulas for Milnor’s triple linking number”.
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. This was a warm-up exercise for the paper “Poincaré duality angles for Riemannian manifolds with boundary”.