# Research

My work uses differential geometry to solve topological and physical problems. My main current interest is in developing exact mathematical models of topologically constrained random walks and polymer networks using Riemannian and symplectic geometry.

## Publications

Note: the first appearance of each collaborator’s name is linked to his/her website (or the nearest approximation thereof). Click the thumbnails for a more detailed description. See also my arXiv author page.

1. “The Geometry of Constrained Random Walks and an Application to Frame Theory”
2018 IEEE Statistical Signal Processing Workshop (SSP), to appear

2. “Random triangles and polygons in the plane”
Joint with Jason Cantarella, Tom Needham, and Gavin Stewart.
The American Mathematical Monthly, to appear
arXiv: 1702.01027 [math.MG]

3. “Spherical geometry and the least symmetric triangle”
Joint with Laney Bowden, Andrea Haynes, and Aaron Shukert.
Geometriae Dedicata (2018), https://doi.org/10.1007/s10711-018-0327-4 (journal link)
doi: 10.1007/s10711–018–0327–4; arXiv: 1708.01559 [math.MG]

4. “Homotopy string links and the $$\kappa$$-invariant”
Joint with Frederick R. Cohen, Rafał Komendarczyk, and Robin Koytcheff.
Bulletin of the London Mathematical Society 49 (2017), no. 2, 246–260 (journal link)
MR: 3656294; doi: 10.1112/blms.12025; arXiv: 1504.03233 [math.GT]

5. “A fast direct sampling algorithm for equilateral closed polygons”
Joint with Jason Cantarella, Bertrand Duplantier and Erica Uehara.
Journal of Physics A: Mathematical and Theoretical 49 (2016), No. 27, 275202 (journal link)
Selected as a 2016 Highlight of J. Phys. A
MR: 3512103; Zbl 1342.82063 doi: 10.1088/1751–8113/49/27/275202; arXiv: 1510.02466 [cond-mat.stat-mech]

6. “The symplectic geometry of closed equilateral random walks in 3-space”
Joint with Jason Cantarella.
Annals of Applied Probability 26 (2016), no. 1, 549–596 (journal link).
MR: 3449326; Zbl: 06554528 doi: 10.1214/15-AAP1100; arXiv: 1310.5924 [math.DG]; INI Preprint #NI13054-TOD

7. “The expected total curvature of random polygons”
Joint with Jason Cantarella, Alexander Y. Grosberg, and Robert Kusner.
American Journal of Mathematics 137 (2015), no. 2, 411–438 (journal link)
MR: 3337799; Zbl: 06434908; doi: 10.1353/ajm.2015.0015; arXiv: 1210.6537 [math.DG]; INI Preprint #NI12084-TOD

8. “Homotopy Brunnian links and the $$\kappa$$-invariant”
Joint with Frederick R. Cohen and Rafał Komendarczyk.
Proceedings of the American Mathematical Society 143 (2015), no. 3, 1347–1362 (journal link)
MR: 3293747; Zbl: 06393823; doi: 10.1090/S0002–9939–2014–12331–8; arXiv: 1208.4587 [math.GT]; KITP Preprint #NSF-ITP–12–163

9. “Probability theory of random polygons from the quaternionic viewpoint”
Joint with Jason Cantarella and Tetsuo Deguchi.
Communications on Pure and Applied Mathematics 67 (2014), no. 10, 1658–1699 (journal link)
MR: 3251909; Zbl: 06353278; doi: 10.1002/cpa.21480; arXiv: 1206.3161 [math.DG]

10. “The symplectic geometry of polygon space”
Joint with Jason Cantarella.
Oberwolfach Reports 10 (2013), no. 2, 1347–1350 (journal link)
doi (for full proceedings): 10.4171/OWR/2013/22

11. “Generalized Gauss maps and integrals for three-component links II”
Joint with Dennis DeTurck, Herman Gluck, Rafał Komendarczyk, Paul Melvin, Haggai Nuchi, and David Shea Vela-Vick.
Algebraic & Geometric Topology, 13 (2013), no. 5, 2897–2923 (journal link)
MR: 3116307; Zbl: 06198033; doi: 10.2140/agt.2013.13.2897; arXiv: 1207.1793 [math.GT]

12. “Poincaré duality angles and the Dirichlet-to-Neumann operator”
Inverse Problems 29 (2013), no. 4, 045007 (journal link)
MR: 3042083; Zbl: 06213991; doi: 10.1088/0266–5611/29/4/045007

13. “Generalized Gauss maps and integrals for three-component links”
Joint with Dennis DeTurck, Herman Gluck, Rafał Komendarczyk, Paul Melvin, and David Shea Vela-Vick.
Journal of Mathematical Physics 54 (2013), no. 1, 013515 (journal link)
MR: 3059903; Zbl: 06255553; doi: 10.1063/1.4774172; arXiv: 1101.3374 [math.GT]

14. “The complete Dirichlet-to-Neumann map for differential forms”
Journal of Geometric Analysis 23 (2013), no. 4, 2063–2080 (journal link)
MR: 3107691; Zbl: 06221533; doi: 10.1007/s12220–012–9320–6; arXiv: 1011.1194 [math.DG]

15. “Legendrian contact homology and nondestabilizability”
Joint with David Shea Vela-Vick.
Journal of Symplectic Geometry 9 (2011), no. 1, 33–44 (journal link)
MR: 2787360; Zbl: 1226.57013; doi: 10.4310/JSG.2011.v9.n1.a3; arXiv: 0910.3914 [math.GT]

Joint with David Shea Vela-Vick.
Proceedings of the American Mathematical Society 139 (2011), no. 4, 1511–1519 (journal link)
MR: 2748445; Zbl: 1221.57038; doi: 10.1090/S0002–9939–2010–10603–2; arXiv: 0801.4022 [math.GT]

17. “Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links”
Joint with Dennis DeTurck, Herman Gluck, Rafał Komendarczyk, Paul Melvin, and David Shea Vela-Vick.
Matemática Contemporânea 34 (2008), 251–283 (journal link)
Special volume in honor of Manfredo do Carmo’s 80th birthday.
MR: 2588614; Zbl: 1194.57007; arXiv: 0901.1612 [math.GT]

18. “Poincaré duality angles for Riemannian manifolds with boundary”
Ph.D. thesis, University of Pennsylvania, 2009.
MR: 2713306; arXiv: 0909.1967 [math.DG]

## Submitted Papers

1. “Symplectic geometry and connectivity of spaces of frames”
Joint with Tom Needham
arXiv: 1804.05899 [math.FA]

## Talks

### Other Talks

• What’s the probability that a random triangle is obtuse? or: What the heck is a random triangle, anyway? — Math 192 Guest Lecture, Colorado State University, Sept. 27, 2016.

• The Symplectic Geometry of Polygon Space and How to Use It — Fragment Seminar, Colorado State University, Nov. 12, 2015.

• A New Algorithm for Sampling Closed Equilateral Random WalksApplied Math Seminar, Colorado State University, Nov. 13, 2014.

• Grassmannians and Random PolygonsPattern Analysis Lab Lecture Series, Colorado State University, Nov. 6, 2014.

• 15 Views of the Hypersphere — Math 192 Guest Lecture, Colorado State University, Oct. 10, 2014.

• The Dirichlet-to-Neumann Operator for Differential Forms — Inverse Problems Seminar, Colorado State University, Sept. 11, 2014.

• Closed Random Walks and Symplectic Geometry — Geometry Seminar, University of Georgia, Nov. 8, 2013.

• Ambidextrous Knots Via Octonions — Geometry Seminar, University of Georgia, Sept. 6, 2013.

• The Total Curvature of Random Polygons — Geometry Seminar, University of Georgia, Mar. 22, 2013.

• Homotopy and Link Homotopy — Topology Seminar, University of Georgia, Aug. 20, 2012.

• Generalized Gauss Maps and Triple Linking Integrals — Geometry Seminar, University of Georgia, Feb. 10, 2012.

• Grassmannians and Random Polygons — Geometry Seminar, University of Georgia, Nov. 11, 2011.

• The Complete Dirichlet-To-Neumann Map for Differential Forms — Geometry Seminar, University of Georgia, Sept. 2, 2011.

• Poincaré Duality Angles for Riemannian Manifolds With Boundary — Geometry Seminar, University of Georgia, Aug. 26, 2011.

• Poincaré Duality Angles for 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.

• 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.

• Geometric Linking Integrals in $$S^n \times \mathbb{R}^m$$ — Pizza Seminar, University of Pennsylvania, Oct. 12, 2007.

• Introduction to Minimal Surfaces — Pre-Colloquium Talk, University of Pennsylvania, Oct. 18, 2006.

• The Four Vertex Theorem and its Converse — Pizza Seminar, University of Pennsylvania, Oct. 6, 2006.

• The Gauss Linking Integral in $$S^3$$ and $$H^3$$ — 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, Apr. 2003.

• Picture Groups for Links — REU final presentation, Louisiana State University, Aug. 2002.

## Notes

Some old research notes that will probably never be published.