# 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.

“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]“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]“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“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

“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“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]“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“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]“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“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]“The complete Dirichlet-to-Neumann map for differential forms”

Joint with Vladimir Sharafutdinov.

*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]“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]“Higher-dimensional linking integrals”

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]“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]“Poincaré duality angles for Riemannian manifolds with boundary”

Ph.D. thesis, University of Pennsylvania, 2009.

MR: 2713306; arXiv: 0909.1967 [math.DG]

## Submitted Papers

“Random triangles and polygons in the plane”

Joint with Jason Cantarella, Tom Needham, and Gavin Stewart.

arXiv: 1702.01027.“Spherical geometry and the least symmetric triangle”

Joint with Laney Bowden, Andrea Haynes, and Aaron Shukert.

arXiv: 1708.01559.

## Talks

### Invited Talks

A Natural Map from Random Walks to Equilateral Polygons in any Dimension — CMO-BIRS Workshop on the Geometry and Topology of Knotting and Entanglement in Proteins, Oaxaca, Mexico, Nov. 7, 2017. Video

A Geometric Approach to Sampling Loop Random Flights — Probability Seminar, University of Colorado Boulder, Oct. 19. 2017.

The Geometry of Polygon Space: Acute Triangles, Convex Quadrilaterals, Flag Means, and More — AMS Special Session on Differential Geometry of Smooth and Discrete Surfaces in Euclidean and Lorentz Spaces, Fall Central Sectional Meeting, Denton, TX, Sept. 9, 2017.

The Geometry of Polygon Space: Acute Triangles, Convex Quadrilaterals, Flag Means, and More — International Workshop on Knots and Polymers: Aspects of topological entanglement in DNA, proteins and graph-shaped polymers, Tokyo, Japan, Aug. 8, 2017.

Polyhedra, Sampling Algorithms for Random Polygons, and Applications to Ring Polymer Models — Minisymposium on Polyhedral and Combinatorial Biology, SIAM Conference on Applied Algebraic Geometry (AG17), Atlanta, GA, Aug. 1, 2017.

Concavity, a question of Sylvester, and how to generate random quadrilaterals — AMS Special Session on Knot Theory and its Applications, Spring Southeastern Sectional Meeting, Charleston, SC, Mar. 12, 2017.

From Obtuse Triangles to DNA Models and Synthetic Polymers: The Geometry of Random Polygons — Mathematics in Science and Society Lecture Series, University of Illinois, Nov. 29, 2016. (poster)

The Symplectic Geometry of Polygon Space and How to Use It — Geometry, Groups, and Dynamics/GEAR Seminar, University of Illinois, Nov. 29, 2016. Video

What’s the Probability That a Random Triangle is Obtuse? — AMS Special Session on Knotting in Physical Systems, in celebration of Kenneth C. Millett’s 75th birthday, Fall Central Sectional Meeting, Minneapolis, MN, Oct. 30, 2016.

Applications of Geometry to Constrained Random Walks and Polymer Models – Geometry for Signal Processing and Machine Learning, Estes Park, CO, Oct. 14, 2016.

What’s the Probability That a Random Triangle is Obtuse? — Geometry–Topology Seminar, University of Pennsylvania, Sept. 15, 2016.

Simulating Constrained Random Walks for Applications to Polymer Models — Minisymposium on Molecular Biosciences and Biophysics – Macromolecular Structures and Interactions, SIAM Conference on the Life Sciences, Boston, MA, July 14, 2016.

Animating Mathematics — Workshop on Illustrating Mathematics, ICERM, June 30, 2016. Slides

Symplectic Geometry and Polygon Sampling: Sometimes Simpler is Also Faster — Geometry Seminar, University of Georgia, Mar. 18, 2016.

What Percentage of Triangles are Obtuse? — Video interview targeted to high school students with Geometry & Topology Today, Nov. 2015.

Random Polygons, Grassmannians, and Polymer Physics — Video interview targeted to graduate students with Geometry & Topology Today, Nov. 2015.

The Symplectic Geometry of Polygon Space and How to Use It — Topology/Virtual Seminar, Louisiana State University, Nov. 4, 2015. Video

A Geometric Perspective on Random Walks with Topological Constraints — Graduate Student Colloquium, Louisiana State University, Nov. 3, 2015. Video

15 Views of the Hypersphere — Undergraduate Student Colloquium, Louisiana State University, Nov. 2, 2015. (You will need the free Wolfram CDF Player to view this file. Also, beware: this is a 6.4 MB file.) Video

The Symplectic Geometry of Polygon Space and How to Use It — Workshop on Symplectic and Algebraic Geometry in the Statistical Physics of Polymers, Simons Center for Geometry and Physics, Oct. 12, 2015. Video

A Geometric Perspective on Random Walks with Topological Constraints — Mathematics Colloquium, Wake Forest University, Sept. 9, 2015.

The Geometric Structure of the Space of Stick Knots — BK21 Seminar, Korea Advanced Institute of Science and Technology (KAIST), Aug. 10, 2015.

The Geometry of Polygon Spaces — Minisymposium on Aspects of Grassmann Manifolds With a View Towards Applications, SIAM Conference on Applied Algebraic Geometry (AG15), Daejeon, South Korea, Aug. 7, 2015.

Minicourse on Differential Geometry and Grassmannians — Universidad de Costa Rica, Apr. 6–17, 2015.

Geometry of Random Polygons, Knots, and Biopolymers — Joint Center for Computational Mathematics and Discrete Mathematics Seminar, University of Colorado Denver, Feb. 2, 2015.

A New Algorithm for Sampling Closed Equilateral Random Walks — Geometry Seminar, University of Georgia, Jan. 16, 2015.

15 View of the Hypersphere — Math Club, University of Georgia, Jan. 15, 2015.

A New Algorithm for Sampling Closed Equilateral Random Walks — AMS Special Session on Knot Theory and Its Applications, Fall Southeastern Sectional Meeting, Greensboro, NC, Nov. 9, 2014.

15 Views of the Hypersphere — Math Club, Colorado State University, Oct. 1, 2014.

Closed Random Walks, Symplectic Geometry, and Ring Polymer Models — Joint Mathematics and Materials Science Colloquia, Colorado State University, Feb. 24, 2014.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, Saint Louis University, Feb. 21, 2014.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, California State University Fullerton, Feb. 18, 2014.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, Ball State University, Feb. 14, 2014.

Unlocking the Geometry of Polygon Space by Taking Square Roots — Mathematics Colloquium, Gettysburg College, Feb. 11, 2014.

Unlocking the Geometry of Polygon Space by Taking Square Roots — Mathematics Colloquium, Amherst College, Feb. 6, 2014.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, Butler University, Feb. 3, 2014.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, University of Rochester, Jan. 28, 2014.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, Utah State University, Jan. 9, 2014.

Grassmannians, Closed Random Walks, and Optimal Reconfiguration — Geometry, Mathematical Physics, and Computer Algebra Seminar, Utah State University, Jan. 9, 2014.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, Wichita State University, Dec. 11, 2013.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, Fordham University, Dec. 5, 2013.

Closed Random Walks and Symplectic Geometry — Mathematics Colloquium, Georgia Southern University, Nov. 22, 2013.

Closed Random Walks and Symplectic Geometry — Geometry–Topology Seminar, University of Pennsylvania, Oct. 31, 2013.

Unlocking the Geometry of Polygon Space by Taking Square Roots — Undergraduate Colloquium, University of Pennsylvania, Oct. 30, 2013.

The Quaternionic Method for Directly Sampling Framed Fixed-Length Polygons — 2013 Georgia Topology Conference, Athens, GA, July 11, 2013.

The Geometry of Random Polygons — Joint Analysis, Geometry & Stochastics and Bioinformatics seminars, Friedrich-Schiller-Universität, Jena, Germany, May 8, 2013.

The Symplectic Geometry of Polygon Space — Workshop on Geometric Knot Theory, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany, Apr. 29, 2013.

The Geometry and Topology of Random Polygons — Topology/Virtual Seminar, Louisiana State University, Baton Rouge, LA, Mar. 13, 2013.

The Geometry of Random Polygons — Geometry Seminar, University of Manchester, Manchester, UK, Dec. 13, 2012.

The Dirichlet-To-Neumann Operator for Differential Forms — Geometry Seminar, University of Manchester, Manchester, UK, Dec. 11, 2012.

The Geometry of Random Polygons — Quantized Flux in Tightly Knotted and Linked Systems, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, Dec. 6, 2012.

Homotopy, Link Homotopy, and (Higher?) Helicity — Topological Dynamics Programme Seminar, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, Oct. 2, 2012.

The Dirichlet-To-Neumann Operator for Differential Forms — Mini-Symposium on Inverse Problems in Geometry, Inverse Problems Conference in honor of Gunther Uhlmann, June 19, 2012.

Homotopy and Link Homotopy — AMS Special Session on Low-Dimensional Topology, Spring Southeastern Section Meeting, Tampa, FL, Mar. 11, 2012.

Grassmannians and Random Polygons — Geometry–Topology Seminar, Georgia Tech, Nov. 7, 2011.

Homotopy Periods of Link Maps and Milnor’s Invariants — AMS Special Session on Geometric Knot Theory and Its Applications, Fall Southeastern Section Meeting, Winston-Salem, NC, Sept. 25, 2011.

Rulings and Augmentations for Bordered Legendrian Knots — AMS Special Session on Low-Dimensional Topology and Geometry, Fall Southeastern Section Meeting, Winston-Salem, NC, Sept. 25, 2011.

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, New Orleans, 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.

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.

### Refereed Talks

Homotopy String Links and the \(\kappa\)-Invariant — IUTAM Symposium on Helicity: Structure and Singularity in Fluid and Plasma Dynamics, Venice, Italy, Apr. 11, 2016.

The Geometry of Random Polygons — AMS Session on Geometry and Differential and Hyperbolic Geometry, Joint Mathematics Meetings, San Diego, Jan. 10, 2013.

Generalized Gauss Maps and Triple Linking Integrals — Workshop on Tangled Magnetic Fields in Astro- and Plasma Physics, International Centre for Mathematical Sciences, Edinburgh, UK, Oct. 18, 2012.

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, Mar. 29, 2008.

### 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 Walks — Applied Math Seminar, Colorado State University, Nov. 13, 2014.

Grassmannians and Random Polygons — Pattern 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.

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 \(S^n \times \mathbb{R}^m\) — 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.

“Linking integrals on \(S^n\)” — A general convolution formula for \(\text{Link}(K,L) + (- 1)^n \mathrm{Link}(K, -L)\), where \(K\) and \(L\) are closed, connected, oriented submanifolds of \(S^n\). This formula is arrived at using invariant forms on the unit tangent bundle for odd \(n\), 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.

“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”.