Department of Mathematics

Colorado State University

Curriculum Vitæ

View CV Google Scholar

About Me

I am a postdoctoral fellow in the Department of Mathematics at Colorado State University.

My research interests primarily lie within random topology and geometry through combinatorial and stat-mech methods, with a view towards knot theory. This research has connections through various lenses to biology, physics, and chemistry. I completed my PhD in May of 2017 under the direction of Jason Cantarella. Recently I have been studying knots from the map-theoretic view of knot diagrams, and my research interests include low-dimensional topology, differential geometry, combinatorics, and probability.

I love to write software for use in research, teaching, and even just for fun. This includes:

During Summer 2012 and the 2012-2013 school year, I was supported by the University of Georgia's VIGRE II grant. I graduated from Bowdoin College in May of 2011 with a B.A. in Mathematics and Computer Science, with Honors in Mathematics.

Teaching

For the semester Fall 2017 at CSU, I'm teaching two sections of MATH 369 (Linear Algebra I).

When I was at UGA, I tought Calculus for Science and Engineering (Spring 2014, Spring 2016, Fall 2016), and Precalculus (Fall 2013, Fall 2015, Spring 2017), and I was a Writing Intensive Program teaching assistant for a robotics-based Calculus I course (Fall 2015).

Contact

Harrison Chapman
Colorado State University
Department of Mathematics
1874 Campus Delivery
Fort Collins, CO 80523-1874

Office: 223C Weber
Email: hchaps [at] gmail.com

Recent Publications

For a full list, see my research page. Entries are also available in BibTeX format.

  • A diagrammatic theory of random knots.
    PhD thesis, University of Georgia.
    Preprint available here

    We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices n. This model is an exciting new model which captures both the random geometry of space curve models of knotting as well as the ease of computing invariants from diagrams. This model of random knotting is similar to those studied by Diao et al., and Dunfield et al. We prove that unknot diagrams are asymptotically exponentially rare, an analogue of Sumners and Whittington's result for self-avoiding walks. Our proof uses the same idea: We first show that knot diagrams obey a pattern theorem and exhibit fractal structure. We use a rejection sampling method to present experimental data showing that these asymptotic results occur quickly, and compare parallels to other models of random knots. We finish by providing a number of extensions to the diagram model. The diagram model can be used to study embedded graph theory, open knot theory, virtual knot theory, and even random knots of fixed type. In this latter scenario, we prove a result still unproven for other models of random knotting. We additionally discuss an alternative method for randomly sampling diagrams via a Markov chain Monte Carlo method.

  • Asymptotic laws for random knot diagrams.
    Journal of Physics A: Mathematical and Theoretical 50 (2017), no. 22, p. 225001.
    DOI: 10.1088/1751-8121/aa6e45
    Preprint: arXiv:1608.02638

    We study random knotting by considering knot and link diagrams as decorated, (rooted) topological maps on spheres and pulling them uniformly from among sets of a given number of vertices n, as first established in recent work with Cantarella and Mastin. The knot diagram model is an exciting new model which captures both the random geometry of space curve models of knotting as well as the ease of computing invariants from diagrams. We prove that unknot diagrams are asymptotically exponentially rare, an analogue of Sumners and Whittington’s landmark result for self-avoiding polygons. Our proof uses the same key idea: we first show that knot diagrams obey a pattern theorem, which describes their fractal structure. We examine how quickly this behavior occurs in practice. As a consequence, almost all diagrams are asymmetric, simplifying sampling from this model. We conclude with experimental data on knotting in this model. This model of random knotting is similar to those studied by Diao et al, and Dunfield et al.

  • Knot probabilities in random diagrams.
    With Jason Cantarella and Matt Mastin.
    Journal of Physics A: Mathematical and Theoretical 49 (2016), no. 40, p. 405001.
    DOI: 10.1088/1751-8113/49/40/405001
    Preprint: arXiv:1512.05749

    We consider a natural model of random knotting—choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to compute exact probabilities for knots in this model. As expected, most diagrams with 10 and fewer crossings are unknots (about 78\% of the roughly 1.6 billion 10 crossing diagrams). For these crossing numbers, the unknot fraction is mostly explained by the prevalence of ‘tree-like’ diagrams which are unknots for any assignment of over/under information at crossings. The data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type, analogous to Zipf’s law for word frequency. The complete tabulation and all knot frequencies are included as supplementary data.

Recent Talks & Conferences

For a full list, see my research page.

Seminar talks and lectures

External

  • Random knots in physics and biology.
    Annual Math and Physics Lecture.
    Piedmont College, Demorest, GA, November 2016.
  • Asmyptotic laws for knot diagrams.
    Geometry Seminar.
    Tulane University, New Orleans, LA, October 2015.
  • Asmyptotic laws for knot diagrams.
    Discrete Math Seminar.
    University of British Columbia, Vancouver, BC, September 2015.

At UGA

  • A Markov chain Monte Carlo sampler for knot diagrams.
    Geometry Seminar.
    University of Georgia, Athens, GA, March 2017.
  • Patterns in knot diagrams.
    Geometry Seminar.
    University of Georgia, Athens, GA, August 2016.
  • The quantum harmonic oscillator.
    Geometry Seminar.
    University of Georgia, Athens, GA, October 2015.

Conference talks

  • A Sumners-Whittington result for knot diagrams.
    Means, Methods, and Results in the Statistical Mechanics of Polymeric Systems II.
    Fields Institute, Toronto, ON, June 2017.
    Video available here.
  • A Markov chain sampler for knot diagrams.
    Special Session on Invariants of Knots, Links, and 3-manifolds (AMS Spring Eastern Sectional Meeting 2017).
    Hunter College, New York, NY, May 2017.
  • Slipknotting in the Knot Diagram Model.
    Special Session on Knot Theory and its Applications (AMS Spring Southeast Sectional Meeting 2017).
    College of Charleston, Charleston, SC, March 2017.