Bill Wood

Scholarship

Discrete Conformal Geometry

• “Discrete Extremal Length and Cube Tilings in Finite Dimensions,” Computational Methods and Function Theory, Vol. 15, pp. 143–149, 2015 

Sa'ar Hersonsky demonstrated a necessary condition called the triple-intersection property for cube tilings to model conformal geometry in the same way square tilings do in two-dimensions. This paper shows no cube tiling satisfies the triple-intersection property, mirroring the classical case where conformal geometry breaks down in dimension three.

• “Bounded Outdegree and Extremal Length on Discrete Riemann Surfaces,” Conformal Geometry and Dynamics, Vol. 14,  pp. 194-201, 2010 

This applies the techniques in the paper below to more general surfaces. Basically combine the same combinatorial methods with the pair-of-pants decomposition to get bounds on extremal lengths and ultimately prove discrete type is preserved under refinement.

• “Combinatorial Modulus and Type of Graphs,” Topology and its Applications, Vol. 156, Issue 17, pp. 2747—2761, 2009 

Every non-compact simply-connected Riemann surface is conformally equivalent to either the plane or the disk -- but which one? This is the type problem, and there is a discrete analog wherein we use triangulation graphs to proxy for surfaces. We can use these triangulations as instructions for prescribing tangencies of circles to give a discrete approximation of the conformal surface. In other words, we can thing of conformal structures by looking at infinitesimal surfaces, and we approximate this with actual circles. We can improve the approximation by adding vertices and edges to the graph to add more circles. But can this change the conformal type? The answer is no, as long as the refinement process isn't too ridiculous. 

This can be proved using quasiconformal techniques if the graph has bounded degree, but this paper removes the bounded degree condition. The key to the proof is that any planar graph can have arrows assigned to its edges so that every vertex has at most 5 arrows coming out of it (the "outdegree"). It turns out this weaker notion of bounded degree shared by all planar graphs is sufficient. This paper is based on my PhD thesis, written at FSU under the direction of Phil Bowers.

• “Evaluation of Three Cortical Surface Flattening Methods,” NeuroImage, vol. 22, no. Supplement 1, pg. S45, CD-Rom Abstract WE-207, 2004 (with L. Ju, M. K. Hurdal, K. Rehm, K. Schaper, J. Stern, D. Rottenberg, refereed poster and abstract) 

This is a project to which I made only small contributions in graduate school, but it's a really neat idea. We were focused on using circle packings to make hyperbolic flat maps of the human brain. See Monica Hurdal's FSU website for more.

Other Publications

• Games
  I am an avid board gamer and I sometimes find opportunities to write on the mathematical depth in puzzles and games.

  • “Managing Expectations in the Game of Pairs,” Iowa Council of Teachers of Mathematics, Winter 2018/19 (with B. Townsend)

  • “Board Games as Platforms for Mathematical Learning,” Iowa Council of Teachers of Mathematics, Winter 2017/18 (with B. Townsend)

  • “Turning Kakuro up to 11,” Focus, Vol. 33, No. 22, April/May 2013
    Kakuro puzzles in base 11

• Course Modules

  • “Introducing Geometry with Spatial Scan Statistics,” DIMACS Educational Module Series 10-3, 2011 (with J. Franko and R. Schwell)
    We look at how we might think geometrically about the problem of detecting an outbreak in a region. This module came out of a terrific workshop at DIMACS.

  • “The Analysis of Games,” Value of Computational Thinking Across Grade Levels 9-12, COMAP, 2016 (with A. Jaggard)

High school level module where students can explore the process of taking apart a game such as Connect Four to look for a winning strategy.

• Student Projects

  • “Constructing Fractals of Specified Dimension,” Pi Mu Epsilon Journal, Spring 2012 (with W. Longley)

I know some things about calculating fractal dimension, so I was curious how exactly to reverse the process. Co-authored with William Longley, then student at Hendrix College. 

Get in touch at bill.wood (at) uni.edu