Mark's Scholarly activities
My scholarly interests include real analysis, fractal geometry, real and complex dynamics, and computer graphics illustrating these ideas. My images have been featured on the covers of The Mathematica Journal, Mathematica in Education and Research, The Mathematical Intelligencer, and Math Horizons.
My publication rate has slowed these past few years while I worked heavily as a consultant for Wolfram Research; mostly, I developed mathematical content for WolframAlpha. I also developed some functionality in Mathematica V10. While I still use Mathematica quite a lot, I'm also trying to expand my horizons to use open tools, such as Python for computation and Javascript for visualization, when appropriate.
In the Summers of 2015 and 2016 I participated in a teacher training workshops organized by FCRSTEM and funded by the US DOE MathScience Partnership. The long term purpose is to study the impact of teacher content knowledge on student learning outcomes. It was loads of fun to develop dynamic illustrations of basic mathematics to support this project; I've archived my tools on this web page.
Publications
Fractals in real and abstract analysis

Fractal Measures on Infinite Dimensional Sets
My 1994 OSU dissertation 
"Entropy dimensions of the hyperspace of compact sets"
The Real Analysis Exchange, V21 (1995/96), #2, pp 194202 
"The Hausdorff dimension of the hyperspace of compact sets"
The Real Analysis Exchage, V22 (1996/97), #2, pp 611625 
"The prevalent dimension of graphs"
The Real Analysis Exchage, V23 (1997/98), #1, pp 241246 
"The Hausdorff dimension of Hilbert's coordinate functions"
The Real Analysis Exchange V24 (1998/99), #2, pp 875883  "The Borel structure of the collections subselfsimilar sets and superselfsimilar sets".
Acta Mathematica UC, V69, #2, (2000) pp 145150
with Bob Vallin 
"Selfsimilar structure in Hilbert's space filling curve".
Mathematics Magazine, V76, #1, (2003) pp 4047
Fractals from a computational perspective

"Inverse iteration algorithms for Julia sets".
Mathematica in Education and Research, V7 (1998), #2, pp 2228 
"Directedgraph iterated function systems".
Mathematica in Education and Research, V9, #2, (2000) 
"Digraph selfsimilar sets and aperiodic tilings".
The Mathematical Intelligencer, V24, #2 (2002), pp 3342 
"Selfsimilar intersections".
Fractals, V16 #2 (2008), pp 187197 
"Images of a vibrating Koch drum".
Computers and Graphics, V32, #6, (2008), pp 711715 
"Generating selfaffine tiles and their boundaries".
The Mathematica Journal, V11, #1, (2009). 
"Parameterized dynamics for the Weierstrass elliptic function
over square period lattices".
Intl. Jour. of Bifurcation and Chaos., V21, #1, (2011), pp 111
with Jane Hawkins 
"Through the LookingGlass, and What the
Quadratic Camera Found There".
The Mathematical Intelligencer., Volume 34, Issue 3, (2011) pp 3034.
with Stan Wagon, Bart de Smit, Willem Jan Palenstijn, and Isaac Sparling
Computational discrete mathematics

"A stochastic cellular automaton for threecoloring Penrose tiles".
Computers and Graphics, V26, #3, (2002) pp 519524 
"Decremental tag systems and random trees".
Complex Systems, V18, #2 (2009)
With Patrick Bahls and Josh Knox.

"Fourcoloring the US counties".
Math Horizons April, 2009 pp 2021
with Stan Wagon
Columns from Mathematica in Education and research, plus one invited review

A review of Stan Wagon's Mathematica in Action, 2nd Ed
V8, issue 3/4 (1999). 
"The connected locus for complex cubic iteration".
V10, #2 (2005) 
"Parametric Lsystems and borderline fractals".
V10, #3 (2005) 
"Steffen's flexible polyhedron" 
V10, #4 (2005) 
"Bifurcation sets and critical curves".
V11, #1 (2006) 
"Newton's method for complex polynomials".
V11, #2 (2006) 
"The ReadBajraktarevic operator".
V11, #3 (2006) 
"The kings problem".
V11, #4 (2006) 
"Boundary scanning and complex dynamics".
V12, #1 (2006) 
"Vibration of the Koch drum".
V12, #2 (2007) 
"Generating Google maps".
V12, #3 (2007)