Research
Metric Space Geometry
My research is primarily focused on the geometry of metric spaces. In particular, I have spent a lot of time studying metric spaces that are highly symmetric in the sense that they admit transitive actions by collections of metrically controlled self-mappings (e.g. isometries, bi-Lipschitz mappings, quasi-symmetric mappings, or something along those lines). More recently, I've been investigating the Lipschitz geometry of quasiconformal arcs and trees in connection with Lipschitz free spaces and Lipschitz dimension. Here is a list of my publications and preprints in these areas:
David Freeman and Chris Gartland, Lipschitz functions on unions and quotients of metric spaces Studia Mathematica 273 (2023), 29-61.
David Freeman and Chris Gartland, Lipschitz functions on quasiconformal trees Fundamenta Mathematicæ 262 (2023), 153-203.
David M. Freeman, Weak quasicircles have Lipschitz dimension 1, Annales Fennici Mathematici 47 (2022), 283-303.
David M. Freeman and Enrico Le Donne, Toward a quasi-Möbius characterization of invertible homogeneous metric spaces, Revista Matemática Iberoamericana 37 (2021), no. 2, 671-722.
Note: some of our results for disconnected spaces (in connection with Cantor-like spaces) are similar in spirit to some of the results found in Characterizing the Cantor bi-cube in asymptotic categories by Taras Banakh. The results are different, but someone who finds our results about disconnected spaces to be interesting may also find the results of Banakh to be interesting.
David M. Freeman, Invertible Carnot groups, Analysis and Geometry in Metric Spaces 2 (2014), 248-257.
A slightly revised version is posted here
David M. Freeman, Transitive bi-Lipschitz group actions and bi-Lipschitz parameterizations, Indiana University Mathematics Journal 62 (2013), no. 1, 311-331.
David M. Freeman, Inversion invariant bilipschitz homogeneity, Michigan Mathematical Journal 61 (2012), no. 2, 415-430.
David M. Freeman, Unbounded bilipschitz homogeneous Jordan curves, Annales Academiæ Scientiarum Fennicæ Mathematica, 36 (2011), no. 1, 81-99.
David M. Freeman, Bilipschitz homogeneous Jordan curves, Möbius maps, and dimension, Illinois Journal of Mathematics 54 (2010), no. 2, 753-770.
David M. Freeman and David A. Herron, Bilipschitz homogeneity and inner diameter distance, Journal d’Analyse Mathématique 111 (2010), no. 1, 1-46.
Continued Fractions
I've also done a bit of work with continued fractions exhibiting a generalized form of palindromic symmetry. In particular, I generalized a transcendence criteria of Adamczewski and Bugeaud. Here's a link:
David M. Freeman, Generalized Palindromic Continued Fractions, Rocky Mountain Journal of Mathematics 48 (2018), 219-236.
Math + Art
My recent exploration of the space between math and art began with some experiments involving epicycloid curves and continued fractions. These experiments led to the following paper and a few images in the Bridges Conference art exhibit:
David M. Freeman, Epicycloid Curves and Continued Fractions, Journal of Mathematics and the Arts 11 (2017), 100-113.
After that, I became interested in aperiodic tilings of the plane. After learning a bit of Python code (via the help of my friend Noah Weaver) I was able to reproduce various tilings appearing in the literature. Here are a few I produced for inclusion in the Bridges exhibit:
As a result of working a bit with tilings, I was asked by a colleague to be interviewed for an article on the Penrose tiling. Here's a link to the article (in French), published by the online magazine of the University of Fribourg:
Math + Philosophy
I've become quite interested in questions regarding the ontology of abstract mathematical objects. I'm particularly interested in possible theological implications of such questions, and vice-versa (along the lines of those examined in William Lane Craig's book God and Abstract Objects). This interest led to the following article:
David M. Freeman, A Belief Expressionist Explanation of Divine Conceptualist Mathematics, Metaphysica 23 (2022), no. 1, 15-26