# Guy C. David

## About

I am an associate professor of mathematical sciences at Ball State University, where I have taught since 2017. From 2014-2017, I worked as a Courant Instructor at the Courant Institute of Mathematical Sciences at New York University. I got my Ph. D. in 2014 from the University of California, Los Angeles under the advisement of Mario Bonk.

(There is another mathematician with the same first and last name as me and some similar research interests. His page is here.)

My mathematical genealogy. (Made by Raanan Schul using geneagrapher, with data from the Math Genealogy Project.)

## Contact information

email: gcdavid at bsu dot edu. (I sometimes don't receive external emails to this address. If you have emailed me and I have not responded, please try guycdavid at gmail.)

mailing address: Department of Mathematical Sciences, Ball State University, Muncie, IN 47306

office: Robert Bell 421

office hours: by appointment, and posted on Canvas for my current students

## Research interests

My research interests are in analysis on metric spaces, Lipschitz mappings, and quasiconformal geometry.

Not everyone knows what "analysis on metric spaces" means. A nice short survey of the area is the article

"Analysis in Metric Spaces" in the Notices of the American Mathematical Society (February 2020).

At a more detailed level are the following survey articles by Juha Heinonen:

and this list of questions by Juha Heinonen and Stephen Semmes:

I am maintaining a page with the current status of the Heinonen-Semmes problems.

## Research publications

(* = undergraduate or master's student co-author)

G. C. David, M. Romney, and R. Schul, Factorization and piecewise affine approximation of bi-Lipschitz mappings on large sets. preprint. arXiv.

G. C. David and S. Eriksson-Bique, Analytically one-dimensional planes and the Combinatorial Loewner Property. preprint. arXiv.

G. C. David and K. Hook*, Quantitative differentiation and the medial axis. preprint. arXiv.

G. C. David and B. Oliva*, Quantitative metric density and connectivity for sets of positive measure. to appear, Studia Mathematica. arXiv.

G. C. David and S. Eriksson-Bique, Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings. Annales de l'Institut Fourier, 74 (2024), no. 3, 973-1016. arXiv.

G. C. David, A non-injective Assouad-type theorem with sharp dimension. The Journal of Geometric Analysis 32 (2024), no. 4, 45. arXiv.

G. C. David and R. Schul, Quantitative decompositions of Lipschitz mappings into metric spaces. Transactions of the American Mathematical Society, 376 (2023), no. 8, 5521-5571. arXiv. video of a talk about this paper.

G. C. David, S. Eriksson-Bique, and V. Vellis. Bi-Lipschitz embeddings of quasiconformal trees. Proceedings of the American Mathematical Society, 151 (2023), no. 5, 2031–2044. arXiv.

G. C. David, M. Kaczanowski*, and D. Pinkerton*, Quantitative straightening of distance spheres. Real Analysis Exchange 48(1): 149-164 (2023). arXiv.

G. C. David and R. Schul, Lower bounds on mapping content and quantitative factorization through trees. Journal of the London Mathematical Society, 106 (2022), no. 2, 1170-1188. arXiv.

G. C. David and V. Vellis. Bi-Lipschitz geometry of quasiconformal trees. Illinois Journal of Mathematics. 66(2): 189-244 (2022) . arXiv.

G. C. David, On the Lipschitz dimension of Cheeger-Kleiner. Fundamenta Mathematicae 253 (2021), no. 3, 417-358. arXiv.

G. C. David and R. Schul, A sharp necessary condition for rectifiable curves in metric spaces. Revista Matemática Iberoamericana 7 (2021), no. 3, 1007-1044. arXiv.

G. C. David and S. Eriksson-Bique, Regular mappings and non-existence of bi-Lipschitz embeddings for slit carpets. Advances in Mathematics, 364 (2020), 107047. arXiv.

G. C. David and E. Le Donne, A note on topological dimension, Hausdorff measure, and rectifiability. Proceedings of the American Mathematical Society, 148 (2020), no. 10, 4299-4304. arXiv.

G. C. David and K. Kinneberg, Lipschitz and bi-Lipschitz maps from PI spaces to Carnot groups. Indiana University Mathematics Journal 69 (2020), no. 5, 1685-1731. arXiv.

G. C. David and B. Kleiner, Rectifiability of planes and Alberti representations. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 19 (2019), no. 2, 723-756. arXiv.

G. C. David and K. Kinneberg, Rigidity for convex-cocompact actions on rank-one symmetric spaces. Geometry & Topology 22 (2018), no. 5, 2757–2790. arXiv.

G. C. David and R. Schul, The Analyst's traveling salesman theorem in graph inverse limits. Annales Academiæ Scientiarum Fennicæ Mathematica, 42: 649-692, 2017. arXiv.

G. C. David, Bi-Lipschitz pieces between manifolds. Revista Matemática Iberoamericana, 32 (1): 175-218, 2016. arXiv.

G. C. David, Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces. Geometric and Functional Analysis, 25 (2): 553-579, 2015. arXiv.

G. C. David, Lusin-type theorems for Cheeger derivatives on metric measure spaces. Analysis and Geometry in Metric Spaces, 3 (1): 296-312, 2015. arXiv.

G. C. David, Lipschitz Maps in Metric Spaces (2014 Ph.D. dissertation at UCLA). ProQuest link.

All my papers on the arXiv can also be found here.

This material is based upon work supported by the National Science Foundation under Grants No. DMS-1664369, DMS-1758709, and DMS-2054004. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

## Student Research

Here are some research projects that students have worked on with me:

Brandon Oliva's 2024 master's thesis is Quantitative metric density and applications. We wrote a joint paper: Quantitative metric density and connectivity for sets of positive measure.

Thomas Maahs completed a reading course with me in Fall 2022, working through the proof of Assouad's Embedding Theorem on embedding metric spaces into Euclidean spaces.

Kevin Hook's 2022 master's thesis is Quantitative differentiation and its applications. He gives a full exposition of a "coarse" differentiation result, and applies it to study the geometry of medial axes of sets. We wrote a paper spinning off the medial axis section: Quantitative differentiation and the medial axis.

Undergraduates McKenna Kaczanowski and Dallas Pinkerton worked with me to extend their Spring 2021 Math 498 project into a paper. Our paper is Quantitative straightening of distance spheres, Real Analysis Exchange 48(1): 149-164 (2023).

Students in my Spring 2021 Math 498 class learned about and developed Mathematica visualizations related to various topics in analysis and geometry, including the Analyst's Traveling Salesman Theorem, quantitative differentiation, metric embeddings, and the Heisenberg group.

Devin Goedeker's 2020 master's thesis is Fractal Spaces and bi-Lipschitz embeddings into Banach spaces. Devin studied some fractals arising as limits of metric graphs, and showed that they cannot be viewed inside Euclidean space (or more general spaces) without drastically distorting their geometry.

Andy Reel's 2020 honors thesis is Square Tilings and Discrete Modulus. Here is a Mathematica notebook Andy wrote that computes the unique square tiling of a rectangle associated to a given triangulation of a planar quadrilateral, which exists by a theorem of Schramm.

## Teaching

In Fall 2024, I am not teaching. Generally, all information about my courses can be found on Canvas.

### Prior teaching at Ball State:

Spring 2024: Two sections of Math 132 (Brief Calculus) and Math 672 (Real Analysis 2)

Fall 2023: Math 267 (Calculus 3) and Math 471/571 (Real Analysis 1)

Spring 2023: Math 166 (Calculus 2) and Math 672 (Real Analysis 2)

Fall 2022: Math 166 (Calculus 2) and Math 471/571 (Real Analysis 1)

Spring 2022: Math 125 (Quantitative Reasoning) and Math 470/570 (Intermediate Analysis)

Fall 2021: Math 165 (Calculus 1) and Math 471/571 (Real Analysis 1)

Spring 2021: Math 470/570 (Intermediate Analysis) and Math 498 (Senior Seminar)

Fall 2020: Three sections of Math 125 (Quantitative Reasoning)

Spring 2020: Math 267 (Calculus 3) and Math 677 (Graduate Complex Variables I)

Fall 2019: Two sections of Math 165 (Calculus 1) and Math 377 (Complex Analysis)

Spring 2019: Math 470/570 (Intermediate Analysis) and Math 675 (Measure Theory)

Fall 2018: Math 165 (Calculus 1) and Math 519 (Quantitative Reasoning for Teachers)

Spring 2018: Math 165 (Calculus 1) and Math 470/570 (Intermediate Analysis)

Fall 2017: Math 165 (Calculus 1)

### Prior teaching at NYU:

Spring 2017: Discrete Mathematics

Fall 2016: Linear Algebra

Spring 2016: Calculus 3

Fall 2015: Analysis (Math 325)

Spring 2015: Discrete Mathematics

Fall 2014: Calculus 1

As a graduate student (2009-2014), I was also a teaching assistant for a number of courses at UCLA.

## Conferences

I had a very small role in the planning of the Quasiworld 2023 workshop in Helsinki, which celebrated the contributions of my Ph.D. advisor Mario Bonk. The workshop was supported in part by National Science Foundation grant DMS-2246679 ("Conference: Quasiworld Workshop").

Co-organizer (with Florent Baudier, Chris Gartland, and Jing Wang), Concentration Week on Geometry and Analysis in Nonsmooth Spaces, Texas A&M University, August 8-12, 2022.

Co-organizer (with Matthew Badger and Lisa Naples), Special Session on Geometry of Measures and Metric Spaces, AMS Spring Central Sectional Meeting, online, March 26-27, 2022.

Co-organizer (with John Dever), Special Session on Analysis and Probability on Metric Spaces and Fractals, AMS Fall Central Sectional Meeting, University of Wisconsin-Madison, September 14-15, 2019.

Co-organizer (with Jay Bagga, Lowell Beineke, Dan Rutherford, and Brad Shutters), The 10th International Workshop on Graph Labeling, Ball State University, September 26-28, 2018.

Kublai Khan does not necessarily believe everything Marco Polo says when he describes the cities visited on his expeditions, but the emperor of the Tartars does continue listening to the young Venetian with greater attention and curiosity than he shows any other messenger or explorer of his. In the lives of emperors there is a moment which follows pride in the boundless extension of the territories we have conquered, and the melancholy and relief of knowing we shall soon give up any thought of knowing and understanding them. There is a sense of emptiness that comes over us at evening, with the odor of the elephants after the rain and the sandalwood ashes growing cold in the braziers, a dizziness that makes rivers and mountains tremble on the fallow curves of the planispheres where they are portrayed, and rolls up, one after the other, the dispatches announcing to us the collapse of the last enemy troops, from defeat to defeat, and flakes the wax of the seals of obscure kings who beseech our protection, offering in exchange annual tributes of precious metals, tanned hides, and tortoise shell. It is the desperate moment when we discover that this empire, which had seemed to us the sum of all wonders, is an endless, formless ruin, that corruption's gangrene has spread too far to be healed by our scepter, that the triumph over enemy sovereigns has made us the heirs of their long undoing. Only in Marco Polo's accounts was Kublai Khan able to discern, through the walls and towers destined to crumble, the tracery of a pattern so subtle it could escape the termites' gnawing.

- Italo Calvino, from Invisible Cities (trans. W. Weaver)

(picture of the Weierstrass function from here.)