Open Challenges in Illustrating Geometry & Topology

Visualization plays a central role in our understanding of objects from geometry and topology (not to mention other areas of mathematics). It can inspire new conjectures and methods of proof—or simply expose the beauty of the subject to a broader audience. Some historical examples include sphere eversions, Costa's minimal surface, Thurston's circle packing conjecture, and surfaces arising from dimer coverings, to name just a few. Participants at the 2019 ICERM Workshop on Illustrating Geometry & Topology were asked to identify the next generation of challenges in geometric and topological visualization: *which objects should we visualize?*This page serves as a record of that list, and a way to keep track of new developments.

Send email to kmcrane@cs.cmu.edu for corrections, clarifications, or updates.

- The conifoldnone known
- Hilbert modular varietiesnone known
- \(S^3 \hookrightarrow S^7 \to S^4 \)none known
- “Geometry” in positive characteristicnone known
- Eversion of \(S^6\) in \(\mathbb{R}^7\)none known
- A smooth Alexander horned sphere (except at the tips)none known
- Isometric embedding of \(K_3\) metricnone known
- A picture of spheres of radius \(R \gg 0\) in complex or quaternionic hyperbolic spaces \(Sl_3\mathbb{R}/SO(3)\mathbb{R}\)none known
- The construction of the Picard-Manin space (with level sets of the intersection form) associated to the Cremona group (birational transformations of the complex proj. space)none known
- Diestel-Leader Graph (and lots of other graphs…) in 3Dnone known
- The target manifold of the Jones polynomialnone known
- Closed hyperbolic surface of large injectivity radiusnone known
- Conformally correct embedding of Klein quartic in \(\mathbb{R}^3\) w/ tetrahedral symmetrynone known
- Fractal nature of 6j-symbols when reduced mod \(p\)none known
- Flat surfaces with conical singularitiesnone known
- \(B\#B\#B\#B \underset{\text{reg. hom.}}{\xrightarrow{\hspace{.35in}}} \bar{B}\#\bar{B}\#\bar{B}\#\bar{B}\) (Regular homotopy between direct sum of four Boy's surfaces \(B,\bar{B}\) of different chirality)none known
- \(\bar{T}\#\bar{T} \underset{\text{reg. hom.}}{\xrightarrow{\hspace{.35in}}} T^2\) (Regular homotopy between direct sum of two nonstandard tori \(\bar{T}\) and the double torus \(T^2\))none known
- \(B\#B \underset{\text{reg. hom.}}{\xrightarrow{\hspace{.35in}}} K\) (Regular homotopy between the direct sum of two Boy's surfaces \(B\) and the Klein bottle \(K\))none known
- Design software for drawing (some restricted class of) locally CAT(0) 2-complexes (e.g., square complexes embedded in \(\mathbb{R}^3\))none knownThere are many algorithms for, and theorems about, drawing (a) planar graphs in the plane, (b) non-planar graphs in the plane (with crossings), and (c) general graphs in \(\mathbb{R}^3\). Of course there are also open problems, like determining the fewest number of crossings in case (b). But the algorithms work pretty well, most of the time. Drawing 2-complexes in \(\mathbb{R}^3\) seems to be harder, even when the 2-complexes embed in \(\mathbb{R}^3\) (e.g. square-tiled surfaces). Are there any algorithms that work pretty well most of the time? If the 2-cells are all simplices, can they be filled in in a reasonably nice way? If there are squares, can each square be nearly planar? Feel free to restrict the class of 2-complexes under consideration; one interesting category is embeddable locally CAT(0) square complexes. (Probably square-tiled surfaces are not the place to start, since we already have high standards for what we want those to look like.)
- Locally ringed spaces with weird rings of functions, e.g., superfunctions (\(\bigwedge \mathbb{R}^n\), \(\mathbb{R}[\varepsilon]/\varepsilon^2\), …), formal power series (\(\mathbb{C}[\![ z ]\!]\)), discrete functions (\(\mathbb{Z},\mathbb{Z}_p\))…none known
- Visualize the infinitesimal (nonstandard analysis) in or with the standard finitistic universe.none known
- “Native” \(\mathbb{R}P^3\) viewer w/ Cayley-Klein supportnone known
- Visualization of singular points in area minimizers:

Simons' cone \(\{(x,y) \in \mathbb{R}^4 \times \mathbb{R}^4 | x_1^2 + x_2^2 + x_3^2 + x_4^2 = y_1^2 + y_2^2 + y_3^2 + y_4^2\}\)

An area minimizer of codimension two \(\{(z,w) \in \mathbb{C} \times \mathbb{C} | z^2 = w^3\}\)none known - Penner cell decomposition of Teichmüller space for a surface with a small number of punctures.none known

Pictured above: Saul Schleimer at the ICERM Workshop on Illustrating Geometry and Topology.