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.
1. The conifoldnone known
2. Hilbert modular varietiesnone known
3. $$S^3 \hookrightarrow S^7 \to S^4$$none known
4. “Geometry” in positive characteristicnone known
5. Eversion of $$S^6$$ in $$\mathbb{R}^7$$none known
6. A smooth Alexander horned sphere (except at the tips)none known
7. Isometric embedding of $$K_3$$ metricnone known
8. 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
9. 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
10. Diestel-Leader Graph (and lots of other graphs…) in 3Dnone known
11. The target manifold of the Jones polynomialnone known
12. Closed hyperbolic surface of large injectivity radiusnone known
13. Conformally correct embedding of Klein quartic in $$\mathbb{R}^3$$ w/ tetrahedral symmetrynone known
14. Fractal nature of 6j-symbols when reduced mod $$p$$none known
15. Flat surfaces with conical singularitiesnone known
16. $$B\#B\#B\#B \underset{\text{reg. hom.}}{\xrightarrow{\hspace{.35in}}} \bar{B}\#\bar{B}\#\bar{B}\#\bar{B}$$ (Regular homotopy between direct sum of four Boy's surfaces $$B,\bar{B}$$ of different chirality)none known
17. $$\bar{T}\#\bar{T} \underset{\text{reg. hom.}}{\xrightarrow{\hspace{.35in}}} T^2$$ (Regular homotopy between direct sum of two nonstandard tori $$\bar{T}$$ and the double torus $$T^2$$)none known
18. $$B\#B \underset{\text{reg. hom.}}{\xrightarrow{\hspace{.35in}}} K$$ (Regular homotopy between the direct sum of two Boy's surfaces $$B$$ and the Klein bottle $$K$$)none known
19. Drawing software for (some restricted class of) locally CAT(0) 2-complexes (e.g., square complexes embedded in $$\mathbb{R}^3$$)none known
20. 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
21. Visualize the infinitesimal (nonstandard analysis) in or with the standard finitistic universe.none known
22. “Native” $$\mathbb{R}P^3$$ viewer w/ Cayley-Klein supportnone known
23. 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
24. 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.