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.
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  1. The conifoldR. Fieldnone known
  2. Hilbert modular varietiesJ. Quinnnone known
  3. \(S^3 \hookrightarrow S^7 \to S^4 \)H. Toddnone known
  4. “Geometry” in positive characteristicD. Dumasnone known
  5. Eversion of \(S^6\) in \(\mathbb{R}^7\)A. Chéritatnone known
  6. A smooth Alexander horned sphere (except at the tips)A. Chéritatnone known
  7. Isometric embedding of \(K_3\) metricA. Hansonnone known
  8. A picture of spheres of radius \(R \gg 0\) in complex or quaternionic hyperbolic spaces \(Sl_3\mathbb{R}/SO(3)\mathbb{R}\)I. Chatterjinone 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)R. Coulonnone known
  10. Diestel-Leader Graph (and lots of other graphs…) in 3DA. Holroydnone known
  11. The target manifold of the Jones polynomialJ. Levittnone known
  12. Closed hyperbolic surface of large injectivity radiusR. Kenyonnone known
  13. Conformally correct embedding of Klein quartic in \(\mathbb{R}^3\) w/ tetrahedral symmetryH. Segermannone known
  14. Fractal nature of 6j-symbols when reduced mod \(p\)J. Carternone known
  15. Flat surfaces with conical singularitiesB. Muetzelnone 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)U. Pinkallnone 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 \(T^2\))A. Chernnone 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 \(K\))A. Chernnone known
  19. Design software for drawing (some restricted class of) locally CAT(0) 2-complexes (e.g., square complexes embedded in \(\mathbb{R}^3\))A. Abramsnone 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.)
  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 (\(\mathbb{Z},\mathbb{Z}_p\))…A. Fenyesnone known
  21. Visualize the infinitesimal (nonstandard analysis) in or with the standard finitistic universe.M. Flashmannone known
  22. “Native” \(\mathbb{R}P^3\) viewer w/ Cayley-Klein supportC. Gunnnone 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 \(\{(z,w) \in \mathbb{C} \times \mathbb{C} | z^2 = w^3\}\)
    Z. Zhaonone known
  24. Penner cell decomposition of Teichmüller space for a surface with a small number of punctures.K. Cranenone known
Pictured above: Saul Schleimer at the ICERM Workshop on Illustrating Geometry and Topology.