Open Challenges in Illustrating Geometry & Topology

The conifold R. Field none known Hilbert modular varieties J. Quinn none known \(S^3 \hookrightarrow S^7 \to S^4 \) H. Todd none known “Geometry” in positive characteristic D. Dumas none known Eversion of \(S^6\) in \(\mathbb{R}^7\) A. Chéritat none known A smooth Alexander horned sphere (except at the tips) A. Chéritat none known Isometric embedding of \(K_3\) metric A. Hanson none known A picture of spheres of radius \(R \gg 0\) in complex or quaternionic hyperbolic spaces \(Sl_3\mathbb{R}/SO(3)\mathbb{R}\) I. Chatterji 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) R. Coulon none known Diestel-Leader Graph (and lots of other graphs…) in 3D A. Holroyd none known The target manifold of the Jones polynomial J. Levitt none known Closed hyperbolic surface of large injectivity radius R. Kenyon none known Conformally correct embedding of Klein quartic in \(\mathbb{R}^3\) w/ tetrahedral symmetry H. Segerman none known Fractal nature of 6j-symbols when reduced mod \(p\) J. Carter none known Flat surfaces with conical singularities B. Muetzel none 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) U. Pinkall 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 \(T^2\)) A. Chern 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 \(K\)) A. Chern none known Design software for drawing (some restricted class of) locally CAT(0) 2-complexes (e.g., square complexes embedded in \(\mathbb{R}^3\)) A. Abrams none known 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. Fenyes none known Visualize the infinitesimal (nonstandard analysis) in or with the standard finitistic universe. M. Flashman none known “Native” \(\mathbb{R}P^3\) viewer w/ Cayley-Klein support C. Gunn none 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 \(\{(z,w) \in \mathbb{C} \times \mathbb{C} | z^2 = w^3\}\) Z. Zhao none known Penner cell decomposition of Teichmüller space for a surface with a small number of punctures. K. Crane none known

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 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.