Topics in Geometry and Dynamics

The PI plans to continue his research in geometry and dynamical systems. One part of his proposed research, concerning optimal paper geometry, builds on his recent successful solution of the long-standing optimal paper Moebius band conjecture. Another part of the proposed research, concerning geometric dynamics, deals with simple processes which start with a shape like a polygon and apply some transformation to it over and over again. Such processes, like the pentagram map, often have deep connections to other areas in science like water waves and physics-based integrable systems. The PI also proposes to work on a number of more exploratory projects, such as the connection between the famous four color theorem and triangulations of the sphere based on fullerenes. This project will also support the PI's continued impacts on society, through public lectures, many colorful graphical user interfaces, and children's mathematics books. In more technical terms, the PI hopes to prove the knotted paper Moebius conjecture, which states that any sequence of aspect ratio minimizing embedded and knotted paper Moebius bands converges, in the Hausdorff topology, to a folded ribbon knot whose underlying shape is a regular pentagon. Relatedly, the PI hopes to prove the optimality of the newly discovered Hennessey-Neinhaus construction, which provides embeddings of paper Moebius bands with arbitrarily high twisting number and uniformly bounded aspect ratio. The conjecture is that if a rectangle can be folded in space so as to make Moebius bands of arbitrarily high twisting number, then the aspect ratio of the rectangle exceeds the square root of twenty seven. For both these results, the PI hopes to leverage additional topological constraints in a geometric way. For example, the high twisting number combined with the small aspect ratio ought to imply that the paper Moebius band must coil very tightly in certain regions, creating a kind of trap. The existence of this trap places constraints on the rest of the Moebius band, possibly leading to a lower bound on its aspect ratio. In the direction of geometric dynamics, the PI plans to continue his exploration of pentagram-like maps which act nicely on certain open subsets of the moduli space of polygons modulo projective transformations. The next step in the analysis is to show that the orbits in these subspaces are pre-compact modulo projective transformations and to try to understand their collapse point via analogues of Glick's collapse point formula for the pentagram map. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.