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Organizing Committee
Technische Universität Berlin
James Madison University
The Florida State University
Harvard University
University of Florida
Abstract
This workshop brings together researchers from three distinct streams of mathematics: the classical rigidity theory of bar-joint and tensegrity frameworks in combinatorics and discrete geometry; the theory of generalized circle packing that arose from the study of 3-manifolds in geometric topology, extending to sphere packing and jamming; and discrete differential geometry. A scattering of results in recent years has started to forge connections among these fields.
Since the discovery that circle packings from triangulations could be used as a scheme for approximating the Riemann mapping of a simply connected proper domain in the plane to the unit disk, the theory of circle packing has enjoyed enormous development and has found widespread theoretical and practical applications. In the theoretical realm, circle packing provides a discrete analytic function theory that is faithful to its continuous cousin and it is closely associated with studies of hyperbolic and projective polyhedra. The theory has been applied to build discrete minimal surfaces, conformally flatten curved and crumpled surfaces, and push further our understanding of the rigidity and existence of non-compact hyperbolic polyhedra.
Recent work has shown that Euclidean frameworks have an affinity with circle and sphere packings, with Euclidean rigid motions replaced by Mobius transformations, and that the language and methods of rigidity theory could provide a guide for how to approach certain problems in circle packings, sphere packings, and jamming. For example, working in the space of circles in which circles are points, coaxial families of circles are lines, and certain bundles of circles are planes, a 3-dimensional incidence geometry appears that behaves in many ways as the usual Euclidean 3-dimensional space.
The main aim of the workshop is to encourage the cross-fertilization of such ideas, with particular emphasis on the rigidity of inversive distance packings. Participants will attend presentations on cutting-edge research and initiate new collaborations.
Since the discovery that circle packings from triangulations could be used as a scheme for approximating the Riemann mapping of a simply connected proper domain in the plane to the unit disk, the theory of circle packing has enjoyed enormous development and has found widespread theoretical and practical applications. In the theoretical realm, circle packing provides a discrete analytic function theory that is faithful to its continuous cousin and it is closely associated with studies of hyperbolic and projective polyhedra. The theory has been applied to build discrete minimal surfaces, conformally flatten curved and crumpled surfaces, and push further our understanding of the rigidity and existence of non-compact hyperbolic polyhedra.
Recent work has shown that Euclidean frameworks have an affinity with circle and sphere packings, with Euclidean rigid motions replaced by Mobius transformations, and that the language and methods of rigidity theory could provide a guide for how to approach certain problems in circle packings, sphere packings, and jamming. For example, working in the space of circles in which circles are points, coaxial families of circles are lines, and certain bundles of circles are planes, a 3-dimensional incidence geometry appears that behaves in many ways as the usual Euclidean 3-dimensional space.
The main aim of the workshop is to encourage the cross-fertilization of such ideas, with particular emphasis on the rigidity of inversive distance packings. Participants will attend presentations on cutting-edge research and initiate new collaborations.
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Workshop Schedule
Monday, February 10, 2025
8:50 - 9:00 AM EST
Welcome
Welcome - 11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EST
Why Does Rigidity Matter?
11th Floor Lecture Hall
Speaker
Robert Connelly, Cornell University
Session Chair
Meera Sitharam, University of Florida
Abstract
Rigidity of frameworks, especially tensegrity frameworks appear in many natural and unexpected places. A tensegrity is just a configuration of points in some Euclidean space, together with upper distance constraints (cables) or lower distance constraints (struts) for some pairs of the vertices. I'll show physical tensegrities with chains for cables and steel tubes for struts, where you can "feel" the rigidity with your hands, and other tensegrities, where the rigidity is lurking in the background. I will also show connections of rigidity to circle and sphere packings, where configurations in a lower dimension are rigid, universally, in any higher dimension, where rigidity is related to polyhedra, and questions where certain kinds natural rigidity are not known, .. yet.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EST
Experimental Takes: circle packing and discrete schwarzians
11th Floor Lecture Hall
Speaker
Kenneth Stephenson, University of Tennessee
Session Chair
Meera Sitharam, University of Florida
Abstract
This talk will emphasize the value of experimental approaches to a variety of topics in conformal geometry, particularly using the software "CirclePack". Emerging notions related to discrete Schwarzian derivatives will be illustrated.
11:30 AM - 12:15 PM EST
Smooth Isometries of Quad Nets
11th Floor Lecture Hall
Speaker
Andy Sageman-Furnas, North Carolina State University
Session Chair
Meera Sitharam, University of Florida
Abstract
We introduce a new notion of isometry between immersions of a quad graph in Euclidean 3-space. In contrast to bar-joint frameworks, corresponding edges may not have the same length in 3-space. Instead, one compares using certain lifts to 4-space. This allows to capture the structure of smooth isometries, independent of resolution. Exemplifying this perspective, we discuss computational experiments immersing a 5x7 quadrangulation of a torus, which led to the discovery of analytic compact Bonnet pairs: two noncongruent tori that correspond via a mean-curvature preserving isometry (Bobenko, Hoffmann, S.-F., 2023).
More generally, we develop spin (conformal) transformations, where an invariant spin cross-ratio generalizes the complex cross-ratio. This allows us to define non-planar quad nets in conformal coordinates, such as discrete Bonnet pairs. We also investigate one-parameter isometric associated families of discrete minimal surfaces.
This is joint work with Tim Hoffmann and Max Wardetzky.
12:30 - 2:30 PM EST
Lunch/Free Time
2:30 - 3:15 PM EST
Infinite circle packings on surfaces with conical singularities
11th Floor Lecture Hall
Speaker
Lorenzo Ruffoni, Binghamton University
Session Chair
John Bowers, James Madison University
Abstract
One of the fundamental problems in the theory of circle packings concerns the existence of geometric structures that admit a circle packing with prescribed combinatorics. The case of constant curvature metrics on closed surfaces is well-understood. In this talk we consider non-compact surfaces of finite type and triangulations that are infinite (i.e. punctures are not vertices of the triangulations). We show that given such a triangulation and a collection of target angles satisfying a Gauss-Bonnet inequality, there exists a singular hyperbolic metric admitting a circle packing with the prescribed combinatorics and having the prescribed conical singularities. Moreover, we construct uncountably many examples of hyperbolic metrics that admit a packing with a fixed combinatorics over a fixed punctured Riemann surface, in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case. This is joint work with P. Bowers.
3:30 - 4:00 PM EST
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EST
Optimal disc and sphere packings
11th Floor Lecture Hall
Virtual Speaker
Daria Pchelina, ENS Lyon
Session Chair
John Bowers, James Madison University
Abstract
How can we arrange an infinite number of spheres to maximize the proportion of space they occupy? Kepler conjectured that the "cannonball" packing is the optimal way to do it. This conjecture remained unproved for almost 400 years until Hales and Ferguson provided a 250-page proof accompanied by hundreds of thousands of lines of computer code.
Given an infinite number of coins of three fixed radii, how can we place them on the plane to maximize the proportion of the covered surface? A disc packing is called triangulated if its contact graph is triangulated. We identified optimal packings for several triplets of disc sizes, all of which are triangulated. Conversely, we also showed that for certain other triplets, no triangulated packing is optimal.
Building on our expertise in multi-size disc packings, we extend our research to two-sphere packings. Simplicial sphere packings are those whose contact graphs form pure simplicial 3-complexes. We consider the only ratio of sphere sizes that allows such packings which are conjectured to be optimal.
5:00 - 6:30 PM EST
Reception
11th Floor Collaborative Space
Tuesday, February 11, 2025
9:00 - 9:45 AM EST
Discrete hyperbolic Laplacian
11th Floor Lecture Hall
Speaker
Wai Yeung Lam, Université du Luxembourg
Session Chair
Steven Gortler, Harvard University
Abstract
The Laplace operator on a Riemannian manifold is a fundamental tool to study the geometry of the manifold. Inspired by electric networks, Laplacians on graphs are defined with edge weights playing the role of conductance. When the edge weights are constant, the graph Laplacian becomes the combinatorial Laplacian and is known to reveal rich combinatorial information of the graph. Given a graph embedded on a surface, it is natural to consider a geometric Laplacian, where edge weights are adapted to the geometry. For the 1-skeleton graph of a geodesic triangulation on a Euclidean surface, there is a “cotangent formula” relating the edge weights to the Euclidean metric. It is known to connect with various problems, e.g. deformations of circle patterns, Delaunay decomposition and discrete harmonic maps. In the talk, we introduce the analogue for hyperbolic surfaces. This is joint work with Ivan Izmestiev.
10:00 - 10:30 AM EST
Coffee Break
11th Floor Collaborative Space
