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Organizing Committee
University of Cambridge
ICERM, Brown University
University of British Columbia
Georgia Institute of Technology
Smith College
University of St Andrews
Abstract
The geometric arrangement of a material’s constituents plays an important role in governing its behaviour. Concepts from discrete mathematics to describe these geometric arrangements, including notions of rigidity and flexibility, can bring fundamental insight into how a material might respond to stress, be designed, be reconfigured, etc. This workshop aims to build connections between the field of mathematical rigidity theory, and other areas of applied mathematics, science and engineering where such ideas might be used.
Rigidity and flexibility form a dichotomy: either a system’s constraints allow for a non-trivial deformation under an external load or they don’t. However, there is a long history, dating to before the Industrial Revolution, of designing mechanisms by starting from a system that has the minimal number of constraints required for rigidity and then removing one of them. One degree of freedom mechanisms obtained this way are a rich class. In recent years, there has been a revival of building deployable mechanisms, inspired by the Japanese arts of origami and kirigami. Generalizing these systems and the mathematics behind their rigidity and deformation has captured the interest of physicists interested in topological materials.
On a different lengthscale, our bodies also play out the dance of building rigidity and compliance into different organs and systems, from the packing of cells during embryogenesis to networks of collagen and cartilage that support our skeletons, to the dynamic networks that are formed and broken within our cells. Even continuum systems can have notions of rigidity and flexibility, both intrinsic and extrinsic to their geometry. Beyond materials, the notions of network rigidity can be applied to more abstract networks and geometries, such as those found in data science. Join us in exploring the many ways that rigidity and flexibility influence the world around us and within us.
Rigidity and flexibility form a dichotomy: either a system’s constraints allow for a non-trivial deformation under an external load or they don’t. However, there is a long history, dating to before the Industrial Revolution, of designing mechanisms by starting from a system that has the minimal number of constraints required for rigidity and then removing one of them. One degree of freedom mechanisms obtained this way are a rich class. In recent years, there has been a revival of building deployable mechanisms, inspired by the Japanese arts of origami and kirigami. Generalizing these systems and the mathematics behind their rigidity and deformation has captured the interest of physicists interested in topological materials.
On a different lengthscale, our bodies also play out the dance of building rigidity and compliance into different organs and systems, from the packing of cells during embryogenesis to networks of collagen and cartilage that support our skeletons, to the dynamic networks that are formed and broken within our cells. Even continuum systems can have notions of rigidity and flexibility, both intrinsic and extrinsic to their geometry. Beyond materials, the notions of network rigidity can be applied to more abstract networks and geometries, such as those found in data science. Join us in exploring the many ways that rigidity and flexibility influence the world around us and within us.
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Workshop Schedule
Monday, April 07, 2025
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
Higher Order Rigidity and Higher Order Derivative Tests
11th Floor Lecture Hall
Speaker
Steven Gortler, Harvard University
Session Chair
Sabetta Matsumoto, Georgia Institute of Technology
Abstract
We describe an energy-based approach for defining the ``rigidity order'' of a bar and joint framework. We also describe higher order (multivariate) generalizations of the second derivative test from calculus. Together, this gives a new proof of Connelly's theorem, that the lack of a "second order flex" implies rigidity. It gives a new proof of V. Alexandrov's theorem, that when the dimension of non-trivial infinitesimal flexes equals 1, then the lack of a "kth order flex" implies rigidity. When Alexandrov's theorem applies, the rigidity order can be computed just using a sequence of linear algebra computations.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
The 2d ideal glass is an amorphous crystal
11th Floor Lecture Hall
Speaker
Eric Corwin, University of Oregon
Session Chair
Zeyuan He, University of Cambridge
Abstract
Systems governed by a free energy landscape can be trapped forever in a local minimum, unable to find more favorable configurations. However, the introduction of transient degrees of freedom can connect otherwise distant regions of the energy landscape and allow for the discovery of rare minima with unusual properties. Using conventionally prepared over-jammed disk and sphere packings as a starting point, we show that the introduction of particle radius as a transient degree of freedom allows for the construction of “ideal” disk packings in 2d and exactly hyperuniform sphere packings in dimensions 3 and higher. Ideal disk packings are the highest density non-overlapping configurations of disks possible for a given set of particle radii. Such structures have nearly all of the thermal and mechanical properties of crystals while entirely devoid of crystalline order.
11:30 AM - 12:15 PM EDT
Backscattering-free edge states below all bands in two-dimensional auxetic media
Virtual
Virtual Speaker
Xiaoming Mao, University of Michigan
Session Chair
Zeyuan He, University of Cambridge
Abstract
Unidirectional and backscattering-free propagation of sound waves is of fundamental interest in physics and highly sought-after in engineering. Current strategies utilize topologically protected chiral edge modes in bandgaps, or complex mechanisms involving active constituents or nonlinearity. Here we propose passive, linear, one-way edge states based on spin-momentum locking of Rayleigh waves in two-dimensional media in the limit of vanishing bulk to shear modulus ratio, which provides perfect unidirectional and backscattering-free edge propagation that is immune to any edge roughness and has no limitation on its frequency (instead of residing in gaps between bulk bands). We further show that such modes are characterized by a topological winding number that protects the linear momentum of the wave along the edge. These passive and backscattering-free edge waves have the potential to enable phononic devices in the form of lattices or continua that work in previously inaccessible frequency ranges.
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Auxetic mechanisms
11th Floor Lecture Hall
Speaker
Henry Segerman, Oklahoma State University
Session Chair
Sabetta Matsumoto, Georgia Institute of Technology
Abstract
I'll talk about various theoretical and (via 3D printed models) practical designs for two- and three-dimensional auxetic mechanisms, each with a single degree of freedom. Auxetic mechanisms expand in all directions when pulled on. The classic example is the Hoberman sphere, a network of hinged arms formed into a sphere whose radius increases as the mechanism flexes. I'll talk about examples which are truly three-dimensional auxetics, tiling three-dimensional space rather than a two-dimensional surface. I'll also talk about recent work based on rack and pinion gearings in place of linkages.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
The Crystal Isometry Principle infers chemistry from geometry
11th Floor Lecture Hall
Speaker
Vitaliy Kurlin, University of Liverpool
Session Chair
Sabetta Matsumoto, Georgia Institute of Technology
Abstract
Structures of solid crystalline materials (periodic crystals) are determined in a rigid form and hence keep all their
properties under rigid motion within the same ambient environment. However, crystal structures that have different rigid shapes can substantially differ by properties and hence should be reliably distinguished, for example, polymorphs with different solubility. Conventional representations based on reduced cells discontinuously change under almost any perturbation of atoms, which led to the accumulation of near-duplicates in major databases of experimental structures [1]. This ambiguity was resolved by generically complete and continuous invariants that distinguish all non-duplicate periodic crystals in major databases within a few minutes on a modest desktop [2]. Now any dataset of experimental or simulated crystals can be visualized on maps with continuous and analytically defined invariant coordinates [3]. These invariants can be inverted to any generic periodic structure in 3 dimensions, uniquely under any distance-preserving transformation. Inspired by Richard Feynman's hint in Fig.1-7 of his first lecture on physics, the resulting Crystal Isometry Principle says that any real periodic material is uniquely determined by precise enough geometry of only atomic centers without chemical elements, under the same ambient conditions such as room temperature and normal pressure.
[1] O.Anosova, V.Kurlin, M.Senechal. The importance of definitions in crystallography. IUCrJ 11 (4), 453-463 (2024). Many more near-duplicates are summarized at https://kurlin.org/projects/ultra-fast-duplicate-detection.pdf.
[2] D.Widdowson, V.Kurlin. Resolving the data ambiguity for periodic crystals. Advances in Neural Information Processing Systems (NeurIPS), v.35, p.24625-24638 (2022). Extended version at https://arxiv.org/abs/2108.04798.
[3] D.Widdowson, V.Kurlin. Continuous invariant-based maps of the Cambridge Structural Database. Crystal Growth & Design, 24(13), 5627–5636 (2024). More maps of the materials spaces are at http://arxiv.org/abs/2410.13796.
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, April 08, 2025
9:00 - 9:45 AM EDT
Exploring Damage in Mechanical Metamaterials
11th Floor Lecture Hall
Speaker
Marcelo Dias, The University of Edinburgh
Session Chair
Louis Theran, University of St Andrews
Abstract
Engineers have long sought to improve mechanical performance and damage tolerance by combining different materials. In this talk, we extend these efforts to the realm of mechanical metamaterials and examine the challenges inherent in designing advanced damage mechanisms. We will focus on structures on the verge of mechanical instability (also known as isostatic or Maxwell lattices), where stress concentrations are dominated by robust topology and geometry-dependent parameters. The damage mechanisms in Maxwell lattices arise from the localization of surface floppy modes and states of self-stress, which can be controlled through topological mechanical polarization related to the unit-cell geometry. By combining different topologically distinct lattices, we will demonstrate how to manipulate the damage path and fracture toughness.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
Request Reimbursement
This section is for general purposes only and does not indicate that all attendees receive funding. Please refer to your personalized invitation to review your offer.
- ORCID iD
- As this program is funded by the National Science Foundation (NSF), ICERM is required to collect your ORCID iD if you are receiving funding to attend this program. Be sure to add your ORCID iD to your Cube profile as soon as possible to avoid delaying your reimbursement.
- Acceptable Costs
- 1 roundtrip between your home institute and ICERM
- Flights on U.S. or E.U. airlines – economy class to either Providence airport (PVD) or Boston airport (BOS)
- Ground Transportation to and from airports and ICERM.
- Unacceptable Costs
- Flights on non-U.S. or non-E.U. airlines
- Flights on U.K. airlines
- Seats in economy plus, business class, or first class
- Change ticket fees of any kind
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- Advance Approval Required
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- Travel Maximum Contributions
- New England: $350
- Other contiguous US: $850
- Asia & Oceania: $2,000
- All other locations: $1,500
- Note these rates were updated in Spring 2023 and supersede any prior invitation rates.
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Refer to the back of your ID badge for more information. Checklists are available at the front desk and in the Reimbursement section of Cube.
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- Reimbursement Timing
6–8 weeks after all documentation is sent to ICERM. All reimbursements are reviewed by multiple Brown University offices.
- Reimbursement Deadline
Submissions must be received within 30 days of ICERM departure to avoid applicable taxes. No submissions are accepted after six months.
