Current research projects

Recently, Tropical Geometry has played a pivotal part in the study of Geometric Rigidity Theory. My current research on this link can be split into three separate branches.

• Understanding and constructing flexible linkages.

• Computing the number of edge-length equivalent realisations for a given graph.

• Investigating connections between extremal tropical varieties and rigidity.

Research papers on topic:

1. Flexing infinite frameworks with applications to braced Penrose tilings. with J. Legerský. Discrete Applied Mathematics 324, 1–17. https://doi.org/10.1016/j.dam.2022.09.002

2. Flexible placements of graphs with rotational symmetry. with G. Grasegger, J. Legerský. 2nd IMA Conference on Mathematics of Robotics, 89–97. https://doi.org/10.1007/978-3-030-91352-6_9

3. Flexible placements of periodic graphs in the plane. Discrete and Computational Geometry 66, 1286–1329. https://doi.org/10.1007/s00454-021-00328-x

4. Computing animations of linkages with rotational symmetry (media exposition). with G. Grasegger, J. Legerský. 36th International Symposium on Computational Geometry (SoCG 2020). https://doi.org/10.4230/LIPIcs.SoCG.2020.77

I am interested in understanding what occurs for standard rigidity results when the very idea of distance is altered in some way. This can be either switching to an alternative norm/metric, or by changing what is being measured (such as angles or volume).

Research papers on topic:

1. Rigid graphs in cylindrical normed spaces. with D. Kitson. http://arxiv.org/abs/2305.08421 

2. Coincident-point rigidity in normed planes. with J. Hewetson, A. Nixon. Ars Mathematica Contemporanea 24:1, #P1.10. https://doi.org/10.26493/1855-3974.2826.3dc

3. Uniquely realisable graphs in analytic normed planes. with J. Hewetson, A. Nixon. https://arxiv.org/abs/2206.07426

4. Infinitesimal rigidity and prestress stability for frameworks in normed spaces. Discrete Applied Mathematics 322, 425–438. https://doi.org/10.1016/j.dam.2022.09.001

5. Generalised rigid body motions in non-Euclidean planes with applications to global rigidity. with A. Nixon. Journal of Mathematical Analysis and Applications 514:1, 126259. https://doi.org/10.1016/j.jmaa.2022.126259

6. Which graphs are rigid in Lpd? with D. Kitson, A. Nixon. Journal of Global Optimization 83, 49–71. https://doi.org/10.1007/s10898-021-01008-z

7. Equivalence of continuous, local and infinitesimal rigidity in normed spaces. Discrete and Computational Geometry 65, 655–679. https://doi.org/10.1007/s00454-019-00135-5

8. Infinitesimal rigidity in normed planes. SIAM Journal on Discrete Mathematics 34:2, 1205–1231. https://doi.org/10.1137/19M1284051

The contact structure of convex body packings (including sphere packings, cube packings, circle packings, square packings, etc.) is often poorly understood and difficult to determine. I am interested in determining contact graphs for these packings when: (i) every object has some fixed size, or (ii) every object has a random size. Convex body packings appear in multiple scientific areas, including Colloidal Matter Physics, Crystallography and Cryptography.

Research papers on topic:

1. On the uniqueness of collections of pennies and marbles. with G. Grasegger, K. Kubjas, F. Mohammadi, A. Nixon. http://arxiv.org/abs/2307.03525

2. Identifying contact graphs of sphere packings with generic radii is NP-Hard. https://arxiv.org/abs/2302.12588

3. How many contacts can exist between oriented squares of various sizes? Discrete Mathematics 347:4, 113879. https://doi.org/10.1016/j.disc.2024.113879 

4. Homothetic packings of centrally symmetric convex bodies. Geometriae Dedicata 216, 11. https://doi.org/10.1007/s10711-022-00675-w

A natural problem in Statistics is obtaining maximum likelihood estimators (MLEs) for various Gaussian graphical models; multivariate normal distributions where known conditional independences are modelled as non-edges. Uhler (2012) investigated what is known as the maximum likelihood threshold (MLT) of Gaussian graphical models: the minimum number of samples to almost surely guarantee the existence of an MLE. Gross and Sulivant (2018) proved that this value can bounded by the lowest dimension where the graph has only stress-free realisations. 

Recently, myself and others proved that the MLT of a graph is directly linked to the highest dimension where the exist internally-stressed generic realisations with a positive semi-definite associated stress matrix. This understanding has lead to many breakthroughs on the topic, including exact computations of the MLT for large families of graphs.

I am currently determining exact MLT values for large families of graphs (such as those with low genus), and seeking further problems in algebraic statistics that can be considered through the lens of Geometric Rigidity Theory.

Research papers on topic:

1. Maximum likelihood thresholds via graph rigidity. with D.I. Bernstein, S.J. Gortler, A. Nixon, M. Sitharam, L. Theran. Annuals of Applied Probability (accepted). https://arxiv.org/abs/2108.02185

2. Computing maximum likelihood thresholds using graph rigidity. with D.I. Bernstein, S.J. Gortler, A. Nixon, M. Sitharam, L. Theran. Algebraic Statistics (forthcoming). https://arxiv.org/abs/2210.11081