William Cunningham

William Cunningham's picture

Area of Research:
Email: wcunningham@perimeterinstitute.ca
Phone: 519-569-7600 x7552

Research Interests

The rapid development of computer technology over the past half century has allowed computational science to flourish as a field in its own right. At the same time, the advancement of a handful of non-perturbative approaches to quantum gravity, and even discrete geometry itself, has made clear the importance of high performance algorithms in theoretical research. Often when we get stuck trying to solve a problem analytically, we may gain insight from large-scale simulations - insight which then guides us in the right direction to solve the problem. In the past, it was commonplace to wait weeks or months for simulations to produce viable results, but today, if we are clever, we can redesign algorithms to get the same results in seconds or minutes.

The consequence is far greater than simply an accelerated pace of scientific achievements: we can now study areas of science previously inaccessible. In the causal set approach to quantum gravity, one models a discrete spacetime using an ensemble of partial orders, or equivalently directed acyclic graphs. On the kinematics side, there are dozens of results on the convergence of discrete observables to their expected continuum values, while on the dynamics side the use of the recently discovered Benincasa-Dowker-Glaser action in Monte Carlo experiments has allowed us to understand the statistical physics of these ensembles. Motivated by such results, other recent work has included the study of causal set boundaries and their geometry, both numerically and analytically.

Similarly, in the causal dynamical triangulation (CDT) approach to quantum gravity, one models a discrete spacetime by foliated simplicial complexes. More progress has been made in this approach, with recent numerical results uncovering the four phases, critical exponents, and spectral geometry of the full CDT ensemble. Others have also studied coarse-graining to understand the renormalization group of the de Sitter phase and potentially identify non-trivial ultraviolet fixed points, hence an indication that the theory is asymptotically safe.

Meanwhile, the path integral approach to loop quantum gravity called spin foams has made similar progress in numerical experiments. The introduction of the fusion basis for 3D spin foams allows us to directly model curvature and torsion excitations of a discrete 3D spacetime, leading to a coarse-graining procedure defined by a tensor network renormalization algorithm. Similarly to the other two approaches, we aim to understand the continuum theory as well as the renormalization group and phase structure.

Finally, in the field of random geometric graphs (RGGs), one aims to understand the convergence of geometric observables in random graphs uniformly embedded in Riemannian manifolds. These spatial networks are important for learning about the emergence of spatial geometry from randomness, and therefore equally important for quantum gravity in the sense that they describe spatial hypersurfaces in a Lorentzian spacetime. Recent interesting work includes the study of information routing on the Internet using an embedding into 2D hyperbolic space, as well as the study of information routing in various Lorentzian manifolds.

My current research projects include the following:

Phase Structure of 2D and 3D Causal Sets with Variable Topology.
We study causal sets using a 2D or 3D lattice gas model with periodic spatial boundaries. Causal sets elements on the lattice undergo random ergodic moves in a Metropolis Monte Carlo experiment whose measure is furnished by the Benincasa-Dowker-Glaser action. Aside from comparing the phase structure to that of the 2D orders previously studied by Glaser, O'Connor, and Surya, we also aim to understand the structure imposed by the lattice, including when we allow for random ergodic changes in the topological structure of the lattice itself. In allowing for such changes, we can measure observables using the path integral formulation, this time taking into account all allowed topologies with S1 boundary conditions. This lattice gas model is appealing compared to the 2D orders, since it can easily be extended to higher dimension, though we note the runtime is still O(N^5), making it difficult to reaches large sizes (N>500). Done in collaboration with Professor Sumati Surya, Raman Research Institute.

Ollivier Curvature in 2D Riemannian Manifolds.
We study random geometric graphs in the torus (flat curvature), sphere (positive curvature), and Bolza surface (negative curvature). After constructing the graphs, we measure the Ollivier-Ricci curvature which is speculated to be the only graph curvature measure which converges exactly to the manifold curvature. The Ollivier-Ricci curvature is furnished by the Wasserstein metric, which is a measure of distance between probability distributions on a metric space. Therefore, we are able to study the convergence of this curvature observable in three 2D manifolds by generating random geometric graphs, measuring pairwise distances of node neighborhoods via a parallel Dijkstra algorithm, and measuring the Wasserstein distance by solving a linear convex optimiztion problem. Results have applications in discrete geometry, causal set quantum gravity, and also CDTs, where a similar observable was used to define the ``quantum Ricci curvature''. Done in collaboration with Professor Dmitri Krioukov and postdoc Pim van der Hoorn, Northeastern University.

Coarse-Graining in 3D Spin Foams.
We study the coarse-graining procedure defined by the theory of tensor network renormalization applied to 3D spin foams in the fusion basis. Spin foam amplitudes are represented by high-dimensional tensors, and they are glued together using a very specific gluing algorithm. Every six iterations one cubic building block is transformed into a larger, coarser cubic building block, and so the computational requirements are intense: not only are many iterations required to study the convergence of the singular values and extract the phase structure (with simulations lasting weeks), but the high-dimensional tensors use an extreme amount of memory (often more than 1TB when torsion excitations are allowed). Future work includes the study of various relevant observables. Done in collaboration with Professor Bianca Dittrich and postdoc Sebastian Steinhaus, Perimeter Institute.

Timelike Hypersurfaces in Causal Sets.
It is currently known the Benincasa-Dowker-Glaser action for causal sets diverges for sprinklings into manifolds with timelike boundaries. At the same time, it is unknown how to measure the volume or extrinsic curvature, hence the boundary term for the action. We believe these two problems are intimately related. We study the precise nature of the divergence and use approximations for small causal diamonds to measure the extrinsic curvature. An open problem is how to accurately measure the length of non-geodesic paths in discrete settings (where paths are represented via partitions). Done in collaboration with postdoc Ian Jubb, Dublin Institute of Advanced Studies.

Other projects in earlier stages of development:

Evolutionary Algorithms for Foundational Physics (with Lee Smolin, Perimeter Insitute; Jaron Lanier and Dave Wecker, Microsoft Research; Stephon Alexander, Brown University; and Michael Toomey and Stefan Stanojevic, Brown University)
Replica Exchange Algorithms for Spin Glasses (with Francesco Caravelli and Yigit Subasi, Los Alamos National Laboratory).
Quantum Growth Dynamics for Causal Set Dynamics (with Lisa Glaser, University of Vienna; Ian Jubb, Dublin Institute of Advanced Studies; and Bohdan Kulchytskyy, 1QBit).

Other research interests:
- Combinatorics and optimization
- Graph theory and complex systems
- Graphons and large deviations theory
- Computer architecture and micro-optimization
- GPU computing
- Quantum algorithms and universal quantum computing
- Machine learning
- F Theory
- Lattice QCD

Positions Held

  • 2018-2021 Perimeter Institute for Theoretical Physics, Postdoctoral Fellow
  • 2015-2018 Northeastern University, Ph.D. Candidate
  • 2013-2015 Northeastern University, Graduate Student
  • 2009-2013 Rensselaer Polytechnic Institute, Undergraduate Student

Recent Publications

  • W.J. Cunningham, B. Dittrich & S. Steinhaus. Tensor Network Renormalization with Fusion Charges: Applications to 3D Lattice Gauge Theory. Universe 6, 97 (2020). arXiv: 2002.10472
  • W.J. Cunningham & S. Surya. Dimensionally Restricted Causal Set Quantum Gravity: Examples in Two and Three Dimensions. Class. Quant. Grav. 37, 054002 (2020). arXiv: 1908.11647
  • J. Carifio, W.J. Cunningham, J. Halverson, D. Krioukov, C. Long & B.D. Nelson. Vacuum Selection from Cosmology on Networks of String Geometries. Phys. Rev. Lett. 121, 101602 (2018). arXiv: 1711.06685
  • W.J. Cunningham & D. Krioukov. Causal Set Generator and Action Computer. Comput. Phys. Commun. 233, 123 (2018). arXiv: 1709.03013
  • W.J. Cunningham. Inference of Boundaries in Causal Sets. Class. Quant. Grav. 35, 094002 (2018). arXiv: 1710.09705
  • W.J. Cunningham, D. Rideout, J. Halverson & D. Krioukov. Exact Geodesic Distances in FLRW Spacetimes. Phys. Rev. D 96, 103538 (2017). arXiv: 1705.00730
  • W. Cunningham, K. Zuev & D. Krioukov. Navigability of Random Geometric Graphs in the Universe and Other Spacetimes. Sci. Rep. 7, 8699 (2017). arXiv: 1703.09057
  • W. Cunningham & J. Giedt. Eguchi Kawai Reduction with One Flavor of Adjoint Mobius Fermion. Phys. Rev. D 93, 045006 (2016). arXiv: 1401.0054
  • P. van der Hoorn, W.J. Cunningham, G. Lippner, C. Trugenberger & D. Krioukov. Ollivier-Ricci Curvature Convergence in Random Geometric Graphs (2020). arXiv: 2008.01209
  • B. Bahr, W.J. Cunningham, B. Dittrich, L. Glaser, D. Lang, E. Schnetter & S. Steinhaus. Data on Sharing Data. Nat. Phys. 15, 724 (2019).
  • W.J. Cunningham. High Performance Algorithms for Quantum Gravity and Cosmology. Ph.D. Dissertation, Northeastern University (2018). arXiv: 1805.04463


  • Rensselaer Polytechnic Institute, Computational Geometry for Quantum Gravity (Jun. 2020)
  • CP3-Origins, Quantum Dynamics of Total Orders (Apr. 2020)
  • CP3-Origins, Classical and Quantum Growth Models for Causal Sets (Feb. 2020)
  • Los Alamos National Laboratory, Classical and Quantum Growth Models for Discrete Spacetime (Dec. 2019)
  • Radboud Universiteit Nijmegen, Restricted Sample Spaces in Causal Set Theory (Sept. 2019)
  • Okinawa Institute of Science and Technology, Timelike Hypersurfaces in Causal Sets (Jul. 2019).
  • Rensselaer Polytechnic Institute, Why Computer Architecture Matters for HPC (Jul. 2019)
  • Rensselaer Polytechnic Institute, An Overview of Computational Linear Algebra (Jul. 2019)
  • NetSci2018, Inference of Boundaries in Causal Sets (Jun. 2018)
  • Universitat Heidelberg, Inference of Boundaries in Causal Sets (Jun. 2018)
  • Nordita, Deep Learning in Quantum Gravity (Mar. 2018)
  • Rensselaer Polytechnic Institute, Vacuum Selection from Cosmology Using Networks of String Geometries (Jan. 2018)
  • Northeastern University, Introduction to Network Science (Nov. 2017)
  • Raman Research Institute, Timelike Boundary Terms in the Causal Set Action (Dec. 2016)
  • Northeastern University, An Introduction to Parallel Programming: OpenMP, SSE/AVX, and MPI (Apr. 2016)
  • PIRSA:17120015, The Big Data Approach to Quantum Gravity, 2017-12-14, Quantum Gravity