Past

TBA

Multiparameter persistent homology (MPH) associates to a vector-valued filter function on a simplicial complex a multiparameter persistence module. In this talk, I will present recent joint work with Heather Harrington and Ulrike Tillmann on the inverse problem for MPH: given a persistence module, what can we say about the data that generated it? To study the geometry of the fibers of MPH, we equip both the space of filters and the moduli space of essentially finite persistence modules with natural stratifications that are preserved by the map. Over each stratum in its image, MPH restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We also bound the dimension of MPH fibers in terms of multigraded Betti numbers, recovering the one-parameter bound of Leygonie and Tillmann as a special case.

Come along for free pizza and to hear about the Mathematrix events this term.

Junior Algebra & Representation Theory starts with a week 1 social. Join us in the Quillen Room!

In this talk, we introduce deep learning schemes for forward–backward systems whose backward component is a backward stochastic Volterra integral equation (BSVIE). BSVIEs generalize classical BSDEs by introducing a second time variable, and arise naturally in recursive utilities with memory, time-inconsistent stochastic control, and dynamic risk measures.

Building on deep solvers for BSDEs, we develop backward learning schemes adapted to the two-time structure of BSVIEs. When the forward process is Markovian, the solution is given by deterministic functions of the two time variables and the forward state, which we approximate with neural networks. We prove the algorithm's convergence by decomposing the error into a time-discretization error and a neural network approximation error, and validate it numerically.

When the forward process is itself of Volterra type, the Markovian structure is lost, and the solution depends on the entire past trajectory. Using a path-dependent PDE representation, we approximate the solution with neural networks taking path signatures as inputs, and test the method on examples with explicit solutions.

Based on joint works with Nacira Agram, and with Nacira Agram and Mounir Zebbar.

Ruth Misener will talk about: 'Optimizing over graphs: Challenges, Formulations, and Applications'

Applications involving optimization over graphs include molecular design, graph neural network verification, neural architecture search, etc. This talk discusses formulating graph spaces using mixed-integer optimization and incorporating application-specific constraints. We discuss computational challenges with these mixed-integer optimization formulations and zoom in on the practical implications for these applications. We mention what has been done (by both ourselves and others) and what other research still needs to be done.

Co-authors: Shiqiang Zhang, Yilin Xie, Christopher Hojny, Juan Campos, Jixiang Qing, Christian Feldmann, David Walz, Frederik Sandfort, Miriam Mathea, Calvin Tsay

This talk is hosted by Rutherford Appleton Laboratory, Harwell Campus

I will present my recent results on global regularity for a canonical shell model of the Navier-Stokes equations, covering the unforced equation and certain classes of forcing. I will discuss the distinction between the forced and unforced regularity problems in this model, with the corresponding Navier-Stokes questions as motivation.

Internal gravity waves (IGWs) are oscillatory modes of motion that exist in the interior of a stably stratified fluid with gravity as the restoring force. In Earth's oceans, these waves play a fundamental role in the transport and mixing of energy, momentum, heat, carbon and nutrients. Much like ocean surface waves, IGWs become unstable and break when their amplitude is large and density interfaces become steeply inclined. This talk will explore the evolution and fate of such waves in two different scenarios.

I will first focus on IGWs that propagate into horizontally sheared currents, explaining how the fate of these waves depends both on their geometry and the strength of the shear flow, and why the total energy dissipated by ensuing turbulence is unexpectedly large. Second, I will discuss the dynamics of internal bores – long, large-amplitude solitary waves that propagate along sharp density interfaces. I will outline the mathematical links between internal bores and gravity currents and, by studying observations off the coast of California, demonstrate their relevance to understanding transport and dissipation on the continental shelf.

David Niederkofler (TU Wien) is going to talk about: "Optimal Time-Adaptivity for Parabolic Problems"

Since the first optimality proofs for adaptive mesh refinement algorithms in the early 2000s, the theory of optimal mesh refinement for PDEs was inherently limited to stationary problems. The reason for this is that time-dependent problems usually do not exhibit the necessary coercive structure that is used in optimality proofs to show a certain quasi-orthogonality, which is crucial for the theory. Recently, by using a new equivalence between quasi-orthogonality and inf-sup stability of the underlying problem, it was shown that an adaptive Crank-Nicolson scheme for the heat equation is optimal under a severe step size restriction. In this work, we use this new approach towards quasi-orthogonality together with Radau IIA methods of any order larger than one to obtain the first adaptive time stepping method for non-stationary PDEs that is provably rate optimal with respect to number of time steps vs. approximation error.

We show that if a group admits an action with no algebraicity, then all of its mixed identities are singular. This generalises a theorem of Abért showing such groups are lawless. Our result confirms a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups, and it also applies to several groups arising in topological contexts. I'll also discuss how ideas from independence relations in model theory arise in the proof and yield more general results, also applying to infinite-dimensional general (and projective) linear groups over arbitrary fields, recovering results of Bradford, Schneider, and Thom.

This is joint work with Michael Pinsker.

In this talk I will discuss homological and homotopical isoperimetric inequalities, including those of higher rank.  I will emphasise what is known in the non-positive curvature case.

The anyons of a (2+1)d topological order are described mathematically by a modular tensor category (MTC). Given an MTC, any n-component link becomes an n-tensor that stores some information about this MTC. For example, the 1-component unknot (twisted once) gives a vector "T" that stores the topological spins of all anyons; the 2-component Hopf link evaluates to matrix "S", from which the Verlinde formula recovers the fusion rules of the anyons. Do T and S store all the data? The answer is no: there are different MTCs with the same S and T. So the question becomes what other links to include in this list of tensors to tell MTCs apart? 
I will start by telling you how to turn any link you can drawn on a piece of paper (or a whiteboard) in to a tensor, then review the T and S story and then discuss some recent work where people considered adding the Whitehead link and the Borromean rings. There will be lots of drawings!

Kesten's theorem characterises amenability in terms of spectral radii: a symmetric random walk on a group G has spectral radius 1 if and only if it generates an amenable subgroup.

We prove a quantitative "1%" version of this result for G a free group, or a fully residually free group: for every ε > 0, there is δ > 0 such that for every probability measure μ on G with convolution operator norm at least ε, there is a co-set of an amenable subgroup of G of mass > δ under μ.

This answers a question of Emmanuel Breuillard. We believe the same result to hold in a broader setting of linear groups, with δ depending on the dimension. The proof uses uniform exactness and a generalisation of the Tits alternative.

Joint work with Emmanuel Breuillard.
 

Riemannian manifolds whose holonomy group lies inside the exceptional Lie group G_2 are called G_2-manifolds. These manifolds have several interesting properties (they are Ricci-flat) and are of interest in geometric analysis as well as in mathematical physics and other fields. In this seminar, we will give an introduction to the theory of G_2-manifolds and discuss an associated non-linear Laplacian-type operator whose kernel essentially determines whether a compact manifold is G_2. We will present uniqueness and existence results for the Poisson’s equation of this operator on homogeneous and cohomogeneity-one manifolds.

In 1989, Benjamin Weiss proved that for a countable group G, there exists a global minimal G-flow which is a topological model for every essentially free p.m.p. Borel G-action. In this talk, I will present our vast generalization, asserting that any essentially free Borel measured action (without the p.m.p. assumption), admits a minimal compact model. Moreover, we show that given a random walk on the group, there exists a global minimal topological model for all essentially free stationary Borel actions.

Based on joint work with Yair Hartman.

In their recent ICM address, Dan Halpern-Leistner and Jarod Alper posed the question: does the stack of canonically polarized curves admit a theta-stratification whose semistable locus is the moduli space of stable curves? In other, perhaps more familiar, language: is there some notion for curves that understands instability the way Harder-Narasimhan type does for vector bundles/coherent sheaves?

I will explain some joint work with Dave Swinarski that solves the analogous problem for certain rational curves, and then sketch out an argument that I think should solve the problem in general.

This talk presents a geometric machine learning framework for resolving algebraic decision problems, with a focus on the Word Problem and post-quantum cryptography. Unreduced words in the Baumslag-Solitar group BS(1,2) and Artin groups are mapped into graphs constructed by their specific generators and defining relations.

To construct a continuous representation of the underlying group, a Graph Neural Network is trained using contrastive triplets (W,P,N). In this setup, W represents a base word, P is a positive example (a word algebraically equivalent to W), and N acts as a negative decoy (a non-equivalent word). The network embeds these graphs into a 128-dimensional unit sphere, forcing algebraically identical elements to cluster together while separating distinct elements.

This learned geometric space offers direct computational solutions to classical group-theoretic problems. It is used to successfully launch a cryptanalytic attack on the Wagner-Magyarik cryptosystem. Furthermore, by directly linking the continuous embedding back to the discrete metric space of the group, a variant network architecture accurately predicts the reduced geodesic length of randomly generated sequences.

The framework is subsequently applied to Right-Angled Artin Groups (RAAGs) and random Artin groups to evaluate the Nielsen equivalence of subgroup generating sets. The presentation concludes by outlining how these structural graph representations can be adapted to estimate Gromov hyperbolicity and extract other coarse geometric invariants from infinite groups.
 

Conlon and Gowers in 2016 described a general approach to proving sparse random analogues of extremal results in combinatorics, such as bounding the minimum and maximum number of triangles in any subgraph of G(n,p) with a given number of edges. The general part of this approach is a functional-analytic statement which, given a sparse setting, constructs a 'dense model'. However there is a condition which must be shown to hold with high probability to apply the dense model theorem. In Conlon and Gowers' work, there is a technical difficulty with the probabilistic part which leads to a rather involved proof, which applies only in a restricted setting (for example, they can handle triangles but not triangles with a pendant edge), and with quite poor bounds on 'high probability'.

Around the same time Schacht, with a very different method, was able to prove a related result, which is applicable in a much more general setting and which has optimal bounds on 'high probability': but Schacht's result does not provide sharp lower bounds on the number of triangles, and does not provide any upper bounds. What it does do is identify the number of edges in a subgraph of G(n,p) which guarantee that triangles will appear; subsequently the 'hypergraph container method' provides another approach to proving this kind of result, but again does not provide sharp lower bounds or any upper bounds.

We revisit Conlon and Gowers' approach, and show how to avoid their technical problem, giving a simpler proof of their counting result which applies in the general setting and with optimal probability bounds. As a corollary, we prove the 'Counting KLR' theorem of Conlon, Gowers, Samotij and Schacht, but for general hypergraphs and with optimal probability bounds. This is joint work with Julia Boettcher, Joanna Lada and Domenico Mergoni.

I will describe a method that deforms analytic cycles to algebraic cycles. This will be done using global derived analytic geometry and analytic (Efimov) K-theory. I will conjecture how this is related to the Tate and Hodge conjectures. If time permits, I will speculate about a global arithmetic site. This is joint work in progress with Federico Bambozzi, Jack Kelly, and Devarshi Mukherjee.

Calibrated fibrations are expected to play an important role in exceptional holonomy, much as special Lagrangian fibrations do in the SYZ picture for Calabi--Yau manifolds. Singular fibres are expected to be essential, and a natural question is whether they are forced by the geometry. In his PhD thesis, Baraglia showed that coassociative fibrations of compact full-holonomy G_2-manifolds must have singular fibres. The analogous problem for Cayley fibrations remained open for many years and turns out to be substantially more difficult, with its resolution relying on deep results from 4-manifold topology.

The proof takes some unexpected twists and turns, involving a Diophantine equation arising from the Spin(7)-structure, families Seiberg--Witten theory and parametrized homotopy theory. After a short crash course on Spin(7) geometry, I will explain how these pieces fit together. This is joint work with Jianfeng Lin and Viktor Majewski.

Professor Lorenzo Pareschi is going to talk about; 'Bridging high-order numerics and machine learning for kinetic plasma simulation'

Reliable uncertainty quantification is a central challenge in kinetic plasma simulation, where high dimensionality, multiple physical scales, and sensitivity to uncertain inputs make repeated high-fidelity computations prohibitively expensive. This is particularly relevant in fusion-oriented applications, for which accurate predictions require sophisticated numerical solvers but direct sampling is often out of reach.

In this talk, I will present a multifidelity framework for the Vlasov–Poisson–Landau system designed to combine, rather than replace, high-order numerical simulation with machine learning. At the high-fidelity level, asymptotic-preserving and structure-aware solvers provide accurate kinetic descriptions across different regimes. These are coupled with reduced plasma models and tensor neural surrogates constructed through a micro–macro decomposition, so that the dominant physical structure is treated analytically and numerically, while learning is used only for the lower-complexity kinetic correction.  The resulting hierarchy produces inexpensive low-fidelity samples that remain strongly correlated with the high-fidelity kinetic solution.  When used as control variates, these models yield substantial variance reduction and computational savings while retaining the high-order solver as the reference description.

Beyond the specific plasma application, the main message is that classical numerical analysis and machine learning need not be competing approaches. High-order solvers can provide structure, reliability, and asymptotic consistency, while learned models provide efficient approximations that can be exploited within rigorous multifidelity estimators. This interaction offers a general route toward trustworthy machine learning for computational science.
 

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from (for example) kites and darts, squares and triangles, rhombi or shield-shaped tiles and can have a variety of different symmetries. However, almost all quasicrystals occurring in soft matter are of the dodecagonal (12-fold rotation symmetry) type, and many can be described in terms of square and equilateral triangular tiles. Here, we explore what contributes to the thermodynamic stability of soft-matter quasicrystals, both in two dimensions and in three, and how the details of how soft-matter particles interact leads to different kinds of aperiodic tilings. Although dodecagonal quasicrystals are the most common, this work points to how more general (beyond dodecagonal) quasicrystals can be designed in soft matter.

Erion Morina is going to talk about: 'Learning PDE-based models from data: An analysis-driven perspective on identifiability, consistency, and interpretability'

Learning governing equations from data is a central problem in scientific machine learning. Given noisy and incomplete observations of a physical process, the goal is to recover the underlying law. This is often approached by fitting a neural network or a dictionary of candidate terms to the observed dynamics. A good fit, however, does not determine identifiability of the law, its approximability by what is computed, physical consistency, or interpretability as a formula. These depend largely on how the learning problem is posed.

This talk formulates the learning task as a regularized inverse problem in function space and studies how these properties can be established. Over a parameterized class of candidate laws, one minimizes a model residual, a data misfit, and a regularizer, with the class and the regularizer as the design choices. With a sufficiently expressive class and suitable regularization, one obtains a regularization-based notion of identifiability, under which the regularization-minimizing law consistent with the data is unique. As the approximation scale grows and the regularization parameters are chosen accordingly, minimizers of the parameterized problem converge to that law. For classes carrying the structural constraints of the underlying physical model, the learned models are physically consistent and well-posed by construction at every finite approximation scale. For symbolic networks built from rational building blocks, the recovered law is a readable formula. The emphasis of this talk is analytical, and first numerical experiments illustrate the recovery in practice. 

Overall, the results show that identifiability, consistency, and interpretability are not competing objectives, but can be unified in one analysis-driven framework.

Professor Hong Yan is going to talk about; 'Efficient Machine Learning Methods Based on Matrix and Tensor CUR Decomposition'

The training of large deep neural networks can be very expensive in term of long computing time, which requires costly equipment and high energy consumption. Vector, matrix and tensor multiplications are among the most time-consuming tasks. In this talk, a CUR decomposition-based method will be presented to reduce the computational complexity. Our basic strategy is to approximate a large low-rank matrix as a product of three much smaller matrices. Then multiplications with these matrices become very efficient. This can increase the speed of the self-attention operations in transformers significantly. Another reason for the long computing time in neural network training is that a large training dataset is needed. Our research group has recently studied a pattern matching method that requires only a small number of training samples. To achieve high accuracy and reliability, we use high-order pattern matching based on hypergraph and tensor models. A difficulty with this approach is that the high-order compatibility tensor is very large. Employing tensor CUR decomposition, we have tackled this problem and proposed an efficient pattern matching algorithm. Based on pattern matching, we have recently developed an intelligent indoor positioning and navigation system, which will be demonstrated in this talk.

Let $(M,\varphi)=(M_1,\varphi_1)*(M_2,\varphi_2)$ be a free product of non-trivial von Neumann algebras equipped with arbitrary faithful normal states. In 2011, Ueda showed that the diffuse summand of $M$ is a factor if and only if the maximum vector space dimension of $M_1$ and $M_2$ is at least 3, and in this case one further has that the diffuse summand has no asymptotically central sequences. Now consider the von Neumann algebra $N$ generated by non-trivial subalgebras $N_i\leq M_i$ with $\varphi_i$-preserving expectations for $i=1,2$, and let $z \in N$ be the central projection supporting the diffuse summand of $N$. Ueda's result characterizes when $Nz$ is a (full) factor, and, in fact, several cases treated in his proof yield the stronger property that $zMz$ lacks asymptotically $(Nz)$-central sequences. Thus, is natural to wonder: is this stronger property equivalent to the maximum vector space dimension of $N_1$ and $N_2$ being at least 3? In this talk, I will make use of amalgamated free products to answer this question in the affirmative. These same techniques can also be used to analyze the centralizer $M^\varphi = \{ x \in M \colon \varphi(xy) = \varphi(yx)\ \forall y\in M\}$ of the free product state, generalizing a separate 2011 result of Ueda from almost periodic states to arbitrary states. This is based on joint work with Aldo Garcia Guinto, Fehmi Ekin Giritlioglu, Yoonkyeong Lee, and Rahul K. Ramachandran.

(Joint work with Brent Nelson) In this talk we investigate the anticoarse space of the von Neumann algebras generated by the kernel and the domain of a closable derivation. We show that when a tuple (x_i)_{i\in I} admits a conjugate system, then for any proper subset J \subset I  with |J| \geq 2 the inclusion W*(x_j :j \in j) \subset W*(x_i: i \in I) is irreducible, infinite index and non-regular.

In this talk, I will start by presenting what KK-rigidity means and how I together with Chris Schafhauser have modified it to the dynamical setting for strongly outer actions on classifiable C*-algebras. This includes an application to the uniqueness of the Bernoulli shift on the CAR-algebra and Jiang-Su algebra for torsion-free amenable countable discrete groups. Then I will move towards the machinery leading up to the rigidity theorem, which includes some exposition on the cocycle category and classification of equivariant KK-lifts

Tumor growth and angiogenesis drive complex spatiotemporal variation in micro-environmental oxygen levels. Previous experimental studies have observed that cancer cells exposed to chronic hypoxia retained a phenotype characterized by enhanced migration and reduced proliferation, even after being shifted to normoxic conditions, a phenomenon which we refer to as hypoxic memory. However, because dynamic hypoxia and related hypoxic memory effects are challenging to measure experimentally, our understanding of their implications in tumor invasion is quite limited. Here, we propose a novel phenotype-structured partial differential equation modeling framework to elucidate the effects of hypoxic memory on tumor invasion along one spatial dimension in a cyclically varying hypoxic environment. We incorporated hypoxic memory by including time-dependent changes in hypoxic-to-normoxic phenotype transition rate upon continued exposure to hypoxic conditions. Our model simulations demonstrate that hypoxic memory significantly enhances tumor invasion without necessarily reducing tumor volume. This enhanced invasion was sensitive to the induction rate of hypoxic memory, but not the dilution rate. Further, shorter periods of cyclic hypoxia contributed to a more heterogeneous profile of hypoxic memory in the population, with the tumor front dominated by hypoxic cells that exhibited stronger memory. Overall, our model highlighted the complex interplay between hypoxic memory and cyclic hypoxia in shaping heterogeneous tumor invasion patterns.

Keywords: Tumor invasion, cyclic hypoxia, hypoxic memory, phenotype-structured model

Professor  Chunmei Su will talk about: 'Temporal high-order structure-preserving parametric finite element methods for curvature flows'

The quality of the mesh is crucial for simulating curvature flows, as standard approaches may fail due to mesh distortion. We first present a series of high-order parametric finite element methods based on the Barrett-Garcke--Nurnberg formulation for solving various types of flows involving curves and surfaces. Extensive numerical experiments demonstrate the anticipated high-order accuracy while maintaining favorable mesh quality throughout the evolution process. Secondly, for flows involving multiple geometric structures, such as surface diffusion—which reduces area while preserving volume—we propose a type of structure-preserving method that incorporates two scalar Lagrange multipliers along with two evolution equations related to area and volume, respectively. These schemes effectively preserve the geometric structure at a fully discrete level. Comprehensive numerical experiments illustrate that our methods achieve the desired temporal accuracy, while simultaneously preserving the geometric structure of the surface diffusion.
 

In this talk, we study some Prandtl-type boundary layer models, including the two-dimensional MHD boundary layer equations and the Prandtl–Shercliff model. For small perturbations of a tangential background magnetic field, we establish the global-in-time existence and uniqueness of solutions to the MHD boundary layer equations in Sobolev spaces. The proof relies on a novel combination of the well-known cancellation mechanism and the concept of linearly good unknowns. We also investigate the Prandtl–Shercliff model. In the two-dimensional case, we establish global-in-time well-posedness in Sobolev spaces without imposing any structural assumptions on the initial data. Moreover, we show that solutions exhibit a global analytic regularization effect in all variables, up to the boundary and for all times. The proofs rely crucially on the intrinsic nonlocal diffusion induced by the Shercliff boundary layer.

In this talk we summary some recent progress on limit theorems for the Wasserstein distance of empirical measures of Markov processes. For symmetric diffusion processes on Riemannian manifold possibly with reflecting or killing boundary, the sharp convergence rate is derived with renormalization limit formulated by using the spectrum of the generator. Moreover, a general framework is established to estimate the convergence rate in Wasserstein distance of empirical measures for ergodic Markov processes.

Professor Yuhua Zhu will talk about; 'A New Framework for Reinforcement Learning in the Physical World'
 

We study reinforcement learning in the physical world, where the underlying dynamics evolve according to an unknown stochastic differential equation, while only discrete-time data are available. Existing RL algorithms typically ignore this SDE structure, which can limit their effectiveness in physical-world settings. We develop a systematic approach for adapting existing RL algorithms to this setting with minimal modifications, by leveraging the smoothness of the underlying continuous-time dynamics. In particular, for the LQR setting, we show that our framework can recover the exact continuous-time optimal control with only discrete-time information. We further identify a fundamental trade-off between discretization error and statistical error that is intrinsic to RL in the physical world. Finally, we extend the framework to mean-field optimal control.

Professor Andrew Saxe will talk about; 'Demystifying depth: principles of learning in deep neural networks' 

Deep neural networks have revolutionized artificial intelligence, yet their inner workings remain poorly understood. This talk presents mathematical analyses of the nonlinear dynamics of learning in several solvable deep network models, offering theoretical insights into the role of depth. These models reveal how learning algorithms, data structure, initialization schemes, and architectural choices interact to produce hidden representations that afford complex generalization behaviours. A recurring theme across these analyses is a neural race: competing pathways within a deep network vie to explain the data, with an implicit bias toward shared representations. These shared representations in turn shape the network’s capacity for systematic generalization, multitasking, and transfer learning. I will show how such principles manifest across diverse architectures—including feedforward and linear attention networks. Together, these results provide analytic foundations for understanding how environmental statistics, network architecture, and learning dynamics jointly structure the emergence of neural representations and behaviour.

I will describe a circle-valued Morse theory for simplicial complexes. The central objects of study are partial matchings which admit certain zigzag cycles; these cyclic matchings lift canonically to acyclic matchings on the infinite cyclic cover of the underlying simplicial complex. From the lifted acyclic matchings, we obtain a finitely generated Morse chain complex defined over the Novikov ring, which consists of power series in one variable with finite negative support. We then establish a quasi-isomorphism between this Morse-Novikov complex and the simplicial chain complex of the cyclic cover, duly completed over the Novikov ring. As a pleasant consequence, we can define new computable invariants to detect (obstructions to) the fiberedness of tame knots.

Stochastic search is ubiquitous in biology and ecology, from synaptic transmission and intracellular signaling to predators seeking prey and the spread of disease. In dynamic systems like these, the number of 'searchers' is rarely constant: new agents may be recruited while others can abandon the search. Despite the ubiquity of these dynamics, their combined influence on search times remains largely unexplored. In this talk we will introduce a general framework for stochastic search in which agents progressively join and leave the process, a mechanism we term 'dynamic redundancy and mortality'. Under minimal assumptions on the underlying search dynamics, our framework yields the exact distribution of the first-passage time to a target region and further reveals surprising connections to stochastic search with stochastic resetting, wherein a single searcher is randomly 'reset' to its initial state. We will then treat the target region as a queue, which we show has interarrival times governed by a thinned nonhomogeneous Poisson process. Altogether this work provides a rigorous foundation for studying stochastic search processes with a fluctuating number of searchers. This work is in collaboration with Dr. Aanjaneya Kumar (Santa Fe Institute) and José Giral-Barajas (Imperial College London).