Tony’s Take November 2021

This month’s topics:

Algebraic topology and the weather

The Lorenz attractor is probably the most famous chaotic dynamical system. It is the trajectory of the solution to a nonlinear system of three differential equations, discovered by Edward Lorenz (1963) in his attempts to model weather; it has come to epitomize the “butterfly effect.” (In this awesome animation you can watch the trajectory take shape). The trajectory of a three-dimensional chaotic dynamical system like the Lorenz system evolves in a region of 3-space that can be collapsed onto a 2-dimensional branched manifold (a smooth surface except along a 1-dimensional branching locus where it has Y-shaped cross-sections).

lorenz attractor
The Lorentz attractor (Image by Dan Quinn used under CC BY-SA 3.0) and its underlying branched manifold. The arrows show the direction of the flow; the central point is a singularity.

In an article in Chaos published October 12, 2021 (see the arXiv preprint), an international team (Gisela D. Charó, Mickaël D. Chekroun, Denisse Sciamarella, and Michael Ghil) examined random time-dependent perturbations of the Lorenz system. For each instant $t$, their program produces a 2-dimensional cell complex (“[a] set in phase space that robustly supports the point cloud associated with the system’s invariant measure”) that plays the role of the underlying branched manifold. They track changes to the topology of that complex by computing the rank of its first homology group (here’s where algebraic topology enters the picture). For 2-dimensional sets like these, that rank is just the number of “holes.” For example, the branched manifold of the original Lorenz attractor can be continuously squeezed down to a figure-eight. Topologically speaking, it has 2 holes, so the rank of its first homology group is two.

three snapshots of Lorenz with noise
Three nearby snapshots from a simulation of the Lorenz system with added stochastic noise. (a), (b) and (c) are the point-clouds generated by the perturbed system; (d), (e) and (f) the corresponding 2-dimensional cell complexes. The rank of the first homology group is 3 for (d), 10 for (e) and 4 for (f). Image from Chaos 31 103115 (2021), used in accordance with CC BY license. The total complexity of this phenomenon can be appreciated from the video generated by a related study.

The authors consider applications to Lorenz’s original problem, weather: “A fairly straightforward application [of this methodology] to the climate sciences might clarify […] the role of intermittent vs. oscillatory low-frequency variability in the atmosphere. […] Such phenomena include the so-called blocking of the westerlies and intraseasonal oscillations with periodicities of 40–50 days. They remark: “The framework introduced in this article to characterize such changes in topological features appears to hold promise for the understanding of topological tipping points in general.” (An online news service put two and two together and came up with: “Algebraic Topology Could be Used to Predict if and When Earth’s Climate System will Tip.”)

In conclusion: “We have concentrated throughout much of this paper on problems related to the climate sciences […]. With all due modesty, it is not unlikely — considering the great generality of topological methods — to expect the results obtained herein to have some applicability to other areas of the physical, life and socioeconomic sciences.”

Topology in children’s visual perception

It has been known at least since Jean Piaget’s 1948 work with Bärbel Inhelder, La représentation de l’espace chez l’enfant, that children start with basic topological distinctions (circle $\neq$ annulus) before more detailed or quantitative ones (circle $\neq$ triangle). So it is surprising to learn, from an article in Child Development (September 27, 2021), that in their peripheral vision, children under 10 do not take advantage of topological cues the way older children and adults do. As the authors, a Shenzhen-based team led by Lin Chen and Yan Huang, put it: “This study demonstrates that the peripheral vision of children aged 6–8 years functions differently than for adults when discriminating geometric properties of objects, that is, topological property (TP) and non-TP shapes.” The team worked with a large group of subjects: 773 children aged 6-14 and 179 adults. In their main experiment, they used two types of figures (one resembling letters and the other made of arrows and triangles) to test “topological and nontopological discrimination in the central and peripheral visual fields.”

letters and figures
Topological-property discrimination was tested using one set of figures resembling letters in the Latin alphabet (experiment 1a), and another using variously oriented triangles and arrows (experiment 1b). Images from the open-access article Childhood Development 92 1906-1918.

Stimuli were presented in pairs, one stimulus on each side of the central fixation point. Participants, keeping their focus on the central point, had to record whether the two stimuli were the same or not by pressing one of the two specified keys on the keyboard. Reaction times were measured for each trial. In total, there were four types of combinations, that is, two eccentricities (central and peripheral) and two discrimination types (TP and non-TP).

central vision test

peripheral vision test
The four types of combinations, illustrated with the letter-like shapes.

The variable that the team measured, the normalized TP priority effect, was “computed from normalized reaction time differences between non-TP and TP trials,” so it measured how much a topological distinction speeded up the discrimination between two shapes.

partial results of experiment 1b
A sample of the team’s results: experiment 1b. Error bars represent SEM (Standard Error of the Mean), ** $p < 0.01$, *** $p < 0.001$.

As announced, the results show that whereas adults and children over 10 process TP differences faster than non-TP differences, both in their central and peripheral visual fields, this effect is almost completely absent in the peripheral vision of children aged 6-8.