{"id":504,"date":"2021-12-02T16:13:43","date_gmt":"2021-12-02T21:13:43","guid":{"rendered":"https:\/\/blogs.ams.org\/mathmedia\/?p=504"},"modified":"2021-12-02T16:12:10","modified_gmt":"2021-12-02T21:12:10","slug":"tonys-take-november-2021","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathmedia\/tonys-take-november-2021\/","title":{"rendered":"Tony&#8217;s Take November 2021"},"content":{"rendered":"<div style=\"margin-top: 0px; margin-bottom: 0px;\" class=\"sharethis-inline-share-buttons\" ><\/div><h2>This month&#8217;s topics:<\/h2>\n<ul>\n<li><a href=\"#one\">Algebaic topology and the weather<\/a><\/li>\n<li><a href=\"#two\">Topology in children&#8217;s visual perception<\/a><\/li>\n<\/ul>\n<h3><a name=\"one\"><\/a>Algebraic topology and the weather<\/h3>\n<p>The Lorenz attractor is probably the most famous chaotic dynamical system. It is the trajectory of the solution to a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Lorenz_system\">nonlinear system of three differential equations<\/a>, discovered by Edward Lorenz (1963) in his attempts to model weather; it has come to epitomize the &#8220;butterfly effect.&#8221; (In this <a href=\"https:\/\/youtu.be\/dP3qAq9RNLg\">awesome animation<\/a> 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 <i>branched manifold<\/i> (a smooth surface except along a 1-dimensional branching locus where it has Y-shaped cross-sections).<\/p>\n<figure id=\"attachment_508\" aria-describedby=\"caption-attachment-508\" style=\"width: 402px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-508\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/mathmedia\/files\/2021\/11\/lorenz2.png?resize=402%2C198\" alt=\"lorenz attractor\" width=\"402\" height=\"198\" \/><figcaption id=\"caption-attachment-508\" class=\"wp-caption-text\">The Lorentz attractor (Image by <a href=\"https:\/\/en.wikipedia.org\/wiki\/Lorenz_system#\/media\/File:A_Trajectory_Through_Phase_Space_in_a_Lorenz_Attractor.gif\">Dan Quinn<\/a> 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.<\/figcaption><\/figure>\n<p>In an article in <i>Chaos<\/i> published October 12, 2021 (see the <a href=\"https:\/\/arxiv.org\/abs\/2010.09611\">arXiv preprint<\/a>), an international team (Gisela D. Char\u00f3, Micka\u00ebl 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 (&#8220;[a] set in phase space that robustly supports the point cloud associated with the system\u2019s invariant measure&#8221;) that plays the role of the underlying branched manifold. They track changes to the topology of that complex by computing the rank of its <i>first homology group<\/i> (here&#8217;s where algebraic topology enters the picture). For 2-dimensional sets like these, that rank is just the number of &#8220;holes.&#8221; 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.<\/p>\n<figure id=\"attachment_509\" aria-describedby=\"caption-attachment-509\" style=\"width: 617px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-509\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/mathmedia\/files\/2021\/11\/three-snapshots.png?resize=617%2C304\" alt=\"three snapshots of Lorenz with noise\" width=\"617\" height=\"304\" \/><figcaption id=\"caption-attachment-509\" class=\"wp-caption-text\">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 <i>Chaos<\/i> <b>31<\/b> 103115 (2021), used in accordance with CC BY <a href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">license<\/a>. The total complexity of this phenomenon can be appreciated from the <a href=\"https:\/\/vimeo.com\/240039610.\">video<\/a> generated by a <a href=\"https:\/\/doi.org\/10.1016\/j.physd.2011.06.005\">related study.<\/a><\/figcaption><\/figure>\n<p>The authors consider applications to Lorenz&#8217;s original problem, weather: &#8220;A fairly straightforward application [of this methodology] to the climate sciences might clarify [&#8230;] the role of intermittent vs. oscillatory low-frequency variability in the atmosphere. [&#8230;] Such phenomena include the so-called <a href=\"https:\/\/journals.ametsoc.org\/view\/journals\/atsc\/6\/2\/1520-0469_1949_006_0068_asoteo_2_0_co_2.xml\">blocking of the westerlies<\/a> and intraseasonal oscillations with periodicities of 40\u201350 days. They remark: &#8220;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.&#8221; (An <a href=\"https:\/\/www.azocleantech.com\/news.aspx?newsID=30435\">online news service<\/a> put two and two together and came up with: &#8220;Algebraic Topology Could be Used to Predict if and When Earth\u2019s Climate System will Tip.&#8221;)<\/p>\n<p>In conclusion: &#8220;We have concentrated throughout much of this paper on problems related to the climate sciences [&#8230;]. With all due modesty, it is not unlikely \u2014 considering the great generality of topological methods \u2014 to expect the results obtained herein to have some applicability to other areas of the physical, life and socioeconomic sciences.&#8221;<\/p>\n<h3><a name=\"two\"><\/a>Topology in children&#8217;s visual perception<\/h3>\n<p>It has been known at least since Jean Piaget&#8217;s 1948 work with B\u00e4rbel Inhelder, <i>La repr\u00e9sentation de l&#8217;espace chez l&#8217;enfant,<\/i> 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 <a href=\"https:\/\/srcd.onlinelibrary.wiley.com\/doi\/10.1111\/cdev.13629\">article in <i>Child Development<\/i><\/a> (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: &#8220;This study demonstrates that the peripheral vision of children aged 6\u20138 years functions differently than for adults when discriminating geometric properties of objects, that is, topological property (TP) and non-TP shapes.&#8221; 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 &#8220;topological and nontopological discrimination in the central and peripheral visual fields.&#8221;<\/p>\n<figure id=\"attachment_511\" aria-describedby=\"caption-attachment-511\" style=\"width: 339px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-511\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/mathmedia\/files\/2021\/11\/letters-figures.png?resize=339%2C102\" alt=\"letters and figures\" width=\"339\" height=\"102\" \/><figcaption id=\"caption-attachment-511\" class=\"wp-caption-text\">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 <i>Childhood Development<\/i> <b>92<\/b> 1906-1918.<\/figcaption><\/figure>\n<p>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).<\/p>\n<p><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-512 aligncenter\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/mathmedia\/files\/2021\/11\/central.png?resize=339%2C147\" alt=\"central vision test\" width=\"339\" height=\"147\" \/><\/p>\n<figure id=\"attachment_513\" aria-describedby=\"caption-attachment-513\" style=\"width: 339px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-513\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/mathmedia\/files\/2021\/11\/peripheral.png?resize=339%2C147\" alt=\"peripheral vision test\" width=\"339\" height=\"147\" \/><figcaption id=\"caption-attachment-513\" class=\"wp-caption-text\">The four types of combinations, illustrated with the letter-like shapes.<\/figcaption><\/figure>\n<p>The variable that the team measured, the <i>normalized TP priority effect,<\/i> was &#8220;computed from normalized reaction time differences between non-TP and TP trials,&#8221; so it measured how much a topological distinction speeded up the discrimination between two shapes.<\/p>\n<figure id=\"attachment_515\" aria-describedby=\"caption-attachment-515\" style=\"width: 506px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-515 size-full\" src=\"https:\/\/i0.wp.com\/blogs.ams.org\/mathmedia\/files\/2021\/11\/experiment-b.png?resize=506%2C197\" alt=\"partial results of experiment 1b\" width=\"506\" height=\"197\" \/><figcaption id=\"caption-attachment-515\" class=\"wp-caption-text\">A sample of the team&#8217;s results: experiment 1b. Error bars represent SEM (Standard Error of the Mean), ** $p &lt; 0.01$, *** $p &lt; 0.001$.<\/figcaption><\/figure>\n<p>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.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This month&#8217;s topics: Algebaic topology and the weather Topology in children&#8217;s visual perception 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;&#8230;<\/p>\n","protected":false},"author":4010,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[4],"tags":[],"class_list":["post-504","post","type-post","status-publish","format-standard","hentry","category-tony-phillips-take"],"jetpack_featured_media_url":"","jetpack-related-posts":[],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/504","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/users\/4010"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/comments?post=504"}],"version-history":[{"count":11,"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/504\/revisions"}],"predecessor-version":[{"id":521,"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/504\/revisions\/521"}],"wp:attachment":[{"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/media?parent=504"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/categories?post=504"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ams.org\/mathmedia\/wp-json\/wp\/v2\/tags?post=504"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}