tag:theconversation.com,2011:/global/topics/topology-17014/articlesTopology – The Conversation2023-01-01T19:39:45Ztag:theconversation.com,2011:article/1960532023-01-01T19:39:45Z2023-01-01T19:39:45ZExploring the mathematical universe – connections, contradictions, and kale<figure><img src="https://images.theconversation.com/files/502042/original/file-20221220-22-x7dzft.png?ixlib=rb-1.1.0&rect=443%2C71%2C3293%2C1886&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock/The Conversation</span></span></figcaption></figure><p>Science and maths skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimisation or medical imaging. Pure mathematics often yields unanticipated applications, but without a time machine to look into the future, how do mathematicians like me choose what to study?</p>
<p>Over Thai noodles, I asked some colleagues what makes a problem interesting, and they offered a slew of suggestions: surprises, contradictions, patterns, exceptions, special cases, connections. These answers might sound quite different, but they all support a view of the mathematical universe as a structure to explore. </p>
<p>In this view, mathematicians are like anatomists learning how a body works, or navigators charting new waters. The questions we ask take many forms, but the most interesting ones are those that help us see the big picture more clearly. </p>
<h2>Making maps</h2>
<p>Mathematical objects come in many forms. Some of them are probably quite familiar, like numbers and shapes. Others might seem more exotic, like equations, functions and symmetries.</p>
<p>Instead of just naming objects, a mathematicians might ask how some class of objects is organised. Take prime numbers: we know there are infinitely many of them, but we need a structural understanding to work out how frequently they occur or to identify them in an efficient way.</p>
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<a href="https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A grid of blue dots" src="https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=602&fit=crop&dpr=1 600w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=602&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=602&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=756&fit=crop&dpr=1 754w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=756&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/500648/original/file-20221213-9515-576dx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=756&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">The ‘Ulam spiral’ reveals some structure in the primes. If you arrange the counting numbers in squares spiralling outward, it becomes clear that many prime numbers fall on diagonal lines.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Ulam_spiral#/media/File:Spirale_Ulam_150.jpg">Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>Other good questions explore relationships between apparently different objects. For example, shapes have symmetry, but so do the solutions to some equations. </p>
<p>Classifying objects and finding connections between them help us assemble a coherent map of the mathematical world. Along the way, we sometimes encounter surprising examples that defy the patterns we’ve inferred. </p>
<p>Such apparent contradictions reveal where our understanding is still lacking, and resolving them provides valuable insight.</p>
<h2>Consider the triangle</h2>
<p>The humble triangle provides a famous example of an apparent contradiction. Most people think of a triangle as the shape formed by three connecting line segments, and this works well for the geometry we can draw on a sheet of paper. </p>
<p>However, this notion of triangle is limited. On a surface with no straight lines, like a sphere or a curly kale leaf, we need a more flexible definition. </p>
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Read more:
<a href="https://theconversation.com/pythagoras-revenge-humans-didnt-invent-mathematics-its-what-the-world-is-made-of-172034">Pythagoras’ revenge: humans didn’t invent mathematics, it’s what the world is made of</a>
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<p>So, to extend geometry to surfaces that aren’t flat, an open-minded mathematician might propose a new definition of a triangle: pick three points and connect each pair by the shortest path between them. </p>
<p>This is a great generalisation because it matches the familiar definition in the familiar setting, but it also opens up new terrain. When mathematicians first studied these generalised triangles in the 19th century, they solved a millennia-old mystery and revolutionised mathematics.</p>
<h2>The parallel postulate problem</h2>
<p>Around 300 BC, the Greek mathematician Euclid wrote a treatise on planar geometry called The Elements. This work presented both fundamental principles and results that were logically derived from them. </p>
<p>One of his principles, called the parallel postulate, is equivalent to the statement that the sum of the angles in any triangle is 180°. This is exactly what you’ll measure in every flat triangle, but later mathematicians debated whether the parallel postulate should be a foundational principle or just a consequence of the other fundamental assumptions. </p>
<p>This puzzle persisted until the 1800s, when mathematicians realised why a proof had remained so elusive: the parallel postulate is false on some surfaces. </p>
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<img alt="Image showing that a triangle on the surface of a sphere will have angles that add up to more than 180°, but on a hyperbolic surface will add up to less than 180°." src="https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=265&fit=crop&dpr=1 600w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=265&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=265&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=334&fit=crop&dpr=1 754w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=334&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/502079/original/file-20221220-18-yelmip.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=334&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
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<p>On a sphere, the sides of a triangle bend away from each other and the angles add up to more than 180°. On a rippled kale leaf, the sides bow in towards each other and the angle sum is less than 180°. </p>
<p>Triangles where the angle sum breaks the apparent rule led to the revelation that there are kinds of geometry Euclid never imagined. This is a deep truth, with applications in physics, computer graphics, fast algorithms, and beyond. </p>
<h2>Salad days</h2>
<p>People sometimes debate whether mathematics is discovered or invented, but both points of view feel real to those of us who study mathematics for a living. Triangles on a piece of kale are skinny whether or not we notice them, but selecting which questions to study is a creative enterprise. </p>
<p>Interesting questions arise from the friction between patterns we understand and the exceptions that challenge them. Progress comes when we reconcile apparent contradictions that pave the way to identify new ones. </p>
<p>Today we understand the geometry of two-dimensional surfaces well, so we’re equipped to test ourselves against similar questions about higher-dimensional objects.</p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/corals-crochet-and-the-cosmos-how-hyperbolic-geometry-pervades-the-universe-53382">Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe</a>
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<p>In the past few decades we’ve learned that three-dimensional spaces also have their own innate geometries. The most interesting one is called hyperbolic geometry, and it turns out to act like a three-dimensional version of curly kale. We know this geometry exists, but it remains mysterious: in my own research field, there are lots of questions we can answer for any three-dimensional space … except the hyperbolic ones.</p>
<p>In higher dimensions we still have more questions than answers, but it’s safe to say that study of four-dimensional geometry is entering its salad days.</p><img src="https://counter.theconversation.com/content/196053/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Joan Licata does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Mathematicians are like anatomists learning how a body works, or navigators charting new waters.Joan Licata, Associate Professor, Mathematics, Australian National UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1716512021-11-15T18:32:52Z2021-11-15T18:32:52ZGot $1.2T to invest in roads and other infrastructure? Here’s how to figure out how to spend it wisely<figure><img src="https://images.theconversation.com/files/431755/original/file-20211112-15225-1ya3u2g.jpg?ixlib=rb-1.1.0&rect=458%2C188%2C5005%2C3448&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A collapsed bridge in Atlanta in 2017 backed up traffic for a month.</span> <span class="attribution"><a class="source" href="https://newsroom.ap.org/detail/OverpassCollapseFire/d398f5f0405b4ed8b2fc406ed30fade5/photo?Query=atlanta%20bridge%20collapse&mediaType=photo&sortBy=arrivaldatetime:desc&dateRange=Anytime&totalCount=75&currentItemNo=0">AP Photo/David Goldman</a></span></figcaption></figure><p>The American economy is underpinned by networks. </p>
<p>Road networks carry traffic and freight; the internet and telecommunications networks carry our voices and digital information; the electricity grid is a network carrying energy; financial networks transfer money from bank accounts to merchants. These networks are vast, often global systems – but a local disruption can really block them up. </p>
<p>For example, <a href="http://www.wsbtv.com/news/local/atlanta/gdot-state-offering-31m-in-incentives-to-reopen-i-85-before-june/511832846">the I-85 bridge collapse in Atlanta in 2017</a> snarled that city’s traffic for months. In 2019, <a href="https://www.forbes.com/sites/niallmccarthy/2019/04/03/report-the-u-s-has-over-47000-structurally-deficient-bridges-infographic/?sh=341754a44bdc">a concrete beam fell from a bridge</a> in Chattanooga, Tennessee, resulting in traffic shutdown on one of the nation’s busiest interstate intersections. And in 2021, <a href="https://www.cnn.com/2021/09/02/us/ida-train-stranded-ny-nj/index.html">Hurricane Ida crippled mass transit in New York City</a>, with flash floods overcoming subway lines and trapping people overnight on trains.</p>
<p>As the U.S. government <a href="https://www.reuters.com/world/us/whats-bipartisan-us-1-trillion-infrastructure-bill-2021-10-01/">prepares to spend over $1 trillion on infrastructure projects</a> over the next 10 years, it will be vital to identify which elements are the most crucial to repair or improve. This is important not only for maximizing benefits; it’s also useful in preventing disaster. </p>
<p>Is there, perhaps, a telecommunication line whose destruction would be particularly damaging? Or one road through an area that has an especially large role in keeping traffic flowing smoothly?</p>
<p><a href="http://greatvalley.psu.edu/person/qiang-patrick-qiang">Patrick Qiang</a> <a href="https://scholar.google.com/citations?user=ecFsBp0AAAAJ&hl=en&oi=ao">and I</a> are operations management scholars who have developed <a href="http://dx.doi.org/10.1007/s10898-007-9198-1">a way to evaluate network performance</a> and simulate the effects of potential changes, whether planned – like a highway repair – or unexpected – like a natural disaster. </p>
<p>By modeling the independent behavior of all the users of a network, we can calculate the flow – of <a href="http://dx.doi.org/10.1007/s10898-015-0371-7">freight</a>, <a href="https://dx.doi.org/10.1209/0295-5075/79/38005">commuters</a>, <a href="http://dx.doi.org/10.1007/978-3-540-77958-2_14">money</a> or anything else – across each link, and how other links’ flows will change. This lets us identify where investment will be most beneficial, and which projects shouldn’t happen at all.</p>
<h2>More isn’t always better</h2>
<p>It’s very difficult to measure networks’ performance, in part because they are so complex, but also because people use them differently at different times, and because those choices affect others’ experiences. For example, one person choosing to drive to work instead of taking the bus puts one more car on the road, which might get involved in a crash or otherwise contribute to a traffic jam.</p>
<p>In 1968, German mathematician Dietrich Braess observed the possibility that adding a road to an area with congested traffic <a href="http://dx.doi.org/10.1287/trsc.1050.0127">could actually make things worse</a>, not better. <a href="https://supernet.isenberg.umass.edu/braess/braess-new.html">This paradox</a> can occur when travel times depend on the amount of traffic. If too many drivers decide their own optimal route involves one particular road, that road can become congested, slowing everyone’s travel time. In effect, the drivers would have been better off if the road hadn’t been built.</p>
<p>This phenomenon has been found not only <a href="https://supernet.isenberg.umass.edu/braess/braess-new.html#BraessArticle">in transportation networks</a> and <a href="http://dx.doi.org/10.1239/jap/1032374242">the internet</a>, but also <a href="https://doi.org/10.1209/0295-5075/115/28004">in electrical circuits</a>. </p>
<p>The U.S. shouldn’t waste time and money building or repairing network links a community would be better without. But how can policymakers tell which elements help and which make things worse? </p>
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<iframe width="440" height="260" src="https://www.youtube.com/embed/8mlH9bnvWVE?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Explaining the Braess paradox.</span></figcaption>
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<h2>Calculating efficiency</h2>
<p>The best networks can handle the highest demand at the lowest average cost for each trip – such as a commute from a worker’s home to their office. </p>
<p>Evaluating a network means identifying which locations need to be connected to one another, as well as the volume of traffic among specific places and the various costs involved – such as gas, pavement wear and tear and police services keeping drivers safe.</p>
<p>Once a network is measured in this way, it can be converted into a computerized model with which we can simulate removing links or adding new ones in particular places. Then we can measure what happens to the rest of the network: Does traffic get more congested, and if so, by how much? Or, as in the Braess paradox, do travel times actually get shorter when a link is removed? And how much money does a particular project cost to build and save in time or user expenses?</p>
<p>[<em>Over 115,000 readers rely on The Conversation’s newsletter to understand the world.</em> <a href="https://theconversation.com/us/newsletters/the-daily-newsletter-3?utm_source=TCUS&utm_medium=inline-link&utm_campaign=newsletter-text&utm_content=100Ksignup">Sign up today</a>.]</p>
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<img alt="With the White House in the distance, President Biden speaks at a lectern before a crowd of people" src="https://images.theconversation.com/files/432073/original/file-20211115-23-1psza3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/432073/original/file-20211115-23-1psza3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/432073/original/file-20211115-23-1psza3q.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/432073/original/file-20211115-23-1psza3q.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/432073/original/file-20211115-23-1psza3q.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/432073/original/file-20211115-23-1psza3q.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/432073/original/file-20211115-23-1psza3q.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">President Biden celebrated signing the bipartisan infrastructure bill on Nov. 15.</span>
<span class="attribution"><a class="source" href="https://newsroom.ap.org/detail/Biden/c2622af4cdc44114ad250ac56a49a76d/photo?Query=biden&mediaType=photo&sortBy=arrivaldatetime:desc&dateRange=Anytime&totalCount=63707&currentItemNo=9">AP Photo/Susan Walsh</a></span>
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<h2>Going global</h2>
<p>Our method of measuring a network’s performance has been used to refine
<a href="https://ercim-news.ercim.eu/en79/rd/route-optimization-how-efficient-will-the-proposed-north-dublin-metro-be">the route of a proposed metro line in Dublin, Ireland</a>; <a href="https://www.researchgate.net/publication/313365965_Maritime_Network_Efficiency_Comparison_in_Indonesia_Nusantara_Pendulum_and_Sea_Tollway">to design new shipping routes in Indonesia</a>; to identify which roads in Germany <a href="http://www.cedim.de/download/14_Schulz.pdf">should be first on the maintenance list</a> and <a href="http://dx.doi.org/10.1007/s11069-013-0896-3">to determine the effects of road closures after major fires in regions of Greece</a>.</p>
<p>Our method has also been applied to make supply chains more efficient, both to <a href="http://dx.doi.org/10.1007/978-1-84882-634-2_6">maximize profits</a> and to <a href="http://www.sciencedirect.com/science/article/pii/S0965856412000249">speed disaster relief supplies</a> to people in need.</p>
<p>As the U.S. works to enhance its economic competitiveness, we believe the country will need to invest in many different types of networks to maximize their usefulness and value to Americans. Using measurement methods like ours can guide leaders to wise investments.</p>
<p><em>This is an updated version of an <a href="https://theconversation.com/calculating-where-america-should-invest-in-its-transportation-and-communications-networks-76258">article originally published</a> on April 19, 2017.</em></p><img src="https://counter.theconversation.com/content/171651/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Anna Nagurney does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>As President Biden signs the bipartisan infrastructure bill, it’s important to determine which road, freight and information networks are the most vital to protect.Anna Nagurney, Eugene M. Isenberg Chair in Integrative Studies, UMass AmherstLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1114602019-02-20T12:42:51Z2019-02-20T12:42:51ZBig data is being reshaped thanks to 100-year-old ideas about geometry<figure><img src="https://images.theconversation.com/files/259923/original/file-20190220-148523-1w06d1l.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Elesey/Shutterstock</span></span></figcaption></figure><p>Your brain is made up of billions of neurons connected by trillions of synapses. And how they’re arranged gives rise to the brain’s functionality and to your personality. That’s why <a href="https://bluebrain.epfl.ch/">scientists in Switzerland</a> recently produced the first-ever digital <a href="https://bluebrain.epfl.ch/op/edit/page-158777.html">3D brain cell atlas</a>, a complete mapping of the brain of a mouse. While this is a colossal achievement, the great challenge now lies in learning to decipher the atlas. And it’s a huge one. </p>
<p>Science is full of this kind of problem: how to turn large amounts of information into useful insight. For many years, researchers relied on mathematics and statistics to explore data. The explosion of large datasets created by digital storage, the internet, and cheap sensors has led to the development of new techniques designed specifically to deal with this “<a href="https://theconversation.com/explainer-what-is-big-data-13780">big data</a>”. </p>
<p>And now there is an emerging new approach based on century-old ideas that’s producing superior tools for understanding certain types of big data. Using the mouse’s brain as an example, its physical shape determines its functionality. But a precise description of this shape, which we now have, doesn’t automatically reveal everything about how the brain works.</p>
<p>Behind the physical shape lies a more abstract shape formed by the interconnections within the brain. Capturing aspects of this shape by applying techniques from the study of what’s known as “topology” can help reveal a deeper understanding of the brain’s functioning. This same guiding principle of using topological techniques on big data also has applications in drug development and other cutting-edge endeavours. </p>
<h2>Topology</h2>
<p>Topology is a branch of modern geometry <a href="https://press.princeton.edu/titles/8722.html">with roots</a> going back to a foundational observation by the Swiss mathematician Leonhard Euler (1707-1783) about polyhedra, 3D shapes with flat faces, straight edges and sharp corners or “vertices”. In 1750, <a href="https://topologicalmusings.wordpress.com/2008/03/01/platonic-solids-and-eulers-formula-for-polyhedra">Euler discovered</a> that for any convex (with all its faces pointing outwards) polyhedron, the number of vertices minus the number of edges plus the number of faces <a href="https://redlegagenda.com/2015/09/23/eulers-polyhedron-formula/">always equals two</a>. </p>
<p>You can apply the same formula to other shapes to get what is known as their Euler characteristic. This number doesn’t change no matter how the shape is bent or deformed. And <a href="https://www.livescience.com/51307-topology.html">topology</a> is the study of these kind of constant properties of shapes.</p>
<p>Topology went through rapid development during the 20th century as a prominent subject in pure mathematics. The researchers who created the subject didn’t have real-world applications on their minds, they were just interested in what was mathematically true about shapes under certain conditions.</p>
<p>Yet some of these ideas from topology that have been around for over 100 years are now finding significant applications in data science. Because topology focuses on constant properties, its techniques make it insensitive to various data inaccuracies or “noise”. This makes it ideal for deciphering the true meaning behind the collected data.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/259929/original/file-20190220-148545-1g16rne.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/259929/original/file-20190220-148545-1g16rne.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=401&fit=crop&dpr=1 600w, https://images.theconversation.com/files/259929/original/file-20190220-148545-1g16rne.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=401&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/259929/original/file-20190220-148545-1g16rne.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=401&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/259929/original/file-20190220-148545-1g16rne.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/259929/original/file-20190220-148545-1g16rne.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/259929/original/file-20190220-148545-1g16rne.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A knotty problem.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/female-hands-unravel-black-little-headphones-1184333146?src=_nMmjx-52kYA7yDjzv2OLg-2-54">VIKTORIIA NOVOKHATSKA/Shutterstock</a></span>
</figcaption>
</figure>
<p>You are probably familiar with a common topological phenomenon. Wires placed neatly in your bag in the morning (your earphones or an adapter) have a tendency to produce a horrible mess by midday. A wire is a very simple shape. Whether or not it is knotted is a topological question, and the tendency to arrive at a topological nightmare in your bag is now <a href="https://www.thenakedscientists.com/articles/questions/why-do-wires-tangle">quite well understood</a>. </p>
<p>Millions of years ago, evolution was confronted with a similar problem. <a href="https://www.livescience.com/37247-dna.html">DNA</a> in cells is a molecule composed of two coiled up chains. Each chain is a very long wire, built up from a sequence of small molecules called nucleobases. When a cell divides, these wires unwind, replicate and then coil up again. But just like wires in a bag, the strands of DNA can become tangled, which prevents the cell from dividing and causes it to die.</p>
<p>Special enzymes in the cell called <a href="https://www.sciencedirect.com/topics/neuroscience/topoisomerase">topoisomerases</a> have the task of preventing such a catastrophe. And deliberately disrupting the topoisomerases of bacteria prevents them from spreading and so stops an infection. This means that a better understanding of how topoisomerases prevent the entanglement of DNA could help us design new antibiotics. And since entanglement is a purely topological feature, <a href="https://academic.oup.com/nar/article/36/11/3515/2410103">topological techniques</a> can <a href="https://academic.oup.com/nar/article/47/1/69/5204334">help us do that</a>.</p>
<h2>Drug development</h2>
<p>Topology can also be used to improve the creation of new drugs. Pharmaceutical drugs are chemicals designed to interact with certain cells in the body in a particular way. Specifically, cells have receptors on them that allow molecules of a certain shape to lock onto them, altering the behaviour of the cells. So producing drugs with these shaped molecules enables them to target and affect the right cells.</p>
<p>As it turns out, manufacturing a molecule to have a particular shape is a rather simple process. But the easiest way to get the drug to the target cells is to send them via the bloodstream, and for that, the drug must be water soluble. After a drug with a correct shape is produced, the million pound question is: does it dissolve in water? Unfortunately, this is a very difficult question to answer just from knowing the chemical structure of the molecule. Many drug discovery projects fail because of solubility issues.</p>
<p>This is where topology comes in. “Molecule space” refers to a way of thinking about an entire collection of molecules as a kind of mathematical entity that can be studied geometrically. Having a map of this space would be a tremendous tool for producing new drugs, particularly if the map included landmarks indicating higher chances of solubility.</p>
<p>In <a href="https://jcheminf.biomedcentral.com/articles/10.1186/s13321-018-0308-5">recent work</a>, researchers used topological data analysis tools as a first step to producing such a map. Analysing vast amounts of data linking molecule properties to water solubility, the new approach led to the discovery of new, previously unsuspected, indicators of solubility. This improved ability to produce water-soluble drugs has the potential to significantly shorten the time it takes to create a new treatment, and to make the whole process cheaper. </p>
<p>In more and more realms of science, researchers are finding themselves with more data than they can effectively make sense of. The response of modern mathematicians to meet the <a href="https://ima.org.uk/9104/3rd-ima-conference-on-the-mathematical-challenges-of-big-data/">mathematical challenges of big data</a> is still unfolding – and topology, a theory bound only by the imagination of its practitioners, is bound to help shape the future.</p><img src="https://counter.theconversation.com/content/111460/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ittay Weiss does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Techniques from topology can help us understand DNA and improve drug development.Ittay Weiss, Lecturer in Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1019362018-09-24T10:19:48Z2018-09-24T10:19:48ZThe weird world of one-sided objects<figure><img src="https://images.theconversation.com/files/236749/original/file-20180917-158228-1ljv6dw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A Mobius strip.</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/paper-mobius-strip-on-wooden-board-346572113?src=oLohKklG1bmK9VPbNOyFJA-1-7">cosma/shutterstock.com</a></span></figcaption></figure><p>You have most likely encountered one-sided objects hundreds of times in your daily life – like the <a href="https://www.recyclenow.com/recycling-knowledge/packaging-symbols-explained">universal symbol</a> for recycling, found printed on the backs of aluminum cans and plastic bottles. </p>
<p>This <a href="http://mathworld.wolfram.com/MoebiusStrip.html">mathematical object</a> is called a Mobius strip. It has fascinated environmentalists, artists, engineers, mathematicians and many others ever since its discovery in 1858 by August Möbius, a German mathematician who died 150 years ago, on Sept. 26, 1868.</p>
<p><a href="http://www.learn-math.info/mathematicians/historyDetail.htm?id=Mobius">Möbius</a> discovered the one-sided strip in 1858 while serving as the chair of astronomy and higher mechanics at the University of Leipzig. (Another mathematician named Listing actually described it a few months earlier, but did not publish his work until 1861.) Möbius seems to have encountered the Möbius strip while working on the geometric theory of polyhedra, solid figures composed of vertices, edges and flat faces.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/ZN4TxmWK0bE?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">An animation of ants crawling along a Möbius strip, inspired by M.C. Escher’s artwork.</span></figcaption>
</figure>
<p>A Möbius strip can be created by taking a strip of paper, giving it an odd number of half-twists, then taping the ends back together to form a loop. If you take a pencil and draw a line along the center of the strip, you’ll see that the line apparently runs along both sides of the loop. </p>
<p>The concept of a one-sided object inspired artists like Dutch graphic designer <a href="https://www.mcescher.com/about/biography/">M.C. Escher</a>, whose woodcut “<a href="https://www.mcescher.com/gallery/recognition-success/mobius-strip-ii/">Möbius Strip II</a>” shows red ants crawling one after another along a Möbius strip. </p>
<p>The Möbius strip has more than just one surprising property. For instance, try taking a pair of scissors and cutting the strip in half along the line you just drew. You may be astonished to find that you are left not with two smaller one-sided Möbius strips, but instead with one long two-sided loop. If you don’t have a piece of paper on hand, Escher’s woodcut “<a href="https://www.mcescher.com/gallery/recognition-success/mobius-strip-i/">Möbius Strip I</a>” shows what happens when a Möbius strip is cut along its center line.</p>
<p>While the strip certainly has visual appeal, its greatest impact has been in mathematics, where it helped to spur on the development of an entire field called <a href="http://mathworld.wolfram.com/Topology.html">topology</a>. </p>
<p>A topologist studies properties of objects that are preserved when moved, bent, stretched or twisted, without cutting or gluing parts together. For example, a tangled pair of earbuds is in a topological sense the same as an untangled pair of earbuds, because changing one into the other requires only moving, bending and twisting. No cutting or gluing is required to transform between them.</p>
<p>Another pair of objects that are topologically the same are a coffee cup and a doughnut. Because both objects have just one hole, one can be deformed into the other through just stretching and bending. </p>
<figure>
<img src="https://cdn.theconversation.com/static_files/files/291/Mug_and_Torus_morph.gif?1537204404">
<figcaption>A mug morphs into a doughnut.<a href="https://en.wikipedia.org/wiki/Topology#/media/File:Mug_and_Torus_morph.gif">Wikimedia Commons</a></figcaption></figure>
<p>The number of holes in an object is a property which can be changed only through cutting or gluing. This property – called the “genus” of an object – allows us to say that a pair of earbuds and a doughnut are topologically different, since a doughnut has one hole, whereas a pair of earbuds has no holes.</p>
<p>Unfortunately, a Möbius strip and a two-sided loop, like a typical silicone awareness wristband, both seem to have one hole, so this property is insufficient to tell them apart – at least from a topologist’s point of view. </p>
<p>Instead, the property that distinguishes a Möbius strip from a two-sided loop is called orientability. Like its number of holes, an object’s orientability can only be changed through cutting or gluing. </p>
<p>Imagine writing yourself a note on a see-through surface, then taking a walk around on that surface. The surface is orientable if, when you come back from your walk, you can always read the note. On a nonorientable surface, you may come back from your walk only to find that the words you wrote have apparently turned into their mirror image and can be read only from right to left. On the two-sided loop, the note will always read from left to right, no matter where your journey took you. </p>
<p>Since the Möbius strip is nonorientable, whereas the two-sided loop is orientable, that means that the Möbius strip and the two-sided loop are topologically different. </p>
<figure>
<img src="https://cdn.theconversation.com/static_files/files/292/Mobius.gif?1537204663">
<figcaption>When the GIF starts, the dots listed off clockwise are black, blue and red. However, we can move the three-dot configuration around the Möbius strip such that the figure is in the same location, but the colors of the dots listed off clockwise are now red, blue and black. Somehow, the configuration has morphed into its own mirror image, but all we’ve done is move it around on the surface. This transformation is impossible on an orientable surface like the two-sided loop. Created by David Gunderman.</figcaption></figure>
<p>The concept of orientability has important implications. Take enantiomers. These chemical compounds have the same chemical structures except for one key difference: They are mirror images of one another. For example, <a href="http://scienceblogs.com/moleculeoftheday/2006/10/27/lmethamphetamine-would-you-bel/">the chemical L-methamphetamine</a> is an ingredient in Vicks Vapor Inhalers. Its mirror image, D-methamphetamine, is a Class A illegal drug. If we lived in a nonorientable world, these chemicals would be indistinguishable. </p>
<p>August Möbius’s discovery opened up new ways to study the natural world. The study of topology continues to produce stunning results. For example, last year, topology led scientists to discover <a href="https://www.nature.com/news/all-shook-up-over-topology-1.22322">strange new states of matter</a>. This year’s Fields Medal, the highest honor in mathematics, <a href="https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2018">was awarded to Akshay Venkatesh</a>, a mathematician who helped integrate topology with other fields such as number theory.</p><img src="https://counter.theconversation.com/content/101936/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The inventor of the brain-teasing Möbius strip died 150 years ago, but his creation continues to spawn new ideas in mathematics.David Gunderman, Ph.D. student in Applied Mathematics, University of Colorado BoulderRichard Gunderman, Chancellor's Professor of Medicine, Liberal Arts, and Philanthropy, Indiana UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/958962018-07-16T10:39:19Z2018-07-16T10:39:19ZWhy I teach math through knitting<figure><img src="https://images.theconversation.com/files/225197/original/file-20180627-112628-1tr48e8.jpg?ixlib=rb-1.1.0&rect=0%2C145%2C752%2C598&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Math in yarn.</span> <span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>One snowy January day, I asked a classroom of college students to tell me the first word that came to mind when they thought about mathematics. The top two words were “calculation” and “equation.” </p>
<p>When I asked a room of professional mathematicians the same question, neither of those words were mentioned; instead, they offered phrases like “critical thinking” and “problem-solving.”</p>
<p>This is unfortunately common. What professional mathematicians think of as mathematics is entirely different from what the general population thinks of as mathematics. When so many describe mathematics as synonymous with calculation, it’s no wonder we hear “I hate math” so often. </p>
<p>So I set out to solve this problem in a somewhat unconventional way. I decided to offer a class called “The Mathematics of Knitting” at my institution, Carthage College. In it, I chose to eliminate pencil, paper, calculator (gasp) and textbook from the classroom completely. Instead, we talked, used our hands, drew pictures and played with everything from beach balls to measuring tapes. For homework, we reflected by blogging. And of course, we knit.</p>
<h2>Same but different</h2>
<p>One crux of mathematical content is the equation, and crucial to this is the equal sign. An equation like x = 5 tells us that the dreaded x, which represents some quantity, has the same value as 5. The number 5 and the value of x must be exactly the same. </p>
<p>A typical equal sign is very strict. Any small deviation from “exactly” means that two things are not equal. However, there are many times in life where two quantities are not exactly the same, but are essentially the same by some meaningful criteria.</p>
<p>Imagine, for example, that you have two square pillows. The first is red on top, yellow on the right, green on bottom and blue on the left. The second is yellow on the top, green on the right, blue on bottom, and red on the left.</p>
<p>The pillows aren’t exactly the same. One has a red top, while one has a yellow top. But they’re certainly similar. In fact, they would be exactly the same if you turned the pillow with the red top once counterclockwise.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=281&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=281&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=281&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=353&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=353&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=353&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Rotating two square pillows.</span>
<span class="attribution"><span class="source">Sara Jensen</span></span>
</figcaption>
</figure>
<p>How many different ways could I put the same pillow down on a bed, but make it look like a different one? A little homework shows there are 24 possible colored throw pillow configurations, though only eight of them can be obtained from moving a given pillow. </p>
<p>Students demonstrated this by knitting throw pillows, consisting of two colors, from knitting charts.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=271&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=271&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=271&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=341&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=341&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=341&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A knitting chart for a throw pillow.</span>
<span class="attribution"><span class="source">Sara Jensen</span></span>
</figcaption>
</figure>
<p>The students created square knitting charts where all eight motions of the chart resulted in a different-looking picture. These were then knit into a throw pillow where the equivalence of the pictures could be demonstrated by actually moving the pillow.</p>
<h2>Rubber sheet geometry</h2>
<p>Another topic we covered is a subject sometimes referred to as “rubber sheet geometry.” The idea is to imagine the whole world is made of rubber, then reimagine what shapes would look like. </p>
<p>Let’s try to understand the concept with knitting. One way of knitting objects that are round – like hats or gloves – is with special knitting needles called double pointed needles. While being made, the hat is shaped by three needles, making it look triangular. Then, once it comes off the needles, the stretchy yarn relaxes into a circle, making a much more typical hat. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=899&fit=crop&dpr=1 600w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=899&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=899&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1129&fit=crop&dpr=1 754w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1129&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1129&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Knitting to learn.</span>
<span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This is the concept that “rubber sheet geometry” is trying to capture. Somehow, a triangle and a circle can be the same if they’re made out of a flexible material. In fact, all polygons become circles in this field of study. </p>
<p>If all polygons are circles, then what shapes are left? There are a few traits that are distinguishable even when objects are flexible – for example, if a shape has edges or no edges, holes or no holes, twists or no twists. </p>
<p>One example from knitting of something that is not equivalent to a circle is an infinity scarf. If you want to make a paper infinity scarf at home, take a long strip of paper and glue the short edges together by attaching the top left corner to the bottom right corner, and the bottom left corner to the top right corner. Then draw arrows pointing up the whole way around the object. Something cool should happen. </p>
<p>Students in the course spent some time knitting objects, like infinity scarves and headbands, that were different even when made out of flexible material. Adding markings like arrows helped visualize exactly how the objects were different. </p>
<h2>Different flavors</h2>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=900&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=900&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=900&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1131&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1131&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1131&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">An infinity scarf.</span>
<span class="attribution"><span class="source">Carthage College</span></span>
</figcaption>
</figure>
<p>If the things described in this article don’t sound like math to you, I want to reinforce that they very much are. The subjects discussed here – abstract algebra and topology – are typically reserved for math majors in their junior and senior years of college. Yet the philosophies of these subjects are very accessible, given the right mediums. </p>
<p>In my view, there’s no reason these different flavors of math should be hidden from the public or emphasized less than conventional mathematics. Further, <a href="https://files.eric.ed.gov/fulltext/ED321967.pdf">studies have shown</a> that using materials that can be physically manipulated can improve mathematical learning at all levels of study. </p>
<p>If more mathematicians were able to set aside classical techniques, it seems possible the world could overcome the prevailing misconception that computation is the same as mathematics. And just maybe, a few more people out there could embrace mathematical thought; if not figuratively, then literally, with a throw pillow.</p><img src="https://counter.theconversation.com/content/95896/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sara Jensen works for Carthage College. She is a member of the Mathematical Association of America, and is a Project NExT red dot ('15).</span></em></p>In this professor’s class, there are no calculators. Instead, students learn advanced math by talking, drawing pictures, playing with beach balls – and knitting.Sara Jensen, Assistant Professor of Mathematics, Carthage CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/925032018-02-28T01:16:58Z2018-02-28T01:16:58ZHow art merges with maths to explore continuity, change and exotic states<figure><img src="https://images.theconversation.com/files/208188/original/file-20180227-36693-udldxp.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C3696%2C2345&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The adaptable form of pleated and folded textiles provides a real world view of the mathematical field of topology. </span> <span class="attribution"><a class="source" href="http://art.uts.edu.au/index.php/exhibitions/soft-topologies/">Kate Scardifield</a>, <span class="license">Author provided</span></span></figcaption></figure><p>Imagine you could turn a hollow sphere completely inside out - without making a hole, without cutting the material, without making any creases. </p>
<p>If you could ever do such a thing, it would rely on a field of mathematics called topology. </p>
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Read more:
<a href="https://theconversation.com/the-nobel-prize-for-physics-goes-to-topology-and-mathematicians-applaud-66532">The Nobel Prize for Physics goes to topology – and mathematicians applaud</a>
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<p>Kate Scardifileld’s current exhibition <a href="http://art.uts.edu.au/index.php/exhibitions/soft-topologies/">Soft Topologies</a> at the UTS Gallery is an example of an artist using the metaphor of topology to make artwork and describe her practice. Scardifield’s primary media are textiles, though the exhibition is also a series of “adaptable sculptures” that will be transformed by the artist and a select, multidisciplinary team.</p>
<p>The exhibition is a choreography of Scardifield’s attractively coloured textiles, which spray, flutter, fan, drape, hang and crumple through the space like escaped gestures. These forms are only one version of the installations, limited by the static state convenient for gallery viewing. </p>
<p>The far wall of the exhibition is covered by a grid of yellow tape against bright blue. It’s a comparatively rigid expression of space that offsets the dynamism and intricacy of the textiles scattered about the room. </p>
<h2>What is topology?</h2>
<p>The origins of topology are in the field of geometry, where it is used to understand “<a href="https://theconversation.com/the-nobel-prize-for-physics-goes-to-topology-and-mathematicians-applaud-66532">geometric objects that don’t change when bent or stretched</a>.” </p>
<p>Combined with algebraic geometry, abstract algebra, differential equations and probabilities, topology is now used in many branches of physics to understand “<a href="https://theconversation.com/the-nobel-prize-for-physics-goes-to-topology-and-mathematicians-applaud-66532">exotic states of matter</a>” - that is, states of matter that are not solid, liquid or gas and that have <a href="https://www.school-for-champions.com/science/matter_states_exotic.htm#.WpTm1BNuY6g">very unusual physical characteristics</a>. </p>
<p>But topology has recently made the transition from an exclusively mathematical field of study to something that is more widely used in the analysis and creation of cultural and social phenomena. Some scholars even claim that culture as a whole is “<a href="http://journals.sagepub.com/doi/abs/10.1177/0263276412454552">becoming topological</a>.” </p>
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<iframe width="440" height="260" src="https://www.youtube.com/embed/R_w4HYXuo9M?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">How to turn a sphere inside out, using topological mathematics.</span></figcaption>
</figure>
<p>Scardifield’s exhibition is an example of the way practitioners from different disciplines productively use ideas that originate in vastly different fields of study. </p>
<p>Metaphors forged in one discipline happily circulate through others. “<a href="https://en.wikipedia.org/wiki/Cell_theory">The cell</a>”, for instance, is an architectural term that became common and influential in biological science, the study of social phenomena (terrorist cells) and technology (cell phone). “Immunity” is a term that originally had <a href="https://books.google.com.au/books?id=ifAczQoHyZ0C&pg=PA249&lpg=PA249&dq=1881+immune+metaphor+law&source=bl&ots=AjoxjJOi1X&sig=6yOxSYMeeY22-Fv6Xw5rtKnN_kc&hl=en&sa=X&ved=0ahUKEwjMyO7-q8fZAhVEvrwKHbXOC9oQ6AEILDAB#v=onepage&q=1881%20immune%20metaphor%20law&f=false">legal applications</a> meaning “exception from liability”, which is now more commonly associated with medicine. </p>
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<strong>
Read more:
<a href="https://theconversation.com/when-artists-get-involved-in-research-science-benefits-82147">When artists get involved in research, science benefits</a>
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</em>
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<h2>Explaining relationships</h2>
<p>Topology enables a different understanding of the relationships between things. </p>
<p>Philosopher <a href="https://muse.jhu.edu/article/390243">Ian Hacking</a> suggests that the topological metaphor of a <a href="https://en.wikipedia.org/wiki/Manifold">manifold</a> is an improvement on the optical metaphor of a spectrum for capturing our present awareness of autism. A spectrum “misleadingly suggests a single dimension from severe to high-functioning” whereas manifolds, like kinds of autism, “come in any number of dimensions”.</p>
<p>Topological metaphors are also useful for thinking differently about other relations that are conventionally conceived as spectra: left and right political, male and female gender. </p>
<p>For example, thinking topologically enables the understanding that Greens supporters and Nationals, who are usually imagined to be at opposed ends of the political spectrum, might be aligned on certain issues, such as those to do with the fights against coal mines on farm land. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/207994/original/file-20180227-36674-17twe3x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/207994/original/file-20180227-36674-17twe3x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=835&fit=crop&dpr=1 600w, https://images.theconversation.com/files/207994/original/file-20180227-36674-17twe3x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=835&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/207994/original/file-20180227-36674-17twe3x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=835&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/207994/original/file-20180227-36674-17twe3x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1050&fit=crop&dpr=1 754w, https://images.theconversation.com/files/207994/original/file-20180227-36674-17twe3x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1050&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/207994/original/file-20180227-36674-17twe3x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1050&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Adaptable sculpture in one variation. Accordion pleated spinnaker cloth, sail battens.</span>
<span class="attribution"><span class="source">Robin Hearfield</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<h2>Topology in textiles</h2>
<p>Scardifield’s practice involves folding and distorting cloth so that it takes on a surprising variety of forms. She focuses on the relationship between change and continuity of form through operations of weaving, folding, draping, pleating and layering. These are analogous to the abstract manipulations and speculations that inform the mathematical field of topology. </p>
<p>A flat piece of cloth is a classic, simple two dimensional space. Three dimensional space of a more complicated nature is created when Scardifield deliberately folds cloth in pleats. </p>
<p>Folds of a more random variety are created in the exhibition by allowing the cloth to describe the forms of other objects (draping) or the forces of gravity and tension (hanging). The new forms created in these processes are at once distinct from and yet continuous with the original pieces of cloth.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/207989/original/file-20180227-36706-qzfdfm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/207989/original/file-20180227-36706-qzfdfm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=900&fit=crop&dpr=1 600w, https://images.theconversation.com/files/207989/original/file-20180227-36706-qzfdfm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=900&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/207989/original/file-20180227-36706-qzfdfm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=900&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/207989/original/file-20180227-36706-qzfdfm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1131&fit=crop&dpr=1 754w, https://images.theconversation.com/files/207989/original/file-20180227-36706-qzfdfm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1131&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/207989/original/file-20180227-36706-qzfdfm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1131&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">Hand pleated and heat-set polyester organza, cotton thread, black tourmaline. 185x120cm (full expansion). Adaptable form, Dimensions variable.</span>
<span class="attribution"><span class="source">Kate Scardifield and ALASKA Projects</span>, <span class="license">Author provided</span></span>
</figcaption>
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<p>Author and academic <a href="http://www.stevenconnor.com/topologies/">Steven Connor suggests</a> that “the labile, intermediary form of the textile” provides a better material metaphor for understanding “the contemporary world of communication and information” than previous alternatives of solidity or fluidity. </p>
<p>Textiles are at once pliable and cohesive, and in this sense invite the imagining of forms that dance with subtle fluctuations over time. This is expressed in the following video of Scardifield making a chevron mould.</p>
<p><div data-react-class="InstagramEmbed" data-react-props="{"url":"https://www.instagram.com/p/BfraGw-Hr-t/?taken-by=katescardifield","accessToken":"127105130696839|b4b75090c9688d81dfd245afe6052f20"}"></div></p>
<p>The expression of space and form in terms of continuity and change is central to the individual objects in Soft Topologies and the overall design of the exhibition. Scardifield has deliberately designed the exhibition as a dynamic space, involving multiple participants from different disciplines who rearrange her works for the duration of the show and express her textiles through other media, such as video and photography. </p>
<p>The traces of her work in different media are like the traces left by the folds Scardifield makes in cloth and paper to create new forms. </p>
<p>Scardifield doesn’t use topological mathematics in her exhibition. However, the show is a great example of topology can invite different possibilities for imagining and analysis. </p>
<hr>
<p><em>Soft Topologies is showing 27 February—20 April 2018 at UTS Gallery, Sydney</em></p><img src="https://counter.theconversation.com/content/92503/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Tom Lee works for the University of Technology Sydney and at times receives funding and support for his research and writing.</span></em></p>Art can help us explore and understand some of the more abstract ideas in maths - such a topology.Tom Lee, Lecturer, Faculty of Design and Architecture Building, University of Technology SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/762582017-04-19T22:34:40Z2017-04-19T22:34:40ZCalculating where America should invest in its transportation and communications networks<figure><img src="https://images.theconversation.com/files/165883/original/file-20170419-2410-x9z679.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Which links are most important in road and information networks?</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/network-connection-technology-concept-city-background-436942042">Sahacha Nilkumhang/Shutterstock.com</a></span></figcaption></figure><p>The American economy is underpinned by networks. Road networks carry traffic and freight; the internet and telecommunications networks carry our voices and digital information; the electricity grid is a network carrying energy; financial networks transfer money from bank accounts to merchants. They’re vast, often global systems – but a local disruption can really block them up.</p>
<p>For example, <a href="http://www.wsbtv.com/news/local/atlanta/gdot-state-offering-31m-in-incentives-to-reopen-i-85-before-june/511832846">the I-85 bridge collapse in Atlanta will affect that city’s traffic for months</a>. A seemingly minor train derailment at New York City’s Penn Station resulted in <a href="https://www.nytimes.com/2017/04/04/nyregion/messy-commute-for-nj-transit-and-lirr-riders-a-day-after-derailment.html?_r=0">multiple days of travel chaos</a> in April. </p>
<p>As the Trump administration plans to <a href="http://thehill.com/blogs/congress-blog/economy-budget/328586-an-infrastructure-plan-coming-but-when">invest hundreds of billions in American infrastructure networks</a>, it will be crucial to identify what elements are the most crucial to repair or improve. This is not only important for maximizing benefits; it’s also useful in preventing disaster. Is there, perhaps, a telecommunication line that would be particularly damaging if it were destroyed? Or one road through an area that has an especially large role in keeping traffic flowing smoothly?</p>
<p><a href="http://greatvalley.psu.edu/person/qiang-patrick-qiang">Patrick Qiang</a> and I are operations management scholars who have developed <a href="http://dx.doi.org/10.1007/s10898-007-9198-1">a way to evaluate network performance</a> and simulate the effects of potential changes, whether planned (like a highway repair) or unexpected (like a natural disaster). By modeling the independent behavior of all the users of a network, we can calculate the flow – of <a href="http://dx.doi.org/10.1007/s10898-015-0371-7">freight</a>, <a href="https://dx.doi.org/10.1209/0295-5075/79/38005">commuters</a>, <a href="http://dx.doi.org/10.1007/978-3-540-77958-2_14">money</a> or anything else – across each link, and how other links’ flows will change. This lets us identify where investment will be most beneficial, and which projects shouldn’t happen at all.</p>
<h2>More isn’t always better</h2>
<p>It’s very difficult to measure networks’ performance, in part because they are so complex, but also because people use them differently at different times, and because those choices affect others’ experiences. For example, one person choosing to drive to work instead of taking the bus puts one more car on the road, which might get involved in a crash or otherwise contribute to a traffic jam.</p>
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<figcaption><span class="caption">Explaining the Braess paradox.</span></figcaption>
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<p>In 1968, German mathematician Dietrich Braess observed the possibility that adding a road to an area with congested traffic <a href="http://dx.doi.org/10.1287/trsc.1050.0127">could actually make things worse</a>, not better. <a href="https://supernet.isenberg.umass.edu/braess/braess-new.html">This paradox</a> can occur when travel times depend on the amount of traffic. If too many drivers decide their own optimal route involves one particular road, that road can become congested, slowing everyone’s travel time. In effect, the drivers would have been better off if the road hadn’t been built.</p>
<p>This phenomenon has been found not only <a href="https://supernet.isenberg.umass.edu/braess/braess-new.html#BraessArticle">in transportation networks</a> and <a href="http://dx.doi.org/10.1239/jap/1032374242">the internet</a>, but also, recently, <a href="https://doi.org/10.1209/0295-5075/115/28004">in electrical circuits</a>. </p>
<p>We shouldn’t waste time and money building or repairing network links the community would be better without. But how can we tell which elements help and which make things worse?</p>
<h2>Calculating efficiency</h2>
<p>The best networks can handle the highest demand at the lowest average cost for each trip – such as a commute from a worker’s home to her office. Evaluating a network means identifying which locations need to be connected to each other, as well as the volume of traffic between specific places and the various costs involved – such as gas, pavement wear and tear, and police services keeping drivers safe.</p>
<p>Once a network is measured in this way, it can be converted into a computerized model where we can simulate removing links or adding new ones in particular places. Then we can measure what happens to the rest of the network: Does traffic get more congested, and if so, by how much? Or, as in the Braess paradox, do travel times actually get shorter when a link is removed? And how much money does a particular project cost to build, and save in time or user expenses?</p>
<h2>Going global</h2>
<p>Our method of measuring a network’s performance has been used to refine
<a href="https://ercim-news.ercim.eu/en79/rd/route-optimization-how-efficient-will-the-proposed-north-dublin-metro-be">the route of a proposed metro line in Dublin, Ireland</a>; <a href="https://www.researchgate.net/publication/313365965_Maritime_Network_Efficiency_Comparison_in_Indonesia_Nusantara_Pendulum_and_Sea_Tollway">to design new shipping routes in Indonesia</a>; <a href="http://www.cedim.de/download/14_Schulz.pdf">to identify which roads in Germany should be first on the maintenance list</a>; and <a href="http://dx.doi.org/10.1007/s11069-013-0896-3">to determine the effects of road closures after major fires in regions of Greece</a>.</p>
<p>Our method has also been applied to make supply chains more efficient, both to <a href="http://dx.doi.org/10.1007/978-1-84882-634-2_6">maximize profits</a> and to <a href="http://www.sciencedirect.com/science/article/pii/S0965856412000249">speed disaster relief supplies</a> to people in need.</p>
<p>As the U.S. works to enhance its economic competitiveness, the country will need to invest in many different types of networks, to maximize their usefulness and value to Americans. Using measurement methods like ours can guide leaders to wise investments.</p><img src="https://counter.theconversation.com/content/76258/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Anna Nagurney does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>When planning major infrastructure investments, it’s important to know which road, freight and information networks are most important – and which proposals might make things worse, not better.Anna Nagurney, John F. Smith Memorial Professor of Operations Management, UMass AmherstLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/665432016-10-08T00:51:40Z2016-10-08T00:51:40ZPhysicists explore exotic states of matter inspired by Nobel-winning research<figure><img src="https://images.theconversation.com/files/140958/original/image-20161007-21414-ajgt9v.jpg?ixlib=rb-1.1.0&rect=61%2C267%2C284%2C181&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Things are kind of different on the quantum level.</span> <span class="attribution"><span class="source">Nandini Trivedi</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>The 2016 Nobel Prize in physics has been awarded to <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/thouless-facts.html">David Thouless</a>, <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/haldane-facts.html">Duncan Haldane</a> and <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/kosterlitz-facts.html">Michael Kosterlitz</a>, three theoretical physicists whose research used the unexpected mathematical lens of <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/popular-physicsprize2016.pdf">topology to investigate phases of matter and the transitions between them</a>.</p>
<p>Topology is a branch of mathematics that deals with understanding shapes of objects; it’s interested in “invariants” that don’t change when a shape is deformed, like the number of holes an object has. Physics is the study of matter and its properties. The Nobel Prize winners were the first to make the connection between these two worlds.</p>
<p>Everyone is used to the idea that a material can take various familiar forms such as a solid, liquid or gas. But the Nobel Prize recognizes other surprising phases of matter – called topological phases – that the winners proposed theoretically and experimentalists have since explored.</p>
<p>Topology is opening up new platforms for observing and understanding these new states of matter in many branches of physics. I work with theoretical aspects of cold atomic gases, a field which has only developed in the years since Thouless, Haldane and Kosterlitz did their groundbreaking theoretical work. Using lasers and atoms to emulate complex materials, cold atom researchers have begun to realize some of the laureates’ predictions – with the promise of much more to come.</p>
<h2>Cold atoms get us to quantum states of matter</h2>
<p>All matter is made up of building blocks, such as atoms. When many atoms come together in a material, they start to interact. As the temperature changes, the state of matter starts to change. For instance, water is a liquid until a fixed temperature, when it turns into vapor (373 degrees Kelvin; 212 degrees Fahrenheit; 100 degrees Celsius); and if you cool, solid ice forms at a fixed temperature (273K; 32°F; 0°C). The laws of physics give us a theoretical limit to how low the temperature can get. This lowest possible temperature is called absolute zero (0K) (and equals -460°F or -273°C).</p>
<p>Classical physics governs our everyday world. Classical physics tells us that if we cool atoms to really low temperatures, they stop their normally constant vibrating and come to a standstill.</p>
<p>But really, as we cool atoms down to temperatures approaching close to 0K, we leave the regime of classical physics – quantum mechanics begins to govern what we see. </p>
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<iframe width="440" height="260" src="https://www.youtube.com/embed/nAGPAb4obs8?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Atoms start to behave not as individual particles but as waves in the world of quantum physics.</span></figcaption>
</figure>
<p>In the quantum mechanical world, if an object’s position becomes sharply defined then its momentum becomes highly uncertain, and vice versa. Thus, if we cool atoms down, the momentum of each atom decreases, and the quantum uncertainty of its position grows. Instead of being able to pinpoint where each atom is, we can now only see a blurry space somewhere within which the atom must be. At some point, the neighboring uncertain positions of nearby atoms start overlapping and the atoms lose their individual identities. Surprisingly, the distinct atoms become a single entity, and behave as <a href="https://www.nobelprize.org/nobel_prizes/physics/laureates/2001/">one coherent unit</a> – a discovery that won a previous Nobel.</p>
<p>This new, amazing way atoms organize themselves at very low temperatures results in new properties of matter; it’s no longer a classical solid in which the atoms occupy periodic well-defined positions, like eggs in a carton.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=581&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=581&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=581&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=730&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=730&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140881/original/image-20161007-21416-dcixy9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=730&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Supercooled atoms are highly coherent.</span>
<span class="attribution"><span class="source">Nandini Trivedi</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Instead, the material is now in a new quantum state of matter in which each atom has become a wave with its position no longer identifiable. And yet the atoms are not moving around chaotically. Instead, they are highly coherent, with a new kind of quantum order. Just like laser beams, the coherent matter waves of superfluids, superconductors and magnets <a href="http://doi.org/10.1126/science.275.5300.637">can produce interference patterns</a>.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=586&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=586&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=586&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=737&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=737&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140883/original/image-20161007-21454-154tkox.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=737&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">As temperatures rise, materials lose their quantum order.</span>
<span class="attribution"><span class="source">Nandini Trivedi</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Physicists have known about quantum order in superfluids and magnets in three dimensions since the middle of the last century. We understand that the order is lost at a critical temperature due to thermal fluctuations. But in two dimensions the situation is different. Early theoretical work showed that thermal fluctuations would destroy the quantum order even at very low temperatures. </p>
<p>What Thouless, Haldane and Kosterlitz addressed were two important questions: What is the nature of the quantum ordered state of superfluids, superconductors and magnets in low dimensions? What is the nature of the phase transition from the ordered to the disordered state in two dimensions? </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=317&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=317&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=317&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=399&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=399&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140892/original/image-20161007-21421-1gllk4k.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=399&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The whirl of a topological defect, a vortex or an anti-vortex, can be felt no matter how far you go from the eye of the storm.</span>
<span class="attribution"><span class="source">Nandini Trivedi</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>Thinking about defects</h2>
<p>Kosterlitz and Thouless’s innovation was to show that topological defects – vortex and anti-vortex whirls and swirls – are crucial to understand the magnetic and superfluid states of matter in two dimensions. These defects are not just local perturbations in the quantum order; they produce a winding or circulation as one goes around it. The vorticity, which measures how many times one winds around, is measured in integer units of the circulation.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=248&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=248&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=248&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=312&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=312&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140906/original/image-20161007-21430-w2slpc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=312&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">On the left, a vortex is bound up with an anti-vortex. On the right, more and more defects unbind upon increasing the temperature, and the material enters a disordered state.</span>
<span class="attribution"><span class="source">Nandini Trivedi</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Kosterlitz and Thouless showed that at low temperatures, a vortex is bound up with an anti-vortex so the order survives. As the temperature increases, these defects unbind and grow in number and that drives a transition from an ordered to a disordered state. </p>
<p>It’s been possible to visualize the vortices in cold atomic gases that Kosterlitz and Thouless originally proposed, <a href="http://doi.org/10.1038/nature04851">bringing to life the topological defects they theoretically proposed</a>. In my own research, <a href="http://doi.org/10.1038/nphys983">we’ve been able to extend these ideas</a> to quantum phase transitions driven by increasing interactions between the atoms rather than by temperature fluctuations.</p>
<h2>Figuring out step-wise changes in materials</h2>
<p>The second part of the Nobel Prize went to Thouless and Haldane for discovering new topological states of matter and for showing how to describe them in terms of topological invariants. </p>
<p>Physicists knew about the existence of a phenomenon called the quantum Hall effect, first observed in two dimensional electrons in semiconductors. The Hall conductance, which is the ratio of the transverse voltage and the current, was observed to change in very precise integer steps as the magnetic field was increased. This was puzzling because real materials are disordered and messy. How could something so precise be seen in experiments?</p>
<p>It turns out that the current flows only in narrow channels at the edges and not within the bulk of the material. The number of channels is controlled by the magnetic field. Every time an additional channel or lane gets added to the highway, the conductance increase by a very precise integer step, with a precision of one part in billion. </p>
<p>Thouless’ insight was to show that the flow of electrons at the boundaries has a topological character: the flow is not perturbed by defects – the current just bends around them and continues with its onward flow. This is similar to strong water flow in a river that bends around boulders.</p>
<p>Thouless figured out that here was a new kind of order, represented by a topological index that counts the number of edge states at the boundary. That’s just like how the number of holes (zero in a sphere, one in a doughnut, two in glasses, three in a pretzel) define the topology of a shape and the robustness of the shape so long as it is deformed smoothly and the number of holes remains unchanged. </p>
<h2>Global, not local, properties</h2>
<p>Interacting topological states are even more remarkable and truly bizarre in that they harbor fractionalized excitations. We’re used to thinking of an electron, for instance, with its charge of e as being indivisible. But, in the presence of strong interactions, as in the fractional quantum Hall experiments, the electron indeed fractionalizes into three pieces each carrying a third of a charge! </p>
<p>Haldane discovered a whole new paradigm: in a chain of spins with one unit of magnetic moment, the edge spins are fractionalized into units of one-half. Remarkably, the global topological properties of the chain completely determine the unusual behavior at the edges. Haldane’s remarkable predictions have been verified by experiments on solid state materials containing one-dimensional chains of magnetic ions.</p>
<p>Topological states are new additions to the list of phases of matter, such as, solid, liquid, gas, and even superfluids, superconductors and magnets. The laureates’ ideas have opened the floodgates for prizeworthy predictions and observations of topological insulators and topological superconductors. The <a href="http://doi.org/10.1038/nature13915">cold atomic gases present opportunities</a> beyond what can be achieved in materials because of the greater variety of atomic spin states and highly tunable interactions. Beyond the rewards of untangling fascinating aspects of our physical world, this research opens the possibility of using topologically protected states for quantum computing.</p><img src="https://counter.theconversation.com/content/66543/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Nandini Trivedi does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Forget solid, liquid, gas. This research used advanced math to theorize about topological phases of matter. And over the years experiments with matter and cold atoms have been validating the ideas.Nandini Trivedi, Professor of Physics, The Ohio State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/666132016-10-06T16:02:03Z2016-10-06T16:02:03ZWhy insights of Nobel physicists could revolutionise 21st-century computing<figure><img src="https://images.theconversation.com/files/140743/original/image-20161006-32708-1qazcoz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Control-alt future.</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-136209272/stock-photo-binary-orb-with-orbiting-bits-quantum-computing-concept.html?src=VQ61ULcsbQBfWcgxu1NpPQ-1-4">Mopic/shutterstock.com</a></span></figcaption></figure><p>British scientists David Thouless, Duncan Haldane and Michael Kosterlitz <a href="https://theconversation.com/odd-states-of-matter-how-three-british-theorists-scooped-the-2016-nobel-prize-for-physics-66517">won this year’s</a> Nobel Prize in Physics “for theoretical discoveries of topological phase transitions and topological phases of matter”. The reference to “theoretical discoveries” makes it tempting to think their work will not have practical applications or affect our lives some day. The opposite may well be true. </p>
<p>To understand the potential, it helps to understand the theory. Most people know that an atom has a nucleus in the middle and electrons orbiting around it. These correspond to different energy levels. When atoms group into substances, all the energy levels of each atom combine into <a href="https://www.halbleiter.org/en/fundamentals/conductors-insulators-semiconductors/">bands of electrons</a>. Each of these so-called energy bands has space for a certain number of electrons. And between each band are gaps in which electrons can’t flow. </p>
<p>If you apply an electrical charge (a flow of extra electrons) to a material, its conductivity is determined by whether the highest energy band has room for more electrons. If it does have room, the material will behave as a conductor. If not, you need extra energy to push the current of electrons into a new empty band and as a result the material behaves as an insulator. Understanding conductivity is vital to electronics, since electronic products ultimately rely on components that are electric conductors, semiconductors and insulators. </p>
<p>What Thouless, Haldane and Kosterlitz began to predict in the 1970s and 1980s and <a href="http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.226801">other</a> theorists have <a href="http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.95.146802">since</a> taken <a href="http://science.sciencemag.org/content/314/5806/1757">forward</a> is that certain materials break this rule. Instead of having a gap between bands in which electrons can’t flow, they have a special energy level between their bands where certain unexpected things are possible. </p>
<p><a href="https://theconversation.com/odd-states-of-matter-how-three-british-theorists-scooped-the-2016-nobel-prize-for-physics-66517">This quality</a> only exists on the surface or edge of these materials, and is very robust. It also depends to some extent on the shape of the material – the topology, as we say in physics. It behaves identically for a sphere and an egg, for example, but would be different for something shaped like a doughnut because of the hole in the middle. The first measurements of this kind of behaviour have been taken for a current along <a href="http://science.sciencemag.org/content/314/5806/1757">the boundary of a flat sheet</a>. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=327&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=327&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=327&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=411&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=411&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140745/original/image-20161006-32698-ic3g8y.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=411&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Thouless, Haldane and Kosterlitz.</span>
</figcaption>
</figure>
<h2>Computer power</h2>
<p>The properties of these so-called topological materials could potentially be extremely useful. Electrical currents can move without resistance across their surface, for example, even where a device is moderately damaged. <a href="https://theconversation.com/explainer-what-is-a-superconductor-38122">Superconductors</a> can already do this without having topological properties, but they only work at very low temperatures – meaning you use a lot of energy keeping them cool. Topological materials have the potential to do the same job at higher temperatures. </p>
<p>This has important implications for computing: most of the energy computers currently use is to run ventilators to cool down the heat produced by electrical resistance in the circuits. Remove this heat problem and you potentially make them many times more energy efficient. This could massively reduce their carbon emissions, for instance. It could also lead to batteries with far longer life spans. Researchers are already experimenting with topological materials like cadmium telluride and mercury telluride to bring <a href="http://science.sciencemag.org/content/314/5806/1757">this vision to life</a>. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=471&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=471&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=471&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=592&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=592&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140744/original/image-20161006-32737-u0u6jg.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=592&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Circuits in action.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-403258948/stock-vector-black-abstract-hi-speed-internet-technology-background-illustration-eye-scan-virus-computer.html?src=F50s84SZ_Qp6pxfXBLYszw-1-4">Titma Ongkantong/shutterstock.com</a></span>
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<p>There is also the potential for a major breakthrough in quantum computing. Classical computers encode information by either applying voltage or not applying voltage to a chip. The computer reads this as a 0 or 1 respectively for each “bit” of information. You put these bits together to build up more complex information. This is how the binary system works. </p>
<p>With quantum computing, you deliver information to electrons instead of microchips. The energy levels of these electrons then correspond to zeros and ones just like in classical computers, but in quantum mechanics both can be true at the same time. Without getting into too much theory, this raises the possibility of computers that can process exceedingly large amounts of data in parallel and are therefore much faster. </p>
<p>While the likes of <a href="https://www.newscientist.com/article/mg23130894-000-revealed-googles-plan-for-quantum-computer-supremacy/">Google</a> and <a href="http://www.research.ibm.com/quantum/">IBM</a> are researching how to manipulate enough electrons to create quantum computers that are more powerful than classical computers, one big obstacle is that these computers are very fragile with respect to surrounding “noise”. Whereas classical computers can cope with interference, quantum computers end up producing intolerable numbers of errors because of shaky support frames, stray electrical fields or air molecules hitting the processor even if you hold it in a high vacuum. This is the main reason why we don’t yet use quantum computers in our everyday lives. </p>
<p>One potential solution is to store information in more than one electron, since noise typically affects quantum processors at the level of single particles. Supposing you have five electrons all jointly storing the same bit of information, so long as the majority store it correctly, a disturbance to a single electron won’t undermine the system. </p>
<p>Researchers have been experimenting with this so-called majority voting, but topological engineering potentially offers an easier fix. In the same way as topological superconductors can carry a flow of electricity well enough that it doesn’t get hampered by resistance, topological quantum processors could be robust enough to be insensitive to noise problems. They could yet offer a major contribution to making quantum computing a reality. Researchers in the US <a href="http://www.nature.com/nphys/journal/v5/n1/abs/nphys1151.html">are working</a> on it. </p>
<h2>The future</h2>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=792&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=792&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=792&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=995&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=995&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140748/original/image-20161006-32713-17flzh9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=995&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Superdrugs?</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-400666111/stock-photo-medicine-bottle-for-injection-in-hand-palm-of-a-doctor-medical-glass-vial-for-vaccination-science-equipment-liquid-drug-or-vaccine-from-treatment-flu-in-laboratory-hospital-or-pharma.html?src=gX9lCJJqucwfDZu29qG_FQ-1-17">Funnyangel/shutterstock.com</a></span>
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<p>It might take between ten and 30 years before scientists become sufficiently good at manipulating electrons to make quantum computing possible, but they open up exciting possibilities. They could simulate the formation of molecules, for example, which is numerically too complicated for today’s computers. This could revolutionise drug research by enabling us to predict what will happen during chemical processes in the body. </p>
<p>To give just one other example, quantum computing has the potential to make artificial intelligence a reality. Quantum machines may be better at learning than classical computers, partly because they might be underpinned by much cleverer algorithms. Cracking AI could be a step change in human existence – for better or worse. </p>
<p>In short, the predictions of Thouless, Haldane and Kosterlitz have the potential to help revolutionise 21st-century computer technology. Where the Nobel committee has recognised the importance of their work in 2016, we are likely to be thanking them many decades into the future.</p><img src="https://counter.theconversation.com/content/66613/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Michael Hartmann does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>How Messrs Thouless, Haldane and Kosterlitz could hold the key to the future.Michael Hartmann, Associate Professor of Photonics and Quantum Sciences, Heriot-Watt UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/665322016-10-05T14:56:02Z2016-10-05T14:56:02ZThe Nobel Prize for Physics goes to topology – and mathematicians applaud<figure><img src="https://images.theconversation.com/files/140538/original/image-20161005-14232-9tfp4b.jpg?ixlib=rb-1.1.0&rect=176%2C131%2C923%2C708&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Math doesn't get its own Nobel, but is the foundation for much Prize-winning research.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/tereneta/88098709">Tim Ereneta</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span></figcaption></figure><p><a href="https://sharepoint.washington.edu/phys/people/Pages/view-person.aspx?pid=85">David Thouless</a>, <a href="http://physics.princeton.edu/%7Ehaldane/">Duncan Haldane</a> and <a href="https://vivo.brown.edu/display/jkosterl">Michael Kosterlitz</a> received the <a href="http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/">2016 Nobel Prize for Physics</a> for their work on exotic states of matter. They were inspired by the observation that some materials have unusual electrical properties – and their investigations led them to topology. That’s the branch of mathematics concerned with the properties of geometric objects that don’t change when bent or stretched (though torn would be a different story). As there is no Nobel Prize for mathematics, the topology community is understandably excited by this recognition of the utility of our discipline.</p>
<p>The old saw is that a topologist is a mathematician who cannot tell the difference between a doughnut and a coffee cup. (This joke is getting tiresome, but we stick with it anyway.) Both objects have just one hole and it’s easy to see how to deform one to the other.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/4iHjt2Ovqag?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Mmm…doughnut. (Video by Jim Fowler)</span></figcaption>
</figure>
<p>Topology aims to classify these spaces via indirect means. Since it’s often rather difficult to demonstrate how to deform a particular space to make it look like another, topologists develop mathematical machinery that takes spaces as an input and produces an algebraic object. This output might just be a number or it could be more complicated, but the machine should take spaces that are “the same” and spit out the same result. This allows us to distinguish spaces – two inputs are different if the corresponding outputs are different.</p>
<p>For example, it may seem obvious that a doughnut and a sphere are distinct objects, but just because you cannot see how to deform one to the other it doesn’t follow that it’s impossible. Topology comes to the rescue, however. One of many ways to show that a sphere and a doughnut aren’t the same is to compute their <a href="https://en.wikipedia.org/wiki/Fundamental_group">fundamental groups</a>. This is an algebraic object built from considering loops in the space.</p>
<p>A useful way to visualize loops is to imagine a rubber band lying on the surface of an object. First consider the sphere. Any loop on the sphere contains a disc inside it, and now you can imagine shrinking that loop down to a single point by pulling it through the disc. So there aren’t any interesting loops on the sphere – they are all deformable to a single point.</p>
<p>That’s not true for the doughnut, however. In fact there are lots of interesting loops on its surface (we are dealing with a hollow doughnut; there’s nothing but air inside). One such loop is obtained by drawing a circle around a vertical cross section (the blue loop in the figure below). Another arises from a horizontal cross-section (the red loop). It’s impossible to contract these loops down to a the same single point, so the fundamental groups of the sphere and doughnut aren’t the same and thus, they are different objects.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=334&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=334&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=334&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=419&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=419&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140350/original/image-20161004-20223-148ey14.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=419&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Loops on a doughnut.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:ToricCodeTorus.png">Woottonjames</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>The topology of materials</h2>
<p>Topology works in all dimensions, but physics is mostly concerned with our three-dimensional universe (well, that’s not always true – just ask <a href="https://en.wikipedia.org/wiki/String_theory#Extra_dimensions">string theorists</a>). When studying electrical properties of materials, we are definitely dealing with three dimensions. Even a thin wire has length, width and height. For a fixed electrical conductor, say a copper wire, it’s usually possible to determine the relationship between the voltage placed on the wire and the current that flows. Sometimes, however, materials experience an electrical phase transition (<a href="https://en.wikipedia.org/wiki/Superconductivity#Superconducting_phase_transition">superconductivity</a>, for example, which is obtained by lowering the temperature of the material) and the usual equations governing voltage and current break down. </p>
<p>Thouless, Haldane and Kosterlitz discovered that mathematically these <a href="https://en.wikipedia.org/wiki/Kosterlitz%E2%80%93Thouless_transition">transitions</a> correspond to an abrupt change in the topological type of the material. Certain thin films can be considered as being two-dimensional – imagine a surface that’s only one atom thick – and electrical current often flows in channels on the surface with low resistance. It turns out that there are points where the electrons flow around in a circular motion, sometimes clockwise and sometimes counterclockwise, and the number of such points can change as the material undergoes a phase transition.</p>
<p>Mathematicians immediately recognize this type of space from a first course in algebraic topology – it’s a plane with a few points removed and its fundamental group is very easy to compute. It turns out that the number of these types of points completely determines the topological type of the space.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/140526/original/image-20161005-14246-30gs8d.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">How can you scientifically describe a black hole without math?</span>
<span class="attribution"><a class="source" href="http://www.jpl.nasa.gov/spaceimages/details.php?id=PIA16695">NASA/JPL-Caltech</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>Topology elsewhere in physics</h2>
<p>Einstein’s <a href="https://en.wikipedia.org/wiki/General_relativity">general theory of relativity</a> posits that space-time is curved by gravity. The equations also imply the existence of black holes, which in mathematical terms correspond to <a href="https://en.wikipedia.org/wiki/Singularity_theory">singularities</a>, points in a space where all hell breaks loose (so to speak). A typical example familiar to calculus students is a point on the graph of a function where the derivative fails to exist. Much more complicated examples are possible and the space around such points can have interesting topology. Around ordinary points, space looks like a three-dimensional ball, but around singularities space can be knotted in unusual ways. Of course, we can’t experience this ourselves, but we can model it mathematically.</p>
<p>Topology has provided a framework in physics in other ways, such as the development of <a href="https://en.wikipedia.org/wiki/Topological_quantum_field_theory">topological quantum field theories</a>. <a href="https://en.wikipedia.org/wiki/String_theory">String theory</a> is a generalization of this idea in which particles are modeled by one-dimensional objects called strings. These theories, unlike Einstein’s four-dimensional spacetime, require extra dimensions to be consistent – either 10, 11 or 26 depending on which model you prefer. Why don’t we observe these dimensions? The prevailing interpretation is that they are “small” and curl up on themselves so that we don’t notice. These extra dimensions form a type of space familiar to algebraic geometers called a <a href="https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold">Calabi-Yau manifold</a>. </p>
<p>So it seems that a great deal of theoretical physics is based in sophisticated mathematics. Using ideas from topology, algebraic geometry and abstract algebra, not to mention differential equations and probability, physicists attempt to make sense of our universe. While math may not have its own Nobel Prize, many of the significant advances in other disciplines would not be possible without the development of sophisticated mathematics to provide the proper language for stating the results (Heisenberg’s <a href="https://en.wikipedia.org/wiki/Uncertainty_principle">uncertainty principle</a>, for example).</p>
<p>This is all heady stuff. In the end, though, the discoveries made by Thouless, Haldane and Kosterlitz have led to practical devices currently in use in industry (for example, efficient hard drives in computers) and may lead to advances in <a href="https://en.wikipedia.org/wiki/Quantum_computing">quantum computing</a>. Understanding how electrons move in materials is crucial to building better computers and instruments, and it’s exciting for us mathematicians to know that topology can help get us there.</p><img src="https://counter.theconversation.com/content/66532/count.gif" alt="The Conversation" width="1" height="1" />
There’s no Nobel Prize in mathematics, but math undergirds much high-level science. The 2016 Nobel in Physics rewards work in topology, a branch of math with multiple real world applications.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/665172016-10-04T15:33:08Z2016-10-04T15:33:08ZOdd states of matter: how three British theorists scooped the 2016 Nobel Prize for Physics<p>The Nobel Prize in Physics for 2016 <a href="https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html">has been awarded</a> to three British scientists working in the US for their theoretical work explaining strange states of matter, including superconductors, superfluids and thin magnetic films.</p>
<p>The prize was split between David J Thouless of the University of Washington, Duncan M Haldane of Princeton and J Michael Kosterlitz of Brown University. They will share a sum of US$928,000. Their work has helped shape an enormous amount of research over the past three decades and this well-deserved prize reflects the continuing importance of new discoveries that have and will continue to emerge from it.</p>
<p>“Normal” states of matter are ones you’re likely familiar with: solids, liquids and gases. The transition between these states is characterised by what is referred to as “symmetry breaking”. </p>
<p>For example, in a liquid, atoms are arranged uniformly in space and it looks identical no matter how you rotate it. However, when a liquid turns into a solid the atoms are locked into a crystal lattice. This new state of matter is less symmetrical in the sense that it only looks the same if it is rotated at certain angles. However, Thouless, Haldane and Kosterlitz found that matter is a lot more interesting than this. Their work showed how new phases of matter can occur where no symmetry is broken – and they used a mathematical idea to explain this. What distinguished these phases of matter – which display strange behaviour such as unusual patterns of electrical conductivity – were “topological properties”.</p>
<p>Topology is the mathematical study of how surfaces can be deformed continuously and smoothly. A famous example is the surface of an orange, a croissant, a coffee cup and a doughnut. To a mathematician, all these objects are imagined to be made of a malleable material that we are allowed to deform continuously without cutting or tearing. In this way an orange and croissant are identical, since we could mould both of them into a sphere. Likewise the coffee cup and doughnut are also the same to a mathematician because they both have one hole – the cup has a hole in its handle and the doughnut at its centre.</p>
<p>So, in this abstract sense the orange and croissant are in one distinct class, while the coffee cup and doughnut are in another. The difference between them boils down to whether their surface has a hole in it or not. This is the topological property of the object that is robust to any form of moulding we might do. The work of Thouless, Kosterlitz and Haldane made important steps in understanding how the notion of topology plays a role in the phases of matter. </p>
<p>This connection was exposed by considering the energies that electrons in materials can occupy – which can be plotted as a surface (when presented as a function of their momentum). In the 1980s scientists discovered that electrons in certain two-dimensional thin films move in a strange way when subjected to a strong magnetic field. These electrons order into perfectly conducting channels, located at the edge of the material, based on a <a href="https://theconversation.com/physicists-prove-quantum-spookiness-and-start-chasing-schrodingers-cat-48190">quantum mechanical</a> property known as spin. </p>
<p>What’s more, this conductivity increases in discrete steps as the magnetic field increases – an effect called the <a href="https://theconversation.com/scientists-discover-fundamental-property-of-light-150-years-after-maxwell-43928">quantum Hall effect</a>. Thouless and coworkers found that the “energy surface” for these materials could be described as a doughnut in topological terms, and the channels of energy that were seen were effectively the number of holes in that surface. Along with further work by Kosterlitz and Haldane on other systems, like vortices superconductors and hidden ordering in magnetic materials, their work demonstrated that the idea of topology could be used to predict the behaviour of solids.</p>
<h2>Great promise</h2>
<p>Thouless, Kosterlitz and Haldane’s work has laid the foundations for new emerging fields. In particular they have been crucial to an area of solid state physics called topological insulator materials. These are new three-dimensional materials that carry electricity on the surface but not in their interior. Their energy surface can also be described by topology. These materials have many “spintronic applications”, and heads of hard drives based on this technology are currently used in industry. </p>
<p>Technological applications of materials often rely on how they behave when they are “excited” as a result of some energy transfer. We can imagine an excitation as being a bit like a pulse travelling down a string if we shake it at one end. </p>
<p>One device that is <a href="http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.82.3045">currently being studied</a> is made of topological insulator layered on top of a superconductor (a material with zero electrical resistance at low temperatures). If we poke this system in the right way then it is excited at the interface between the materials. These excitations carry a topological property, like a hole in a doughnut, which is robust to noise and imperfections that might scatter the excitation (which could be some sort of signal).</p>
<p>This effect is potentially very useful for <a href="https://theconversation.com/explainer-quantum-computation-and-communication-technology-7892">quantum computing</a>. The “bits” of data in a normal computer are 1 or 0. However a quantum computer uses quantum bits, which can be in superpositions of states (according to quantum mechanics) – making calculations super fast. Currently scaling quantum computing up to commercially applicable sizes is hampered by noise from the external environment, such as something shaking. However, by exploiting excitations of topological materials, the information encoded in them could be protected and preserved. </p>
<p>This is an exciting avenue of research that could help revolutionise information processing technologies.</p><img src="https://counter.theconversation.com/content/66517/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Stephen Clark does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Don’t worry, understanding the work is a piece of cake. But it may make you hungry.Stephen Clark, Lecturer in Physics, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/395542015-05-18T10:17:24Z2015-05-18T10:17:24ZTopology looks for the patterns inside big data<figure><img src="https://images.theconversation.com/files/81866/original/image-20150515-25428-1mvo0fk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">What good is all this data if we can't figure out how to analyze it?</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/alpha_auer/5068482201">Elif Ayiter</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Big data gets much attention from <a href="http://bits.blogs.nytimes.com/category/data/?_r=0">media</a>, <a href="http://www.forbes.com/sites/louiscolumbus/2014/10/19/84-of-enterprises-see-big-data-analytics-changing-their-industries-competitive-landscapes-in-the-next-year/">industry</a> and <a href="http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=504767">government</a>. Companies and labs generate massive amounts of data associated with everything from weather to cell phone usage to medical records, and each data set may involve hundreds of variables.</p>
<p>These sets are so large and complex that traditional methods of looking for patterns within them can’t make much headway. Often touted as a silver bullet, data analytics certainly have the potential to make inroads into once intractable problems. But we have to be able to figure out what we’re looking at.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=361&fit=crop&dpr=1 600w, https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=361&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=361&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=454&fit=crop&dpr=1 754w, https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=454&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/80084/original/image-20150501-23893-p9bwcb.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=454&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A regression line can show the relationship between height and weight in a group of people.</span>
<span class="attribution"><a class="source" href="http://pgfplots.net/tikz/examples/regression-line/">Jake</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Statistics 101 invariably contains a lecture or two about linear regression – finding the best line that fits a set of points scattered in a plane. These graphs often show up in articles about climate change, for example, where temperature and other weather data are plotted against time, or in economic forecasts where employment or GDP history is used to extrapolate into the future.</p>
<p>But what if the set of points doesn’t lie near a line but instead forms something like a circle? </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=235&fit=crop&dpr=1 600w, https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=235&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=235&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=296&fit=crop&dpr=1 754w, https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=296&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/81900/original/image-20150515-25400-qf3bf1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=296&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A collection of points on the circle (left), and the best-fitting line (right).</span>
<span class="attribution"><span class="source">Kevin Knudson</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>Clearly regression isn’t useful in this context, but we know that only because we can <em>see</em> that the points form a circle.</p>
<p>Now imagine instead a collection of points lying on a circle in a higher-dimensional space. In three dimensions we might be able to see the circle, but if we have more variables, as often happens when examining large data sets, we are in trouble. How could we detect the circle? Better: how could we tell a computer to find the circle?</p>
<p>These are the types of questions arising from the growth of big data – and algebraic topology provides some answers.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/81856/original/image-20150515-25415-19m7w9v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Spheres, cubes, look the same to me.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/phil_shirley/5548073664">Phil Shirley</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<h2>How to make sense of big data spatially</h2>
<p>Topology is sometimes called “rubber sheet geometry.” To a topologist, a sphere and a cube are the same thing. Imagine a cube made from flexible material; inserting a straw and blowing into the cube would puff it out into a sphere. Operations like this are called <em>deformations</em>, and two objects are considered to be the same if one can be deformed to the other.</p>
<p>Topologists study spaces by assigning algebraic objects called <em>invariants</em> to them. They may be as simple as an integer, but they are often more complicated algebraic structures. For data analysis, the invariant of choice is <em>persistent homology</em>. </p>
<p>Ordinary homology measures the number of “holes” that cannot be filled in a space. Let’s think about a sphere again. If we draw a loop on the sphere, it bounds a 2-dimensional disc on the surface; that is, we can fill in any loop on the sphere and so there are no 2-dimensional “holes.” By contrast, the surface of the sphere itself bounds a 3-dimensional “hole” that cannot be filled. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=585&fit=crop&dpr=1 600w, https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=585&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=585&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=735&fit=crop&dpr=1 754w, https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=735&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/81901/original/image-20150515-25415-al1ghj.JPG?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=735&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A closed loop on the surface of a sphere; it bounds a disc and therefore does not add to the first Betti number.</span>
<span class="attribution"><a class="source" href="http://inperc.com/wiki/index.php?title=File:Sphere-with-cycle.JPG#filelinks">Peter Saveliev</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>The Betti numbers of a space count the number of such unfillable holes of each dimension. A sphere has second Betti number equal to 1 (because its interior cannot be filled) and first Betti number equal to 0 (because any loop bounds a disc on the sphere). The zero-th Betti number counts the number of pieces a space has; in the case of the sphere we have one piece. There are higher-dimensional versions of this as well for more complicated spaces.</p>
<p>The problem with using ordinary homology for data analysis is that if we compute the homology of a discrete set of data points, we will be disappointed. There are no holes, only a collection of disconnected points. The zero-th Betti number will count how many points there are, but as there are no loops or spheres in such a set the higher Betti numbers will all be 0. This is where persistent homology enters the story. </p>
<p>We need to take our discrete set of points and join them together. Imagine putting a small ball of radius <em>r</em> around each point in our data set. If <em>r</em> is very small, then none of the balls will intersect and the Betti numbers of all the balls in the set are the same as for the discrete set.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=631&fit=crop&dpr=1 600w, https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=631&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=631&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=793&fit=crop&dpr=1 754w, https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=793&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/77118/original/image-20150406-26473-129q4r1.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=793&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Balls of increasing radius around data points.</span>
<span class="attribution"><span class="source">Kevin Knudson</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>However, if we allow <em>r</em> to grow, then eventually the balls will begin to touch and we will likely get nontrivial higher Betti numbers. In the animation, we see that once <em>r</em> reaches a certain threshold, the balls around the top three points intersect in pairs and therefore contain the triangle joining the three points. Moreover, we cannot fill in the triangle since there’s a small gap in the middle; this means the first Betti number is 1 at that stage. But as <em>r</em> gets a bit bigger, then all three balls intersect at once and we can fill in the triangle; the first Betti number then drops to 0.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=534&fit=crop&dpr=1 600w, https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=534&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=534&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=671&fit=crop&dpr=1 754w, https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=671&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/77124/original/image-20150406-26502-idhbru.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=671&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Barcode associated to the above data. The 0-th Betti number at top decreases from 4 to 0. The first Betti number at bottom shows the emergence of two short-lived 1-dimensional homology classes.</span>
<span class="attribution"><span class="source">Kevin Knudson</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>Persistent homology tracks these numbers as the radius grows; the plot of these numbers against the parameter <em>r</em> is called a <em>barcode</em>. Long bars suggest features in the data that may be significant (they <em>persist</em>, hence the terminology). Short bars often arise from noise in the data and may be disregarded (or not – context is important).</p>
<p>So what we’ve done is pass from a discrete collection of points to a sequence of more complicated spaces (one for each <em>r</em>) which hopefully model the data much better than a simple linear regression might.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/81287/original/image-20150511-19560-gacn18.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A circle persists in the spaces as the radius of the balls increases.</span>
<span class="attribution"><span class="source">Kevin Knudson</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>In the above animation, we show how a few points on a circle might be modeled in this way. We have suppressed the balls around the points, connecting two points when their associated balls overlap, forming triangles when three intersect and so on. A circle persists for quite a long time, leading us to guess that our data lie near a circle.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=412&fit=crop&dpr=1 600w, https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=412&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=412&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=517&fit=crop&dpr=1 754w, https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=517&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/81864/original/image-20150515-25428-9zoj61.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=517&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Topological data analysis led to a new way to compress digital photos.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/pforret/280780627">Peter Forret</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span>
</figcaption>
</figure>
<h2>Applications beyond the theory</h2>
<p><a href="http://math.stanford.edu/%7Egunnar/">Gunnar Carlsson</a> of Stanford University is one of the pioneers of topological data analysis. One of his group’s first successes was the discovery of the topology of the space of natural images. This data set consists of several million 3-pixel by 3-pixel patches sampled from black and white digital photographs. Each pixel is described by a number between 0 and 255, measuring its grayscale value; each 3-by-3 patch then corresponds to a point in a 9-dimensional space, each coordinate giving the numerical value of the associated pixel. After tossing out the constant patches and doing some normalizations, this space lies inside a 7-dimensional sphere. At first glance, the set appears to fill out the sphere, but structure emerges by restricting attention to areas where the points pack closer together.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=1152&fit=crop&dpr=1 600w, https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=1152&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=1152&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1448&fit=crop&dpr=1 754w, https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1448&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/81862/original/image-20150515-25428-xk0r8v.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1448&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A Klein bottle is like a Mobius strip: it has no boundary.</span>
<span class="attribution"><a class="source" href="http://commons.wikimedia.org/wiki/File:Klein_bottle.svg">Tttrung</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Carlsson and his collaborators showed that the data actually lie on a <a href="http://en.wikipedia.org/wiki/Klein_bottle">Klein bottle</a>, a nonorientable 2-dimensional surface embedded in the sphere. They were able to push this further to find a compression algorithm for photos that is slightly better than the industry standard <a href="http://www.jpeg.org/jpeg2000/">JPEG 2000</a>. Carlsson has published an excellent <a href="http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/">survey</a> of this work. </p>
<p>In light of this success, Carlsson and some of his colleagues founded <a href="http://www.ayasdi.com/">AYASDI</a>, a company with a growing roster of clients in banking, finance, government and other industries. They use these and other techniques to analyze <a href="http://www.ayasdi.com/wp-content/uploads/2015/02/JoA_Ayasdi.pdf">diabetes</a>, <a href="http://www.ayasdi.com/wp-content/uploads/2015/02/Topology_Based_Data_Analysis_Identifies_a_Subgroup_of_Breast_Cancer_with_a_unique_mutational_profile_and_excellent_survival.pdf">breast cancer</a> and <a href="http://www.ayasdi.com/wp-content/uploads/2015/01/publication_Using_TDA_for_Diagnosis_Pulmonary_Embolism.pdf">cardiopulmonary disease</a> data. The results are encouraging – certain subgroups of patients with high survival rates, invisible using traditional statistical methods, may be found via these techniques. </p>
<p>The real promise of these methods, however, lies in the possibility of tailoring treatments and solutions to individuals. Analysis of large data sets lets us know, for example, that a drug once thought to be 80% effective is actually 100% effective on 80% of patients, identifiable via some marker. Topological data analysis provides another tool to advance these analytics, often identifying features that were hidden before.</p><img src="https://counter.theconversation.com/content/39554/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Kevin Knudson does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Collect all the data you want, but if you can’t figure out what you’re looking at, it’s useless. Topologists look for spatial relationships to figure out what the data can tell us.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.