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enAbel Prize 2018: the power of asking good questions
https://plus.maths.org/content/abel-prize-2018-power-asking-good-questions
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_2.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In 1967 a 30-year-old American mathematician, still in the early stages of his career, wrote a <a href="http://publications.ias.edu/rpl/paper/43">17 page letter</a> to the eminent French mathematician André Weil. The covering note that he sent with the letter said:
</p><p>
<em>"After I wrote [this letter] I realized there was hardly a statement in it of which I was certain. If you are willing to read it as pure speculation I would appreciate that; if not – I am sure you have a waste basket handy."</em>
</p>
<div class="rightimage" style="max-width:350px"><img src="/content/sites/plus.maths.org/files/news/2018/Math_Robert_Langlands_20160930_DKomoda-0643_web.jpg?s20948d1521730004" alt="Robert Langlands"><p>Robert Langlands (Image <A href="https://www.ias.edu/news/press-releases/2018/abel">Dan Komoda/Institute for Advanced Study</a>)</p></div>
<p>
The letter may have contained many unproved statements, but these turned out to be asking questions that would create a whole area of mathematical research, unmatched in modern mathematics for its scope, deep results and sheer size in terms of the number of mathematicians it has enticed. The author of the letter – <a href="https://www.math.ias.edu/people/faculty/rpl">Robert Langlands</a> – has been awarded the <a href="http://www.abelprize.no/c73016/seksjon/vis.html?tid=73017">2018 Abel Prize</a> for his "visionary program" that bridges previously unconnected areas of mathematics, and is now frequently described as a "grand unified theory of mathematics".
</p><p>
The insights contained in Langlands' letter to Weil, and more fully in his 1970 lecture <a href="http://publications.ias.edu/rpl/paper/47"><em>Problems in the Theory of Automorphic Forms</em></a>, set out a programme of research that is now referred to as the <em>Langlands program</em>. Langlands made a number of conjectures about the connections between two previously unrelated areas: <em>number theory</em> and <em>harmonic analysis</em>.
</p>
<h3>From waves to numbers</h3>
<div class="leftimage" style="max-width:350px">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/tuningfork.png"/ alt="Soundwave of a tuning fork">
<img src="/content/sites/plus.maths.org/files/articles/2017/carola/speech.png" alt="Soundwave of human speech"/>
<p style="max-width:600px">The sound wave from a tuning fork (top), compared with that of human speech (bottom).</p></div>
<!-- tuning fork from http://www.arborsci.com/wordpress/wp-content/uploads/2013/08/3.png - permission granted by email to RT 27/2/2017 -->
<!-- speech is screenshot of plus interview recording in audacity, created by RT for Plus -->
<p>
Harmonic analysis explores how functions and signals can be represented as a sum of waves. For example, the sound wave of the middle A on a tuning fork is a perfect example of a <em>sine wave</em>, written mathematically as <em>sin(x)</em>. The sound wave of speech is more complicated. But any sound wave, indeed any repeating function, can be broken up into a number of sine waves of various frequencies and intensities, thanks to work that started with Joseph Fourier, in the eighteenth century. (You can read more about this <a href="/content/fourier-transforms-images">here</a>.) <em>Automorphic forms</em> are a more generalised version of periodic waves, like the familiar sine wave, which operate in more complicated geometric settings.
</p>
<p>
A large area of number theory – the study of numbers and their properties – is concerned with certain types of solutions to equations. For example, you might remember the quadratic formula from school which gives a way to solve any quadratic equation: the equation <table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/8853921b66e76033963bedce5264de77/images/img-0001.png" alt="\[ ax^2+bx+c=0 \]" style="width:120px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> (where <img src="/MI/8853921b66e76033963bedce5264de77/images/img-0002.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />, <img src="/MI/8853921b66e76033963bedce5264de77/images/img-0003.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" /> and <img src="/MI/8853921b66e76033963bedce5264de77/images/img-0004.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> are real numbers) has the solutions </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/8853921b66e76033963bedce5264de77/images/img-0005.png" alt="\[ x_1 = \frac{-b+\sqrt {b^2-4ac}}{2a} \]" style="width:155px;
height:37px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> and </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/8853921b66e76033963bedce5264de77/images/img-0006.png" alt="\[ x_2 = \frac{-b-\sqrt {b^2-4ac}}{2a}. \]" style="width:160px;
height:37px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table></p>
<p>
Évariste Galois, in the nineteenth century, began the mathematical study of symmetry by looking at the symmetries that are generated by solutions to equations. For example, the quadratic equation <img src="/MI/717388be194f3464b8acf3261588f0a1/images/img-0001.png" alt="$x^2-2=0$" style="vertical-align:0px;
width:75px;
height:14px" class="math gen" /> has two different solutions <img src="/MI/717388be194f3464b8acf3261588f0a1/images/img-0002.png" alt="$\sqrt 2$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> and <img src="/MI/717388be194f3464b8acf3261588f0a1/images/img-0003.png" alt="$-\sqrt 2$" style="vertical-align:-2px;
width:34px;
height:17px" class="math gen" />, but you can see there is a kind of mirror symmetry between them. The <em>Galois group</em> of an equation is the group of symmetries that arise in a mathematical object that is generated from the solutions of the equation. (You can read more <a href="/content/stubborn-equations">here</a>.)
</p><p>
Langlands proposed a bridge between harmonic analysis and number theory built on the relationship between Galois groups and automorphic forms. A particularly famous example of this correspondence comes from Andrew Wiles' proof of Fermat's last theorem, <a href="/content/very-old-problem-turns-20">announced in 1997</a>. Wiles actually proved a more general result known as the <A href="https://en.wikipedia.org/wiki/Modularity_theorem">Taniyama-Shimura-Weil conjecture</a>, that related the number of solutions of a particular type of equation, called <a href="/content/very-old-question-very-latest-maths-fields-medal-lecture-manjul-bhargava">elliptic curves</a>, to a type of automorphic form called <em>modular forms</em>, mathematical objects that are symmetric in an infinite number of ways. (You can read more <a href="/content/fermats-last-theorem">here</a>.)
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2014/bhargava/533px-ellipticcurvecatalog.jpg" width="533" height="515" alt="different elliptic curves"/><p style="width:530px">Elliptic curves are the equations that can be written as <em>y<sup>2</sup>=x<sup>3</sup>+ax+b</em>. The real solutions of these are shown in this figure for different values of <em>a</em> and <em>b</em>. </p></div>
<p>
This year's Abel Prize recognises Langlands' insight and ability to pose such fruitful questions. Since that letter in 1967, hundreds of mathematicians have been inspired by Langlands program. There has been success in proving some aspects of the program, including deep results by <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Drinfeld.html">Vladimir Drinfeld</a>, <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Lafforgue.html">Laurent Lafforgue</a> and <a href="/content/little-lemma-could">Ngô Bào Châu</a> – whose work was recognised by <a href="/content/tags/fields-medal">Fields medals</a>. But core parts of the programme remain unsolved and are still the subject of intense mathematical research. By asking the right questions and providing a map to answer them, Langlands has inspired some of the most fundamental mathematics of the last 50 years.
</p>
<p><em>You can find out more about the Abel Prize and Langlands work at the <a href="http://www.abelprize.no/c73016/seksjon/vis.html?tid=73017&strukt_tid=73016">Abel Prize website</a> and the <A href="https://www.ias.edu/idea-tags/robert-langlands">Institute for Advanced Study</a>.</em></p></div></div></div>Thu, 22 Mar 2018 14:14:43 +0000Rachel7017 at https://plus.maths.org/contenthttps://plus.maths.org/content/abel-prize-2018-power-asking-good-questions#commentsCrystal clear
https://plus.maths.org/content/going-liquid
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/crystal_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Chris Budd</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article is part of our <a
href="/content/material-maths">package</a> on the maths of materials,
which is based on a talk in Chris Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. You can see a video of the talk <a
href="/content/lets-rock#video">below</a>.</em></p>
<h3>Let there be light: Photonic crystals</h3>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/photonic2.jpg" alt="Photonic crystal" width="350" height="350" /><p>A cross section of a photonic crystal.</div>
<!-- image in public domain https://commons.wikimedia.org/wiki/File:Photonic-crystal-fiber-from-NRL.jpg -->
<p>Crystals are fascinating for their beauty and regular structures, but they are also useful in communication. Early in the twentieth century it was discovered that crystals could be used in radios, both to detect signals imposed on radio waves (in the celebrated <a href="https://en.wikipedia.org/wiki/Crystal_radio">crystal radio sets</a> which I used to build) and then to fix the frequency of the same waves. The same technology is used in digital watches to give a highly accurate time piece. </p>
<p>More recently crystals, as well as glass, have been used in fibre optic technology. If you have broadband, the broadband signal is likely to be transmitted along a fibre-optic cable, at least for some of its path. It is a simple fact that light going down such a cable can not only convey much more information than going through a wire, but can also be transmitted without a lot of loss of signal. This observation led the designers of fibre-optic cables to think about whether it is possible to reduce the attenuation (that is, the loss) of the signal still further. This would allow very long distance communication without the need for any units to boost the power. It's especially important for the transmission of signals under the ocean. </p>
<p>
The question has been answered with the use of some very fancy mathematics relating to so-called <em>photonic crystals</em>. An example of such a crystal is shown above. As you can see it is made up of a fibre optic cable with a regular lattice of holes drilled into it. Photonic crystals aren't easily found in nature. Instead, they are a so-called <em>meta material</em>, designed to have specific properties.</p>
<p>To study such crystals we set up the (partial differential) equations for laser light moving through glass.
There are a number of such equations, including the <em>Helmholtz equation</em> and the <em>nonlinear Schrödinger equation</em>, for example. The precise equation used depends upon the intensity of the laser light beam, and also the type of crystal used. The Helmholtz equation is the simplest of these: if <img src="/MI/f7fe946db354755e9dcf7f2889272d02/images/img-0001.png" alt="$u$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is the intensity of the laser light, then it satisfies the partial differential equation</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/9e0de02311a3faea52df87ae367cfce5/images/img-0001.png" alt="\[ \frac{\partial ^2 u}{\partial x^2}+\frac{\partial ^2 u}{\partial y^2}+\frac{\partial ^2 u}{\partial z^2}+ \omega ^2 u(x,y,z)=0, \]" style="width:267px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>where <img src="/MI/9e0de02311a3faea52df87ae367cfce5/images/img-0002.png" alt="$\omega $" style="vertical-align:0px;
width:10px;
height:7px" class="math gen" /> is the frequency of the light wave and <img src="/MI/9e0de02311a3faea52df87ae367cfce5/images/img-0003.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />, <img src="/MI/9e0de02311a3faea52df87ae367cfce5/images/img-0004.png" alt="$y$" style="vertical-align:-3px;
width:8px;
height:10px" class="math gen" /> and <img src="/MI/9e0de02311a3faea52df87ae367cfce5/images/img-0005.png" alt="$z$" style="vertical-align:0px;
width:8px;
height:7px" class="math gen" /> are the three spatial coordinates. </p><p>To solve this equation for <img src="/MI/9e0de02311a3faea52df87ae367cfce5/images/img-0006.png" alt="$u$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> you need to stipulate boundary conditions, and these encode the geometry of the photonic crystal. The solution then shows that at certain frequencies the holes in the crystal act as miniature <em>resonant cavities</em> whose walls reflect the light back and forth. These resonant cavities effectively stop the light beam going through the middle of the crystal from leaking out through the sides. </p>
<p>The effect is an effectively loss free cable, provided that it can be manufactured in the first place. Fortunately this is now possible. The picture above shows an example of just such a photonic crystal. As you can see it is a very complex shape. Until fairly recently it was thought impossible to analyse the behaviour of light in such a crystal without the use of huge, and expensive, computing power. However advances in manufacturing have gone hand in
hand with advances in mathematics. Indeed it is because of the development of new mathematical techniques that we are not only able to analyses these complex shapes, but also to predict their optical performance with some precision. </p>
<p>The new mathematical techniques are called <em>multi-scale homogenisation methods</em> and they work by recognising that in a material such as a photonic crystal several different length scales are important. First, there is a <em>micro scale</em> given by the wavelength of light. Then there is an intermediate <em>meso scale</em> given by the size of the holes of the crystal. And finally, there is a <em>macro scale</em> given by the width of the whole crystal. (Many other materials have several scales, two natural examples are wood, bone and cells). The material behaves differently at these different scales, and its properties result from the way in which these different behaviours interact. Multi-scale mathematical methods allow us to understand this behaviour and thus predict the behaviour of photonic crystals without the need of a super computer. Multi-scale methods now have many more applications, ranging from weather forecasting to structural optimisation.</p>
<h3>Going liquid</h3>
<p>At school we are told that materials are either liquids, solids or gases. This is not strictly true. Depending on the arrangement of their molecules materials can be in an intermediate state between being a solid and a liquid. An important example of such an intermediate state are <em>liquid crystals</em>. These materials play a vital role in modern technology. </p>
<div class="leftimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/nematic.jpg" alt="Nematic phase" width="200" height="313" /><p>The arrangements of molecules for a crystal in the <em>nematic phase</em>, which is between solid and liquid. Image <a href="https://commons.wikimedia.org/wiki/File:LiquidCrystal-MesogenOrder-Nematic.jpg">Kebes</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>.</p></div>
<p>The key feature of a liquid crystal is that its molecules are long and thin, so they come with a direction. Such materials are called <em>anisotropic</em> and are very different from <em>isotropic liquids</em> which have no preferred direction. The properties of anisotropic crystals can all be determined by mathematical arguments. In the figure on the left you can see the arrangements of molecules for a crystal in the <em>nematic phase</em>, which is between solid and liquid. </p>
<p> What makes nematic liquid crystals useful in technology is that the direction of the molecules can be changed by an electrical or a magnetic field, and this change can alter the crystal's conductivity, viscosity, compressibility and, crucially, its <em><a href="https://en.wikipedia.org/wiki/Refractive_index">refractive index</a></em>, which describes how light propagates through the crystal. Their response to light gives liquid crystals many applications in liquid crystal displays (LCDs) which can be found in computer screens and nearly every other type of display medium. (Possibly the first example of LEDs were the screens used in early digital watches.) Nematic liquid crystals are also used in optical switches, sensors and elastomers (a visco-elastic polymer). </p><p>Mathematics is hugely important in the design of LCD devices. In particular in the recent development of <em>bi-stable displays</em> which require no energy to display information, only to change it. The mathematical approach to such a design is to find a formula describing the energy of the liquid crystal state and then proving rigorously that this energetic state has two stable configurations. This takes us back into the realm of partial differential equations, which we will also meet in the next section.</p>
<h3>Crystals and separating phases</h3>
<!--
<div class="leftimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/poly.png" alt="Photonic crystal" width="250" height="257" /><p>A polycrystalline structure. Image: <a href="https://commons.wikimedia.org/wiki/User:Cdang">Cdang</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>.</div>
<p>Most engineered materials, including metals and ceramics, are in fact <em>polycrystalline</em>. An example of a polycrystal is illustrated on the left. Polycrystalline materials are composed of many small crystalline parts (called grains) which are separated by boundaries. They can behave in interesting ways. If, for example, a piece of metal is heated in a furnace, the small grains will disappear, while the big ones will grow. This is due to changes in the boundaries between individual grains. Exactly the same mathematics can be used to describe this process as describes the evolution of froth on a pint of beer, which we looked at in a previous article. This raises the important question of whether we can design new materials by manipulating [?] the shape of the polycrystals.</p>
-->
<p>Crystals themselves have a strong link to mathematics: their shape, their symmetries, their size and their strength can be understood using mathematical methods. A recent example of the relevant of maths to crystals is the celebrated <em>Bucky ball</em>; a form of carbon, in which 60 carbon atoms are arranged geometrically in the shape of a soccer ball made of twenty hexagons and twelve pentagons. It is shown below.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/Bucky.png" alt="A Bucky ball." width="230" height="240" /><p style="max-width: 230px;">A Bucky ball.</div>
<!-- image from Wikipedia in public domain -->
<p>Mathematicians call this shape a <em>truncated icosahedron</em>. Crystals take such symmetric shapes, not because these shapes are particularly beautiful (although of course they are), but because they represent so-called <em>minimal energy states</em> for the molecules. Any material, when left alone, will naturally assume such a state, which out of all states the material could be in requires the minimal amount of energy to maintain. Minimal energy states make the crystals both strong and stable. </p>
<p>Understanding energy turns out to be the key to understanding many different types of material, including the liquid crystals that we looked at in the previous section. To do this mathematically you need to find a formula for the energy of the crystal and then find the configurations of the material's molecules that minimise this energy. This simple idea is the key behind unlocking the secrets of many materials and allows us to predict many varied and exotic shapes of those materials. </p>
<div class="rightimage" style="max-width: 246px;"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/colours.jpg" alt="Photonic crystal" width="246" height="246" /><p>Computer simulation of the two phases in a two-phase material. Image courtesy <a href="https://www.ctcms.nist.gov/fipy/">National Institute of Standards and Technology</a>.</div>
<p>One example of this is a <em>two-phase material</em>, consisting of distinct parts with different chemical or physical properties, such as a cooling metal that is solid is some areas and liquid in others. Such a metal can form exotic patterns like the one illustrated on the right, in which the distinct parts of the material are represented by different colours. It is possible to describe, understand, predict, classify and control these patterns.</p>
<p> One model for pattern formation in a two-phase material is the celebrated <em>Cahn-Hilliard equation</em>:</p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0001.png" alt="\[ \frac{\partial c}{\partial t} = D\nabla ^2 \left(c^3-c-\gamma \nabla ^2 c\right). \]" style="width:202px;
height:35px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> In this partial differential equation <img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0002.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> takes the values of <img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0003.png" alt="$+1$" style="vertical-align:-1px;
width:19px;
height:13px" class="math gen" /> in one of the phases and <img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0004.png" alt="$-1$" style="vertical-align:0px;
width:19px;
height:12px" class="math gen" /> in the other. Here <img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0005.png" alt="$D$" style="vertical-align:0px;
width:14px;
height:11px" class="math gen" /> is a constant that measures how the liquid diffuses, <img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0006.png" alt="$\gamma $" style="vertical-align:-3px;
width:9px;
height:10px" class="math gen" /> is a constant related to the energy of solidification, and <img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0007.png" alt="$\nabla ^2$" style="vertical-align:-1px;
width:19px;
height:15px" class="math gen" /> stands for </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/58b6b375f01be8620757b007aa356423/images/img-0008.png" alt="\[ \nabla ^2=\frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2}. \]" style="width:127px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>The expression describes how the phases evolve in time and in space to give the shapes above. The Cahn–Hilliard equation has applications in polymer science, complex fluids and many other industrial applications and is a major tool in allowing mathematicians to predict the behaviour and form of complex materials which are binary mixtures of two other materials. </p>
<hr/>
<h3>About this article</h3>
<p>This article is based on a talk in Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. A video of the talk is below.</p>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2015/Mornington/chris.jpg" alt="Chris Budd" width="250" height="153" />
<p>Chris Budd.</p>
</div>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
<a name="video"></a>
<iframe width="560" height="315" src="https://www.youtube.com/embed/yDkkJFO0zfI?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe></div></div></div>Fri, 16 Mar 2018 17:11:34 +0000Marianne7003 at https://plus.maths.org/contenthttps://plus.maths.org/content/going-liquid#commentsMaterial maths
https://plus.maths.org/content/material-maths
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/rock_icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div style="float:right"><iframe width="400" height="225" src="https://www.youtube.com/embed/yDkkJFO0zfI?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
</div>
<p>Materials dominate our lives. The clothes that we wear, the tools that we use, the cars that we drive, the aeroplanes that we fly in and the houses that we live in are all made up of materials. Modern electronics would not be possible without the development of new materials that don't occur in nature. Even our history has been defined in terms of the materials that we use, from the stone age, to the bronze age, to the iron age. </p>
<p>In this series of articles Chris Budd looks at the maths of materials, in particular <em>meta materials</em> which can be designed to have the properties we need for whatever context they will be needed in. The articles are based on Chis Budd's <a href="https://www.gresham.ac.uk/lectures-and-events/mathematical-materials">Gresham College lecture</a>. You can see a video of the lecture on the right and find out more about this ongoing lecture series <a href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">here</a>.</p>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/rock_icon.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/lets-rock">Let's rock</a> — Rock and its use for tools and buildings has shaped human civilisation. Here's an introduction to the maths of rocks. </p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/crystal_icon.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/going-liquid">Crystal clear</a> — From communication technology to LCD displays: crystals are a hugely important part of modern life. Find out more about them with this article. </p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/musli_icon.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/maths-minute-brazil-nut-effect">Maths in a minute: The brazil nut effect</a> — A quick look at a phenomenon that may have puzzled you at breakfast: why are there never any of those nice nuts left when you come to the end of your muesli box? </p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/wizard_icon.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/invisibility-cloak">Invisibility cloaks</a> — Fully functioning invisibility cloaks are closer to becoming a reality than you might think. Here's a quick look at the maths and science involved in producing them. </p></div>
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<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/electric_icon.png" alt="" width="100" height="100" /> </div>
<p><a href="/content/getting-electrical">Getting electrical</a> — How do you make a stealth aircraft that's invisible to radar? An area of maths called random matrix theory holds the answer.</p></div>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2015/Mornington/chris.jpg" alt="Chris Budd" width="250" height="153" />
<p>Chris Budd.</p>
</div>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org/">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
</div></div></div>Fri, 16 Mar 2018 11:14:42 +0000Marianne7008 at https://plus.maths.org/contenthttps://plus.maths.org/content/material-maths#commentsGetting electrical
https://plus.maths.org/content/getting-electrical
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/electric_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Chris Budd</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article is part of our <a
href="/content/material-maths">package</a> on the maths of materials,
which is based on a talk in Chris Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. You can see a video of the talk <a
href="/content/lets-rock#video">below</a>.</em></p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/material.jpg" alt="Mixture of aluminium oxide and titanium oxide" width="350" height="259" /><p><p>A mixture of aluminium oxide and titanium oxide.</p></div>
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<p>Suppose that you want to construct a building at an airport. If that building reflects radar waves then you have a problem. The radar at the airport, or from the aircraft landing or taking off there, will reflect off the building at all sorts of angles. The result is that the radar system gets confused and the radar displays get cluttered. Therefore it is a good idea to cover buildings with a material that absorbs the radar waves. The same principle applies if you want to make a stealth aircraft invisible to radar (rather like <a href="/content/invisibility-cloak">Harry Potter's invisibility cloak</a>). During the Second World War it turned out that nature had already solved this problem. The famous <a href="https://en.wikipedia.org/wiki/De_Havilland_Mosquito">de Havilland Mosquito</a> was made of wood, and this made it almost invisible to the German radar systems.</p>
<p> Now, the challenge is to make a synthetic material with the same radar absorbing properties. To do this we make use of the new ideas of <em>complexity</em> and of <em>emergent properties</em>. Loosely speaking, a complex material is something which is made up of many different materials. An example is carbon fibre, which is used in many different applications, from aircraft engines to fishing rods. The key feature of such a complex material is that its properties are much more than the sum of the different component properties. Instead they reflect the way that these different properties interact. Or more poetically, the whole is greater than the sum of the parts. Such properties are then said to <em>emerge</em> from the interactions, and may be very different from the original properties of the component materials. The only way to find out what these emergent properties are is to use quite sophisticated mathematics.</p>
<p>The question of designing such a radar absorbing material came to our team at the University of Bath, and it was decided to make a complex meta-material combining the different electrical properties of aluminium oxide and titanium oxide. On the technical side, one of these behaves like a <em>resistor</em>, and the other like a <em>capacitor</em>. Resistors have electrical properties that are independent of the frequency of the electrical waves that pass through them. In contrast, capacitors conduct electricity well at high frequencies, and badly at low frequencies.</p><p> To create the complex material we took a random mixture of these two different materials mixed in careful proportion. You can see a picture of the resulting mixture above. As you can see it is very complex, and hard to analyse, even using the multi-scale methods described in <a href="/content/going-liquid"><em>Crystal clear</em></a>. Instead we use a branch of mathematics called <em>random matrix theory</em> to study them. </p>
<p>The material above can be modelled by a network of resistors and capacitors, see below.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/network.jpg" alt="Invisibility cloak" width="351" height="297" /><p style="max-width: 351px;"><p>A network of resistors and capacitors. </div>
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<p> Now we use a trick often employed in studying complex problems. Instead of trying to exactly reproduce the parts of the material we instead consider them to be randomly placed according to certain statistical rules. We then look at the statistics of the electrical properties of the resulting network, in particular its conductivity. The network of resistors and capacitors itself can be represented by a <em>matrix</em> (an array) with random entries, and its statistical properties found by the theory of such random matrices. This turns out to be a very powerful technique for predicting the properties of the original material and gives excellent results. (We use the same idea of representing complex processes by random variables in many other applications, including studies of the behaviour of the stock market and the movement of crowds of people.) </p><p>Below you can see the result of our efforts. On the right is the conductivity of the material as a function of frequency, and on the left the paths of the current through the network at the frequency indicated by the arrow.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/conductivity.jpg" alt="Invisibility cloak" width="683" height="376" /><p style="max-width: 351px;"><p></div>
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<p>These results agree very well with experiment. Random matrix methods are now finding many other applications to the design of many other complex materials.</p>
<hr/>
<h3>About this article</h3>
<p>This article is based on a talk in Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. A video of the talk is below.</p>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2015/Mornington/chris.jpg" alt="Chris Budd" width="250" height="153" />
<p>Chris Budd.</p>
</div>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
<a name="video"></a>
<iframe width="560" height="315" src="https://www.youtube.com/embed/yDkkJFO0zfI?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe></div></div></div>Fri, 16 Mar 2018 11:13:14 +0000Marianne7007 at https://plus.maths.org/contenthttps://plus.maths.org/content/getting-electrical#commentsInvisibility cloaks
https://plus.maths.org/content/invisibility-cloak
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/wizard_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Chris Budd</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article is part of our <a
href="/content/material-maths">package</a> on the maths of materials,
which is based on a talk in Chris Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. You can see a video of the talk <a
href="/content/lets-rock#video">below</a>.</em></p>
<p>Many of us will have watched <em>Harry Potter films</em>, or maybe are fans of <em>Star Trek</em>. Both of these contain devices for making you invisible. In the case of <em>Harry Potter</em> a special cloak, and for <em>Star Trek</em> the infamous cloaking device. </p>
<p>Once thought just to be the product of the imagination, invisibility cloaks are getting closer to becoming reality due to the modern science of meta-materials. Early invisibility devices used a combination of lenses and mirrors to bend the light around an object to make it appear invisible, as shown below. Making these required a knowledge of geometrical optics, and in particular the mathematics of angles and trigonometry. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/mirrors.jpg" alt="Invisibility cloak" width="549" height="485" /><p style="max-width: 385px;"><p>In this example the incoming light rays (red) are guided by mirrors around the wizard in the middle of the diagram. The wizard would therefore appear invisible to an onlooker seeing the outgoing light rays — all they would see is the area behind the wizard as reflected by the mirrors.</div>
<!-- image from Wikipedia in public domain -->
<p>However these cloaking devices could only work when viewed from one direction, so there has been intensive research into finding materials which can cloak someone when viewed from many directions. Researchers in the US have now invented a digital cloaking device that does work from many different directions, and which, despite some short-comings, suggests such technology may be ready for practical uses in the real world sooner than you might think.</p>
<p>The reason why we can see an object (or person) is that light rays that encounter it are obstructed and reflected by it. The system developed by the US researchers calculates the direction and position of those light rays and then displays them as if they were unobstructed. As a result, the area behind the display is effectively cloaked. As the viewpoint shifts, the image on the display changes accordingly, keeping it aligned with the background.</p>
<p> Invisibility cloaks themselves rely on <em>meta materials</em>, which have been engineered to produce properties that don't occur naturally. Light is electromagnetic radiation, made up of vibrations of electric and magnetic fields. Natural materials usually only affect the electric component, but meta-materials can affect the magnetic component too, expanding the range of interactions that are possible. The meta materials used in attempts to make invisibility cloaks are made up of a lattice with the spacing between elements less than the wavelength of the light, which can then be bent by the material.</p>
<p>A related development has been the design of clothing which is not only flexible to fit the body, but also has the properties of LCD screens (see <a href="/content/going-liquid"><em>Crystal clear</em></a> to find out more). By using these materials we can envisage a future where the colour of a dress or a suit can be instantly changed to meet the needs of an event, and can even display moving writing if a sponsor desires it. Maths meets fashion indeed.</p>
<p><em>You can read more about invisibility cloaks on <a href="/content/now-you-see-it-now-you-dont">Plus</a></em>.</p>
<hr/>
<h3>About this article</h3>
<p>This article is based on a talk in Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. A video of the talk is below.</p>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2015/Mornington/chris.jpg" alt="Chris Budd" width="250" height="153" />
<p>Chris Budd.</p>
</div>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
<a name="video"></a>
<iframe width="560" height="315" src="https://www.youtube.com/embed/yDkkJFO0zfI?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe></div></div></div>Fri, 16 Mar 2018 11:07:52 +0000Marianne7005 at https://plus.maths.org/contenthttps://plus.maths.org/content/invisibility-cloak#commentsLet's rock
https://plus.maths.org/content/lets-rock
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/rock_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Chris Budd</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article is part of our <a href="/content/material-maths">package</a> on the maths of materials, which is based on a talk in Chris Budd's ongoing <a href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham College lecture series</a>. You can see a video of the talk <a href="/content/lets-rock#video">below</a>.</em></p>
<p>Perhaps the earliest materials that our ancestors made use of were wood and stone. Wooden tools don't survive for long, but there are examples of stone tools which are believed to be over one million years old. Rock is itself a very interesting material, and its use for tools and then later for building, has shaped human civilisation. Indeed we might say that rock is our longest lasting material. </p>
<h3>Springy rocks</h3>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/rock.jpg" alt="Rock formation" width="350px" height="234" /><p>Rock is an amazing material that has shaped human civilisation.</p></div>
<!-- Image from fotolia.com -->
<p>When we look at a rock we see something very solid. Indeed this is its main property as a material. However, appearances can be deceiving, and rock has many other properties. Some of these are very obvious to us all. Anyone who has been in an earthquake will have experienced the fact that rock can behave elastically. An elastic medium is one which can be distorted a small amount by a (suitably large) force, with a displacement proportional to the force. Crucially, when the force is removed, an elastic medium returns to its original position.</p>
<p> An example of an elastic medium is a wire or a spring. The relationship between the extension of a spring and the force on it was, splendidly, first discovered by a predecessor of mine as Gresham Professor of Geometry: the amazing polymath <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Hooke.html">Robert Hooke</a> (1635-1704). One feature of an elastic material is that it admits elastic waves of expansion and compression. In a spring we can see this by sending a pulse along its length. In just the same way rocks admit waves and we see these in the shock waves associated with earthquakes.</p>
<p> As well as being destructive these waves can also be very useful. Indeed much of our knowledge of the Earth's interior comes from studying the way in which these waves are reflected, refracted and diffracted through the rock. By solving an <em>inverse problem</em> (this is a problem in which we try to find the cause of a measured effect) we can then find out the properties of the Earth's core.</p><p> From a mathematical point of view this involves solving a wave equation. A very similar idea is used by oil companies to prospect for oil. To do this they detonate a small explosion (usually with compressed air) and measure the strength of the reflected waves. This allows them to detect oil deposits deep under the sea bed.</p>
<p>The above observations show that over short time scales (which to a rock is anything up to about ten thousand years), a rock will behave elastically. However, over longer timescales rock can behave like a different type of material. We are used to seeing rocks as cold, hard and brittle. However, during the course of their existence, rocks can be hot, under extreme pressure and (over sufficiently long periods) can flow almost like a liquid. This sort of behaviour is called <em>viscous</em>. Unlike an elastic material, in which displacement is proportional to force, a viscous material has a velocity, which is proportional to the applied force.</p>
<h3>Folding rocks</h3>
<p> Understanding all of these aspects of a rock as a material allows mathematics to be applied to a fascinating question: how do rock layers buckle under the application of a large force and over long times? The reason this question is important is that under the effects of continental drift, rock layers are subjected to huge pressures from moving plates, and these forces deform the rock layers to produce mountains. Mathematics can thus be used to predict how rocks fold and how mountains form.</p>
<p>As an example, think of an undeformed layer of rock on which a pressure is acting in the horizontal direction, which we measure using a horizontal <img src="/MI/07f3522b1bdca9a2a656657680cd894b/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />-axis. Any deformation will happen in the vertical direction, which we measure using a vertical <img src="/MI/07f3522b1bdca9a2a656657680cd894b/images/img-0002.png" alt="$u$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />-axis. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/horizontal.jpg" alt="Rock layers" width="500px" height="223px" /><p style="max-width: 500px;"></p></div>
<!-- Image created by MF -->
<p>At position <img src="/MI/e36253a6353ab75ff73e586f63543433/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> the deformation <img src="/MI/e36253a6353ab75ff73e586f63543433/images/img-0002.png" alt="$u$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> that will occur as a result of a pressure <img src="/MI/e36253a6353ab75ff73e586f63543433/images/img-0003.png" alt="$P$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> is given by the (fourth order differential) equation </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/e36253a6353ab75ff73e586f63543433/images/img-0004.png" alt="\[ EI\frac{d^4u}{dx^4}+P\frac{d^2u}{dx^2}+au+bu^2+cu^3=0. \]" style="width:275px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Here, <img src="/MI/1869338b2abc58fc9b48bebdd35b439f/images/img-0001.png" alt="$E$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> is the <a href="https://en.wikipedia.org/wiki/Elastic_modulus">elastic modulus</a> of the rock, which measures its resistance to being deformed, <img src="/MI/07b73b05aad2e47e84ce5ee630d54297/images/img-0001.png" alt="$I$" style="vertical-align:0px;
width:9px;
height:11px" class="math gen" /> is something called the <a href="https://en.wikipedia.org/wiki/Second_moment_of_area">second moment of area</a>, and <img src="/MI/63b22dce10a5b1499242df07c102f6a5/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" />, <img src="/MI/63b22dce10a5b1499242df07c102f6a5/images/img-0002.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" /> and <img src="/MI/63b22dce10a5b1499242df07c102f6a5/images/img-0003.png" alt="$c$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" /> are constants which depend on the rock in question.</p>
<p>This equation is called the <em>beam equation</em> by structural engineers. It is used to predict how well steel (and other) beams in buildings, bridges and other structures will stand up to buckling forces. Solving it accurately is important in determining safety limits for these structures. The fact the we can also apply it to rocks shows the power and flexibility of applied mathematics. Solving it gives us the shape of the rock layers. An example of a solution is a sine wave, which is the same shape as a wave on a string. A typical sine shaped deformation is illustrated below. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/layers2.jpg" alt="Rock layers" width="450px" height="245px" /><p style="max-width: 450px;"></p></div>
<!-- Image created by MF -->
<h3>Layered rocks</h3>
<p>However, not all rocks deform in this way. Many sedimentary rocks (which are typically ancient sea beds) form in layers of different materials. We see layered materials in other areas, one example being the composite materials now used to build aircraft wings. When a layered material deforms what matters most is the way that the layers interact with each other, so that one deformed layer lies on top of the next one. </p>
<p>Some shapes fit together better than others, and this tells us how the sedimentary layers can deform. Not having any rocks to hand at Bath (nor a convenient continental plate to deform them with) we did a set of experiments with compressing a layered material made up of out of date business cards (a suitable end for these we thought). If you look below, then you can see the results of our efforts. They key thing to see is that we don't get the beautiful sine waves of the text books. Instead we get folds which are straight and have sharp bends. This is a consequence of the geometrical fact that sine waves cannot sit exactly on top of each other:
as illustrated by the image above, two sine waves will always be closer to each other at some points than they are at others. Zig-zag shapes, however, do sit on top of each other. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/paper1.jpg" alt="Paper experiment" width="240px" height="360px" /><p style="max-width: 240px;"></p></div>
<!-- Image provided by author -->
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/paper2.jpg" alt="Paper experiment" width="355px" height="237px" /><p style="max-width: 355px;"></p></div>
<!-- Image provided by author -->
<p>So do rocks deform in the same way was paper? Very much so! Here are some lovely examples of folded rocks in Cornwall, the top one can be seen in Bude and the bottom one in Millook Haven. These are very well worth a visit. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/Bude.jpg" alt="Folded rocks in Bude, Cornwall" width="305px" height="228px" /><p style="max-width: 305px;">Folded rocks in Bude, Cornwall</p></div>
<!-- Image provided by author -->
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/Millock.jpg" alt="Folded rocks in Millook Haven, Cornwall" width="361px" height="218px" /><p style="max-width: 361px;">Folded rocks in Millook Haven, Cornwall</p></div>
<!-- Image provided by author -->
<p>If you look at the paper folding examples you will see that there are occasional gaps, or voids, in the layers. Using the mathematical formulation we can precisely predict where these voids will occur. Why should we care? Voids occur in real rock folding, and it is in the voids that minerals get deposited. One of these minerals is gold. So mathematics can help in a geological treasure hunt. Maybe this might make us all rich! </p>
<hr/>
<h3>About this article</h3>
<p>This article is based on a talk in Budd's ongoing <a href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham College lecture series</a>. A video of the talk is below.</p>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2015/Mornington/chris.jpg" alt="Chris Budd" width="250" height="153" />
<p>Chris Budd.</p>
</div>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org/">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
<a name="video"></a>
<iframe width="560" height="315" src="https://www.youtube.com/embed/yDkkJFO0zfI?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
</div></div></div>Fri, 16 Mar 2018 11:04:17 +0000Marianne7000 at https://plus.maths.org/contenthttps://plus.maths.org/content/lets-rock#commentsMaths in a minute: The brazil nut effect
https://plus.maths.org/content/maths-minute-brazil-nut-effect
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/musli_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Chris Budd</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>This article is part of our <a
href="/content/material-maths">package</a> on the maths of materials,
which is based on a talk in Chris Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. You can see a video of the talk <a
href="/content/lets-rock#video">below</a>.</em></p>
<div class="rightimage" style="max-width: 300px;"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/musli.jpg" alt="Muesli" width="300" height="199" /><p>How's your muesli shaking down?</div>
<!-- image from the article "the serious maths of cereal, socialising and starlings -->
<p>It is time for breakfast and you reach for your packet of cereal, which is a mix of muesli and nuts. As you pour the cereal into your breakfast bowl you find, much to your surprise, that you get a bowl full of nuts. That's is odd: have the large nuts have risen to the top of the muesli? If yes, then this seems to contradict our intuition that larger objects should go to the bottom. </p>
<p>What you have seen (literally) is an example of the <em>brazil nut effect</em>. It occurs because your cereal packet comes to you having travelled a considerable distance. (To learn more about the movement of food around the world have a look at <a href="/content/how-much-maths-can-you-eat">this article</a>.) During its travels it gets shaken up. It is the shaking which is the key to the brazil nut effect. As the cereal is shaken the nuts may move up a little, leaving gaps beneath them. The smaller and faster moving muesli then flows into these gaps. As it does so, so the nuts rise upwards. Eventually they get to the top of the cereal.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2018/materials/brazil.png" alt="The Brazil nut effect" width="640" height="206" /><p style="max-width: 230px;">The Brazil nut effect. Image: <a href="https://commons.wikimedia.org/wiki/File:Brazil_nut_effect.svg">Gsrdzl</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA 3.0</a>.</div>
<!-- image from Wikipedia in public domain -->
<p>The brazil nut effect is just one of the counter intuitive properties shared by <em>granular materials</em>, which as the name suggests are materials made up of grains. Granular materials are very important in the food industry. For example, they make up the grain in grain silos, the are the basis of baby food, nuts, rice, coffee, chocolate, custard powder and many cereals such as cornflakes. Granular materials also occur naturally, with snow in large quantities being a good example, and sand another (find out more <a href="/content/cereal-sand-and-snow">here</a>). Many of the fascinating patterns seen in sand dunes are a consequence of sand being a granular material.</p>
<p> Another context in which granular materials play a role is in the manufacture and storage of paint powders, fertiliser, cement and even sand. Mathematics is used to predict how granular materials can flow. In some sense, granular materials do not constitute a single phase of matter, but have characteristics similar to solids, liquids or gases, depending on the average energy per grain. However, in each of these states granular materials also exhibit properties which are unique to them. Indeed, granular materials also exhibit a wide range of pattern forming behaviours when excited (e.g. vibrated as above, or allowed to flow).</p>
<p> Different behaviours arise depending on whether the grains can move freely over each other, or if they can get jammed. The latter is certainly what you don't want in a grain silo. Mathematics plays a critical role in predicting whether jamming will or will not occur, and by doing this helps to feed the world.</p>
<hr/>
<h3>About this article</h3>
<p>This article is based on a talk in Budd's ongoing <a
href="https://www.gresham.ac.uk/series/mathematics-and-the-making-of-the-modern-and-future-world/">Gresham
College lecture series</a>. A video of the talk is below.</p>
<div class="rightimage" style="max-width: 250px;"><img src="/content/sites/plus.maths.org/files/articles/2015/Mornington/chris.jpg" alt="Chris Budd" width="250" height="153" />
<p>Chris Budd.</p>
</div>
<p>Chris Budd OBE is Professor of Applied Mathematics at the University of Bath, Vice President of the <a href="http://www.ima.org.uk/">Institute of Mathematics and its Applications</a>, Chair of Mathematics for the <a href="http://www.rigb.org">Royal Institution</a> and an honorary fellow of the <a href="http://www.britishscienceassociation.org/">British Science Association</a>. He is particularly interested in applying mathematics to the real world and promoting the public understanding of mathematics.</p><p>
He has co-written the popular mathematics book <em><a href="/content/mathematics-galore">Mathematics Galore!</a></em>, published by Oxford University Press, with C. Sangwin, and features in the book <em><a href="https://global.oup.com/academic/product/50-visions-of-mathematics-9780198701811?cc=gb&lang=en&">50 Visions of Mathematics</a></em> ed. Sam Parc.
</p>
<a name="video"></a>
<iframe width="560" height="315" src="https://www.youtube.com/embed/yDkkJFO0zfI?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe></div></div></div>Fri, 16 Mar 2018 11:02:39 +0000Marianne7004 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-brazil-nut-effect#commentsCelebrating Stephen Hawking
https://plus.maths.org/content/celebrating-stephen-hawking
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/hawking_gravity_icon_0.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>"Can you hear me?"</p>
<div class="rightimage"><img src="/content/sites/plus.maths.org/files/news/2018/Hawking/hawking_gravity_small.jpg" width="350" height="233" alt="Hawking"/><p style="width:350px;">Stephen Hawking experiencing zero gravity. Image courtesy of NASA.</p></div>
<p>This was how <a href="http://www.hawking.org.uk ">Stephen Hawking</a>, the former Lucasian Professor of Mathematics, current Director of Research at the <a href="http://www.ctc.cam.ac.uk/">Centre for Theoretical Cosmology</a>, best selling author and world famous science communicator, started his lectures. I first heard Hawking speak at his 60th Birthday Symposium on 11 January 2002, coincidentally my first day officially working for <em>Plus</em>. I was very sad to learn that Stephen Hawking died this morning at home in Cambridge at the age of 76.</p>
<p>In <a href="http://www.ctc.cam.ac.uk/news/Hawking_an_appreciation-Rees.pdf">his tribute to Hawking</a> astronomer <a href="https://www.ast.cam.ac.uk/people/Martin.J.Rees">Sir Martin Rees</a> said that despite Hawking's increasing frailty in his early 30s, when he couldn't even turn the pages of a book without help, he came up with his "best ever idea — encapsulated in an equation that he said he wanted on his memorial stone." </p>
<p>That equation is </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/216cd963c3b4ee2637c719ecdae76ada/images/img-0001.png" alt="\[ S = \frac{\pi A k c^3}{2hG} \]" style="width:81px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>and is known as the formula for the <em>Bekenstein-Hawking entropy</em>. </p><p> Reading about its discovery, and the work of Hawking and <a href="https://en.wikipedia.org/wiki/Jacob_Bekenstein">Jacob Bekenstein</a>, reminded me of the passion and excitement of science, the value of controversy and proving yourself wrong, and the importance of sharing ideas and building bridges between areas. I felt very sad when I heard the news of Hawking's death first thing this morning. Now, after a morning spent learning and about black hole entropy and Hawking radiation, I feel much more cheerful.
</p>
<div class="leftimage" style="width: 214px;"><img src="/content/sites/plus.maths.org/files/articles/2014/bekenstein/bekenstein.jpeg" alt="Jacob Bekenstein" width="214" height="320" />
<p>Jacob Bekenstein (1947-2015).</p>
</div>
<p>One of the results to come out of the rush that was the "golden age of black hole theory" in the 1960s and 70s was Hawking's <em>area theorem</em>. It states that the <em>horizon</em> of a black hole, the surface area of no return, could never decrease. Whatever happens to the black hole, whatever is gobbled up by it, its surface area can only ever become bigger. (You can read more about black holes <a href="/content/what-black-hole">here</a>.)</p>
<p>Jacob Bekenstein had just finished his PhD at Princeton University and was inspired by Hawking's area theorem. He explored the idea that the surface area of a black hole might be analogous to something called <em>entropy</em>. Entropy is a measure of disorder of a physical system and, according to the <em>second law of thermodynamics</em>, it never decreases. We're familiar with this fact from every day life: no system ever tidies itself up, instead things, when left alone, get messier over time (You can read more about entropy and the second law of thermodynamics <a href="/content/maths-minute-second-law-thermodynamics">here</a>.)</p>
<p>Bekenstein conjectured what has become known as the <em>generalised second law of thermodynamics</em>: that the entropy of a system containing a black hole is equal to the entropy of the system outside the black hole plus a constant times the area of a black hole, and that this value never decreases. This meant that entropy of a black hole is</p>
<p> <table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/4d50189badf640359048b9398fa8f507/images/img-0001.png" alt="\[ S_{bh }= \mbox{constant} \times \frac{kA}{L_ p^2}, \]" style="width:157px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> where <img src="/MI/4d50189badf640359048b9398fa8f507/images/img-0002.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" /> is Boltzmann's constant from thermodynamics, and <table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/4d50189badf640359048b9398fa8f507/images/img-0003.png" alt="\[ L_ p=\sqrt {\frac{hG}{2\pi c^3}}=1.62\times 10^{-35cm} \]" style="width:225px;
height:41px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> is the Planck length. (You can read more about Bekenstein's work <a href="/content/bekenstein">here</a>.)</p>
<p>Hawking disagreed with Bekenstein's ideas when they met at a conference in 1972, believing there were instances of black holes that would disprove Bekenstein's conjecture. He apparently went back to Cambridge intent on proving Bekenstein wrong. It was generally accepted that nothing should be able to escape from a black hole, but when Hawking applied quantum theory to black holes he realised that something could escape. Quantum physics predicts that even in a vacuum pairs of quantum particles pop in and out of existence, and that these particles have an energy. Their presence is usually fleeting and they annihilate each other. But when this happens near the horizon of a black hole one of a pair of virtual particles can be captured by the black hole while one escapes. The escaping quantum particles create what is known as <em>Hawking radiation</em>.</p>
<p>Hawking radiation means that black holes are not black – they constantly emit radiation and glow with a, admittedly very low, temperature. The radiation provided the missing piece of black hole thermodynamics. Hawking verified Bekenstein's conjectured generalised second law, provided a physical explanation of the temperature of black holes (their <em>quantum radiation</em>) and provided the missing constant, 1/4, in the entropy formula: </p>
<table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/cb0b04f1e2e7f258b836fa5c79254f27/images/img-0001.png" alt="\[ S_{bh}=\frac{Ak}{4L_ p^2}. \]" style="width:81px;
height:40px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Or written in full, as Hawking preferred: </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/cb0b04f1e2e7f258b836fa5c79254f27/images/img-0002.png" alt="\[ S = \frac{\pi A k c^3}{2hG}. \]" style="width:86px;
height:36px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table>
<p>Hawking described this process of discovery in his <a href="/content/stephen-hawkings-60-years-nutshell">60th birthday lecture</a>:</p>
<div class="rightimage"><img src="/content/sites/plus.maths.org/files/articles/2012/Hawking70/hawking_1980.jpg" width="250" height="359" alt="Hawking"/><p style="width:205px;">Stephen Hawking in the 1980s</p></div><!-- image is in public domain -->
<p>"...the obvious next step would be to combine general relativity, the theory of the very large, with quantum theory, the theory of the very small. I had no background in quantum theory... So as a warm-up exercise, I considered how particles and fields governed by quantum theory would behave near a black hole.
... To answer this, I studied how quantum fields would scatter off a black hole. I was expecting that part of an incident wave would be absorbed, and the remainder scattered. But to my great surprise, I found there seemed to be emission from the black hole. At first, I thought this must be a mistake in my calculation. But what persuaded me that it was real, was that the emission was exactly what was required to identify the area of the horizon with the entropy of a black hole. I would like this simple formula to be on my tombstone."</p>
<p>In his tribute Rees says that the equation encapsulates the notion of Hawking radiation, the contribution Hawking was most proud of, that brought together previously unlinked areas of gravity (signified by Newton's constant <img src="/MI/d441baf6cc34b2d6125e0f957ef7ee1a/images/img-0001.png" alt="$G$" style="vertical-align:0px;
width:13px;
height:11px" class="math gen" /> and the area of the black hole <img src="/MI/d441baf6cc34b2d6125e0f957ef7ee1a/images/img-0002.png" alt="$A$" style="vertical-align:0px;
width:12px;
height:11px" class="math gen" />), quantum physics (the Planck constant <img src="/MI/d441baf6cc34b2d6125e0f957ef7ee1a/images/img-0003.png" alt="$h$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />) and thermodynamics (Boltzmann's constant <img src="/MI/d441baf6cc34b2d6125e0f957ef7ee1a/images/img-0004.png" alt="$k$" style="vertical-align:0px;
width:8px;
height:11px" class="math gen" />). Rees goes on to say that Hawking radiation had very deep implications for mathematical physics and is still a focus of theoretical interest, a topic of debate and controversy more than 40 years after his discovery.</p>
<p>The equation seems an apt inscription for Hawking's tombstone. Its discovery brought together disconnected fields, combining them into an elegant, surprising, and stimulating result. It came about through collaboration, communication and a healthy bit of competition between scientists. And most importantly, it is a human story of great triumph as Hawking himself said in his 70th birthday lecture: "The fact that we humans, who are ourselves mere collections of fundamental particles of nature, have been able to come this close to an understanding of the laws governing us and our Universe is a great triumph."</p><p>
The answer to Hawking's question at the start of his lectures is yes, we can hear you. And your passion and enthusiasm for communicating with everyone from academic peers to the general public, in lectures, books, cartoons and TV shows, has meant you have had an impact on a great many of us indeed. </p>
</div></div></div>Wed, 14 Mar 2018 14:48:00 +0000Marianne7015 at https://plus.maths.org/contenthttps://plus.maths.org/content/celebrating-stephen-hawking#commentsCelebrating Stephen Hawking
https://plus.maths.org/content/rip-stephen-hawking
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/hawking_gravity_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>This morning the world woke up to the sad news that Stephen Hawking has passed away. One of the greatest, and most famous, scientists of modern times Hawking will be sorely missed by his family and friends, his colleagues, the scientific community as a whole, and not least by us. Here is a collection of material published on <em>Plus</em> over the years by or about Hawking and his work.</p>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/hawking_gravity_icon_0.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/celebrating-stephen-hawking">Celebrating Stephen Hawking</a> — In memory of Stephen Hawking we look at the equation that was one of his greatest contributions to physics and cosmology.
</p>
</div>
<h3>Articles by Stephen Hawking</h3>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/news/2018/Hawking/hawking2006.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/brief-history-mine">A brief history of mine</a> — In this excerpt from Stephen Hawking's address to his 70th birthday symposium Hawking looks back over his life and how our understanding of the Universe has changed over the decades.
</p>
</div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/5/10_jan_2012_-_1211/icon.jpg?1326197483" alt="" width="100" height="100" /> </div>
<p><a href="/content/stephen-hawkings-60-years-nutshell">My 60 years in a nutshell</a> — This is an Stephen Hawking's address to his 60th birthday symposium, in which he looks back over some of his favourite achievements.
</p>
</div>
<h3>Articles relevant to Hawking's work</h3>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/abstractpics/5/13_jan_2012_-_1617/icon.jpg?1326471421" alt="" width="100" height="100" /> </div>
<p><a href="/content/planets-universes-part-i">From planets to universes</a> — In this article Astronomer Royal Martin Rees looks back over the developments in astronomy over the decades spanned by Hawking's and his own career. It comes in two parts, the first is <a href="/content/planets-universes-part-i">here</a> and the second <a href="/content/planets-universes-part-i">here</a>, and is based on Rees's lecture at Hawking's 70th birthday symposium.
</p>
</div>
<div style=" float: right; border: thin solid grey; background: #CCCFF; padding: 0.5em; margin-left: 0.8em; margin-right: 1em; margin-bottom: 0.5em; ">
<div class="leftimage" style="width: 100px;"><img src="/content/sites/plus.maths.org/files/news/2018/Hawking/relativity.jpg" alt="" width="100" height="100" /> </div>
<p><a href="/content/what-general-relativity">What is general relativity?</a> — Hawking's work concerned the Universe at large scales: the scale of planets, stars and galaxies. At these scales the dominant force is gravity, which is described by Einstein's general theory of relativity. Here's a quick introduction.
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<p><a href="/content/what-black-hole">What is a black hole - physically?</a> — Hawking is famous for his work on black holes. In this article and video cosmologist Pau Figueras, who worked with Hawking, explains what black holes are, physically, and how we observe them. Be prepared to be spaghettified.
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<p><a href="/content/what-black-hole">What is a black hole - mathematically?</a> — In this article and video cosmologist Figueras looks at black holes from a mathematical point of view.
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<p><a href="/content/stuff-happens-listening-universe">Listening to the Universe: Gravitational waves</a> — The most important breakthrough in 21st century physics was the detection of gravitational waves, caused by the collisions of massive black holes. According to Hawking it revolutionised astronomy. This collection of articles and videos tells you all you need to know about gravitational waves.
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<p><a href="/content/bekenstein">The limits of information</a> — In the 1970s Stephen Hawking predicted that black holes come with a property called <em>entropy</em>. This article explores what this means and how it has led to the astonishing discovery that there's a limit to how much information you can fit into a region of space.
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</div></div></div>Wed, 14 Mar 2018 10:13:21 +0000Marianne7014 at https://plus.maths.org/contenthttps://plus.maths.org/content/rip-stephen-hawking#commentsThe best of both worlds
https://plus.maths.org/content/both-worlds
<div class="field field-name-field-abs-img field-type-image field-label-hidden"><div class="field-items"><div class="field-item even"><img class="img-responsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/warnick_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-inlinec clearfix field-label-inline"><div class="field-label">By </div><div class="field-items"><div class="field-item even">Marianne Freiberger</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
The mathematician <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Hardy.html">GH Hardy</a> famously prided himself on the fact that pure mathematics is a means to its own end, without any application whatsoever to the real world. "The 'real' math of the 'real' mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly 'useless'," he wrote in his 1940 essay <em><a href="/content/node/6725">A mathematician's apology</a></em>. </p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2018/Warnick/Warnick.jpg" alt="Claude Warnick" width="350" height="233" />
<p>Claude Warnick.</p>
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<p>Indeed, to modern eyes the boundary between pure mathematics on the one hand and applied mathematics on the other can seem like a rigid line. Many mathematicians place themselves firmly into one of the two camps, maths and physics (an important area of applications of maths) are taught as separate subjects in schools and universities, and their textbooks sit in different parts of the library. Are there still mathematicians that do
pure maths and think about applications in equal measure?
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<p>The answer is yes. We recently had the pleasure of meeting Claude Warnick, employed jointly by the pure maths department and the applied maths and theoretical physics department at the University of Cambridge. </p>
<div class="leftshoutout">You can see a video interview with Warnick <a href="/content/getting-pure-hands-dirty#video">below</a></div>
<p>Warnick is interested in a topic that genuinely belongs to both pure maths and physics at the same time. "I work on partial differential equations and general relativity," he explains. "Partial differential equations describe how a physical quantity varies depending on two or more continuous quantities. <a href="/content/maths-minute-einsteins-general-theory-relativity">General relativity</a> is our theory to describe [the force of gravity]."</p>
<p>The link between the two rests on a phenomenon we're all familiar with: waves. "The fact that you can hear me is down to the fact that sound waves are propagating through the air in this room," says Warnick. Waves, including sound waves, can be described by partial differential equations, and it's these wave equations that Warnick is interested in. "One of the most interesting things about [general relativity] is that [its <a href="/content/what-general-relativity">central equation</a>] is very similar in character to the wave equation that describes sound," he explains. The equivalent of sound waves moving through air are <em><a href="/content/stuff-happens-listening-universe">gravitational waves</a></em>. First predicted by Einstein around a 100 years ago, these waves are ripples in the fabric of spacetime that originate from gravitational events, such as the collisions of massive black holes. The <a href="/content/spacetime-does-ripple">detection of gravitational waves in 2015</a> was one of the major breakthroughs in modern physics.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/news/2018/Warnick/relativity.jpg" alt="General relativity" width="350" height="204" />
<p>General relativity asserts that massive objects curve the fabric of spacetime. To formulate it Einstein used geometric notions of curvature developed by Riemann in the 19th century.</p>
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<p>"My research [explores] the kind of equations that link these phenomena: wave equations and also problems to do with gravity and general relativity," says Warnick. "The kind of work I am interested in naturally has aspects that are pure mathematical. [In] the study of partial differential equations one has to do a fair amount of analysis and one wants to prove rigorous theorems where possible. On the other hand [what we] are interested in are questions like how black holes behave or how gravitational radiation propagates through the Universe. These are very physical questions, so it's very natural to have a foot in both camps."</p>
<p>In fact, ideas often flow from one camp to the other. General relativity couldn't exist without the supposedly useless (according to Hardy) <a href="/content/kissing-curve-manifolds-many-dimensions">geometric notions</a> developed by <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Riemann.html">Bernhard Riemann</a> in the 19th century. <a href="/content/symmetry-making-and-symmetry-breaking">Considerations of mathematical symmetry</a> have predicted the existence of fundamental particles of nature, including the famous Higgs boson, and continue to lead the way in theoretical physics. And these are just two examples.</p>
<p>Perhaps less famously, ideas have also flowed in the other direction, especially in recent decades. A beautiful example comes from physicists' approach to explore an object by bombarding it with particles and seeing how they scatter, as happens in particle colliders. The idea inspired mathematicians to explore geometric objects by letting hypothetical particles (or strings from <a href="/content/string-theory-newton-einstein-and-beyond">string theory</a>), described by mathematical formulae, move around on them. The approach eventually helped them answer questions in pure geometry that had been open for a hundred years.</p>
<p>Warnick is starting his joint appointment at a time of spectacular advances in his field. "The most exciting recent development in my field is the direct measurement of gravitational waves. [Their existence] has been conjectured for many decades and only now do we have direct experimental evidence to show that [they do] exist. It's a really exciting time because [gravitational waves] open a window on gravitational physics and a window on the Universe. I think a tremendous amount of interesting data will come out of [gravitational wave] experiments in the near future. I'm really looking forward to where this discovery will go in the next few years."</p>
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</div></div></div>Fri, 09 Mar 2018 10:26:28 +0000Marianne7011 at https://plus.maths.org/contenthttps://plus.maths.org/content/both-worlds#comments