tag:theconversation.com,2011:/us/topics/fields-medals-11893/articlesFields medals – The Conversation2018-08-01T20:19:15Ztag:theconversation.com,2011:article/1008872018-08-01T20:19:15Z2018-08-01T20:19:15ZAn Australian takes top honours in the prestigious Fields Medal in mathematics<figure><img src="https://images.theconversation.com/files/230155/original/file-20180801-136655-a4dm6y.jpg?ixlib=rb-1.1.0&rect=4%2C109%2C2912%2C2072&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Australian mathematician Akshay Venkatesh honoured in this year's Fields Medals.</span> </figcaption></figure><p>Australian mathematician Akshay Venkatesh is one of four winners of the Fields Medal, <a href="https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2018">announced overnight</a> at the International Congress of Mathematics, in Brazil.</p>
<p>The 36-year-old now becomes only the second Australian to win a Fields Medal, described as the Nobel Prize for mathematics. The previous Australian winner was <a href="http://www.math.ucla.edu/%7Etao/">Terence Tao</a> in 2006.</p>
<p>The <a href="https://www.mathunion.org/imu-awards/fields-medal">medal</a> is awarded every four years to researchers under 40 years old. It recognises their outstanding mathematical achievement for existing work and for the promise of future achievement. </p>
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<p>Akshay’s medal is in <a href="https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2018/Venkatesh-Citation.pdf">recognition</a> for “his synthesis of analytic number theory, homogeneous dynamics, topology and representation theory”.</p>
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<h2>Showed early promise</h2>
<p>Akshay Venkatesh is currently a <a href="https://mathematics.stanford.edu/people/name/akshay-venkatesh/">professor of mathematics at Stanford University</a>, in the United States, and will soon be moving to join the faculty at the Institute for Advanced Study at Princeton.</p>
<p>But he started out as a student at the University of Western Australia, studying physics and mathematics.</p>
<p>I was doing my undergraduate studies at the same time and he skipped first year mathematics and jumped straight into my second year classes. He was only 13 at the time, while the the rest of the students were 18 and 19 years old. It was clear that he was a very talented individual with exceptional abilities in mathematics.</p>
<p>He topped all his classes and I remember one assignment in third year where he managed to show that something that we had been asked to prove was, in fact, false. The lecturer had missed out an important condition that was required to make it work, but Akshay was the only one to spot it. </p>
<p>Another example of Akshay’s early interest in mathematics comes from one of his mentors at UWA, my colleague Professor Cheryl Praeger, an Australian Academy of Science Fellow. She said:</p>
<blockquote>
<p>At our first meeting I was speaking with Akshay’s mother Svetha, while Akshay was sitting at a table in my office reading my blackboard which contained fragments from a supervision of one of my PhD students, just completed.</p>
<p>At Akshay’s request I explained what the problem was. He coped with quite a lot of detail and I found that he could easily grasp the essence of the research.</p>
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<p>Initially Praeger was worried about the 16-year-old heading to Princeton in 1998, but he soon settled in and it allowed him to return to his love of number theory.</p>
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<span class="caption">Akshay graduated at an early age from UWA before heading to Princeton to study for a PhD.</span>
<span class="attribution"><span class="source">Australian Academy of Science</span></span>
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<h2>A love of number theory</h2>
<p><a href="https://www.britannica.com/science/number-theory">Number theory</a>, Akshay’s main area of research, is one of the central areas of mathematics. It is notorious for having problems that are very easy to state but which turn out to be extremely difficult to solve.</p>
<p>One of Akshay’s major contributions has been to use ideas from many other areas of mathematics (such as representation theory, ergodic theory and topology) to help solve some of these very difficult problems.</p>
<p>One of the fundamental questions of number theory is to understand the distribution of the prime numbers – a <a href="https://www.britannica.com/science/prime-number">prime number</a> is one that is divisible only by 1 and itself, such as 2, 3, 5, 7, 11, 13 and so on. Mathematicians have been trying to understand these numbers since the ancient Greeks.</p>
<p>One of the main tools used in the study of prime numbers is what are called L-functions – the most famous of which is the <a href="https://www.britannica.com/science/Riemann-zeta-function">Riemann zeta function</a> introduced in the 1800s.</p>
<p>One of Akshay’s major results was joint work with the French mathematician Philippe Michel that solved the <a href="https://link.springer.com/article/10.1007/s10240-010-0025-8">“subconvexity problem”</a> for a large family of L-functions. </p>
<p>This problem had been identified in 1999 as one of the four most important problems in the area, and involved providing good upper bounds on the values that these functions can take. He was then able to apply its solution to the study of homogeneous dynamics and the theory of quadratic forms.</p>
<h2>The Fields Medal</h2>
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<span class="caption">The Fields Medal.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:FieldsMedalFront.jpg">Stefan Zachow/International Mathematical Union</a></span>
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<p>The <a href="https://www.mathunion.org/imu-awards/fields-medal">Fields Medal</a>, regarded as one of the most important honours in mathematics, is named after the Canadian mathematician John Charles Fields (1863–1932). He conceived the award to celebrate the great achievements in the area.</p>
<p>In addition to a gold medal, a winner receives CA$15,000 (A$15,500).</p>
<p>It is an incredible honour to receive a Fields Medal, which is made even more difficult by the fact that you need to be under 40 at the start of the year to receive it. </p>
<p>The award is great recognition for the significant work that Akshay has done and hopefully it inspires other students to study mathematics.</p>
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<p>The Fields Medal is usually awarded to two to four mathematics. This year’s other winners are:</p>
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<li><p>Alessio Figalli, 34, professor at ETH in Zurich (Switzerland) who works on differential equations.</p></li>
<li><p>Caucher Birkar, 40, professor at Cambridge University (UK), who works in algebraic geometry.</p></li>
<li><p>Peter Scholze, 30, professor at Bonn University (Germany), who also works in algebraic geometry.</p></li>
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<p class="fine-print"><em><span>Michael Giudici receives funding from the Australian Research Council. </span></em></p>I was in second year at the University of Western Australia when Akshay Venkatesh skipped first year maths and jumped straight into my classes. He was 13 at the time. Now he’s won a prestigious award.Michael Giudici, Professor, The University of Western AustraliaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/956322018-04-29T15:50:17Z2018-04-29T15:50:17ZNavigating the AI maze is a challenge for governments<figure><img src="https://images.theconversation.com/files/216375/original/file-20180425-175038-17d98lt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Finding the optimal route to benefiting from AI is like navigating a maze for most governments. </span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>New developments in artificial intelligence are proceeding apace. As an economist who has <a href="https://economicsofai.com">researched the AI revolution</a>, I see 2018 as similar to 1995 when the commercial internet was born. The technology is advancing rapidly, but most businesses are only just starting to figure out how to put it to work.</p>
<p>While much of the media attention is focused on corporate applications of AI, governments are also increasing their focus on this prediction enabling technology.</p>
<p>In late 2016, just as President Barack Obama was leaving office, his administration <a href="https://obamawhitehouse.archives.gov/blog/2016/10/12/administrations-report-future-artificial-intelligence">published four reports on how best to prepare the American economy for the development and arrival of AI</a>.</p>
<p>Last month, <a href="http://www.sciencemag.org/news/2018/03/emmanuel-macron-wants-france-become-leader-ai-and-avoid-dystopia">France released a comprehensive report on AI</a> chaired by <a href="http://www.fields.utoronto.ca/news/2010-Fields-Medallist-C%C3%A9dric-Villani-Elected-to-French-National-Assembly">Fields Medalist Cédric Villani</a>. President Emmanuel Macron stressed the immediacy of government policy choices to ensure that France is well positioned to benefit from AI innovation.</p>
<h2>Navigating a maze</h2>
<p>To consider the main policy options available to Canada, let’s consider an analogy. Finding the optimal route to benefiting from AI is like navigating a maze. Most countries are just waking up to the size of the prize for navigating the maze quickly and in a manner consistent with their values.</p>
<p>Mazes have sharp and surprising turns. Just because a mouse is close to the cheese, doesn’t mean it will get there first. This is shorthand for saying that it is hard to know what the correct path is — it’s not necessarily the shortest.</p>
<p>What can we do to increase the chance that the mouse (country) will successfully navigate the maze? One option is to increase the size of the cheese. That increases the incentive to move quickly and work hard at navigation.</p>
<p>For AI, this means ensuring that innovators can profit from AI development. To achieve this, we have policy levers such as competitive grants for compelling research proposals, prizes for research results and the removal of trade barriers so that products can be sold worldwide.</p>
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<span class="caption">French President Emmanuel Macron delivers a speech during the Artificial Intelligence for Humanity event in Paris on March 29, 2018.</span>
<span class="attribution"><span class="source">(Etienne Laurent/Pool Photo via AP)</span></span>
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<p>Interestingly, the French report does not spend much time on such possibilities. And we should consider why that is. Put simply, profit-oriented companies already know there is cheese at the end of the maze but they do not know what type of cheese it is.</p>
<h2>Where’s the cheese?</h2>
<p>The government could lower taxes on the income of companies applying AI, but how would they identify such companies, even after the fact? AI is a general purpose technology. It may be used anywhere. Creating an incentive would be like promoting Canadian cheddar, but subsidising thousands of other cheese types.</p>
<p>The second way to improve maze performance is to make the mouse stronger. If a mouse is starving, it may not be equipped to make it through the maze. So, you might fatten the mouse a bit and make it stronger. For AI, this is the world of tax breaks for expenditures on AI, government subsidies for basic AI research and subsidising the training of AI talent to ensure that Canadian companies can get the talent they need.</p>
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<span class="caption">Alan Aspuru-Guznik, an expert in machine learning, is moving from Harvard to the University of Toronto.</span>
<span class="attribution"><span class="source">University of Toronto</span></span>
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<p>Canada is showing itself to have some advantages. Just this month, the <a href="http://www.canada150.chairs-chaires.gc.ca/home-accueil-eng.aspx">Canada 150 Research Chair program</a> led the <a href="http://www.chem.utoronto.ca/edistillations/2018/03/aspuru-guzik-joins-chemistry.html">University of Toronto to hire Alan Aspuru-Guznik</a>, an expert in machine learning, quantum computing and chemistry, from his tenured position at Harvard. He saw Canada as a country consistent with his values. More critically, he joins a growing scientific ecosystem fuelled by initiatives such as the <a href="https://vectorinstitute.ai/">Vector Institute for Artificial Intelligence</a>.</p>
<h2>Removing barriers</h2>
<p>The final way to improve the maze is to remove barriers. While some barriers are the nature of innovation, others are placed there by government policy. The very first proposal of the French AI report deals with this — ensuring data is available to train AI.</p>
<p>Most computer-related projects are hungry for data and knowledge. After all, the web is just a big data transfer engine. But as I outline in my new book, <a href="https://www.amazon.ca/Prediction-Machines-Economics-Artificial-Intelligence/dp/1633695670/ref=sr_1_1?ie=UTF8&qid=1524756045&sr=8-1&keywords=prediction+machines"><em>Prediction Machines: The Simple Economics of Artificial Intelligence</em></a>, when it comes to AI, data is critical. The better, more comprehensive and richer the data, the better the performance of the AI at its main job — prediction.</p>
<p>Just as our ability to predict the weather depends on weather data acquired all over the globe, and our experience in identifying objects comes from a lifetime of experience stored in our memories, AIs need data to build their capabilities.</p>
<p>The problem is that data may be locked down in various silos created for reasons other than AI. This is currently a topical issue with regard to Facebook’s user data. A few years ago, Facebook was freer with its data, which led to a variety of uses — some creative and productive and others unsavoury.</p>
<p>In response to the current crisis, Facebook has now locked this down. You may feel comforted by the privacy that affords, but at the same time, it is just another barrier to data being available for researchers and creators outside of Facebook.</p>
<p>In actuality, if we want to promote AI, we need to encourage rather than discourage companies from sharing data. And in some cases, that data — for instance, health data — rests with governments.</p>
<h2>Making data available</h2>
<p>The sooner governments find a way to make that data available for research and creative applications in a manner that suitably protects the privacy of Canadians, the easier the maze navigation will be for Canadian businesses to leverage this powerful prediction technology to enhance their products and services, making them more globally competitive. </p>
<p>The French approach is to choose key sectors where they will make things easier for businesses — something they call “sandboxes.” They are exploring the removal of certain regulations to encourage development in health (predictive diagnostics, personalized medicine), transport (autonomous vehicles), defence (predicting cyber-attacks) and the environment (predicting problems in the food supply chain).</p>
<p>There is, of course, more to the French report than just encouraging AI development. Regardless of whether they or others develop AI, the report reflects thinking about how to protect French workers from disruptions and ensure that AI does not lead to biases that humans engender — particularly on the dimensions of gender and race.</p>
<p>The Canadian government would benefit from carefully reviewing the French proposal, including the speculative sections that only apply when the mouse finally reaches the cheese.</p>
<p>For the moment, I urge the Canadian government to think about whether that mouse is Canadian or not.</p><img src="https://counter.theconversation.com/content/95632/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Joshua Gans is Chief Economist of the Creative Destruction Lab and is the recipient of funding from the Sloan Foundation.</span></em></p>Most businesses are only just starting to figure out how to put artificial intelligence to work. But governments are also increasing their focus on this prediction enabling technology.Joshua Gans, Professor of Strategic Management, University of TorontoLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/811432017-07-17T23:56:59Z2017-07-17T23:56:59ZMaryam Mirzakhani was a role model for more than just her mathematics<p>On July 14, 2017, Maryam Mirzakhani, Stanford professor of mathematics and the only female winner of the prestigious Fields Medal in Mathematics, died at the age of 40. </p>
<p>In just a few hours, her name, both in her native Farsi (#مریم میرزاخانی) and English (#maryammirzakhani), was trending on Twitter and Facebook. <a href="http://ifpnews.com/exclusive/iran-newspaper-front-page-july-16-2017/">Most major news agencies</a> were <a href="http://www.bbc.com/news/science-environment-40617094">covering the news</a> of <a href="https://www.nytimes.com/2017/07/16/us/maryam-mirzakhani-dead.html">her death</a> as well as recounting <a href="http://news.stanford.edu/2017/07/15/maryam-mirzakhani-stanford-mathematician-and-fields-medal-winner-dies/">her many achievements</a>.</p>
<p>The grief was especially hard-hitting for a generation of younger academics like me who have always held Maryam as a role model whose example is helping redefine women’s status in science and especially mathematics. </p>
<p>The irony was that Maryam always tried to avoid the media’s spotlight. Her modesty and simplicity despite being the only woman to gain such high status in the world of mathematics – winning what’s often called the “Nobel Prize of math” – stood out to those who knew her.</p>
<p>Unfortunately, I did not get the chance to meet Maryam personally. But like many of my Iranian peers in academia, I looked to her example as proof that the world would welcome us and our scientific contributions no matter our skin color, nationality or religion. </p>
<p>As people around the globe grieve the loss of this talented mathematician, Maryam’s life stands as an inspiration for young girls and boys from all walks of life the world over.</p>
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<figcaption><span class="caption">Maryam Mirzakhani in her own words in a video by the Simons Foundation and the International Mathematical Union.</span></figcaption>
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<h2>Steady advances of a hardworking genius</h2>
<p>Despite her calm expression and warm smile, Maryam was a warrior. She and her family, alongside many other Iranians, lived through the hard economic and social transformations after the Iran revolution in 1979 and also survived the eight years of the Iran-Iraq war a few years after that.</p>
<p>Maryam originally wanted to be a writer, a passion of hers that never faded away even during her postgraduate studies. However, she found an even greater joy in how rewarding it felt to solve mathematical problems. As a student, she was the first female member of Iran’s national team to participate in the International Math Olympiad, and she <a href="https://www.imo-official.org/participant_r.aspx?id=926">won two gold medals</a> in two consecutive years – still a record.</p>
<p>She received her bachelor’s degree from Sharif University of Technology in Iran and later a doctorate from Harvard. In 2014, Maryam was <a href="https://theconversation.com/meet-the-winners-of-the-fields-medal-the-nobel-prize-of-maths-30411">recognized with the Fields Medal</a>, the highest-ranking award in mathematics, for her efforts in what’s known as <a href="https://theconversation.com/corals-crochet-and-the-cosmos-how-hyperbolic-geometry-pervades-the-universe-53382">hyperbolic geometry</a>. Her work focused on curved surfaces – such as spheres or donut shapes – and how to understand their properties. Her achievements have applications in other fields of science including quantum field theory, engineering and material science, and could even influence theories around how our universe was born.</p>
<p>Maryam was a “hall of fame” all by herself. She modestly attributed her own success to her perseverance, hard work and patience. <a href="https://www.theguardian.com/science/2014/aug/13/interview-maryam-mirzakhani-fields-medal-winner-mathematician">As she put it</a>:</p>
<blockquote>
<p>“The beauty of mathematics only shows itself to more patient followers.”</p>
</blockquote>
<p>Unfortunately, when she was honored with the Fields Medal, she was already tackling her last challenge, the breast cancer that eventually killed her.</p>
<h2>Who she was, not just what she did, matters</h2>
<p><a href="https://theconversation.com/maryam-mirzakhanis-success-showed-us-the-challenges-women-in-maths-still-face-81193">Maryam’s contributions</a> to the field of mathematics will long be remembered. But just as important is her legacy as a role model. </p>
<p>Maryam was an Iranian, a woman and an immigrant to the United States. Unfortunately, these three words together raise red flags for some in Western countries, particularly in the U.S., in the time of <a href="https://theconversation.com/us/topics/trump-travel-ban-35583">Trump’s proposed travel ban</a>. </p>
<p>Against all odds, Maryam’s talent was nurtured in Iran and later flourished in the U.S. Her successes discredit the xenophobic stereotypes that are encouraged by a politics of fear. Maryam defied expectations and rose above all the labels that make it easy to judge others who are not like “us.”</p>
<p>Maryam’s legend may continue to grow after her early death. Still only 20 percent of full-time math faculty at U.S. universities are women, <a href="http://www.ams.org/profession/data/annual-survey/demographics">according to a 2015 demographic survey</a> of 213 departments by the American Mathematical Society. Research shows that <a href="https://doi.org/10.1177/0361684312459328">stereotyped role models can influence</a> whether people “see themselves” in certain STEM careers. The example of a woman who rose to the top of this still very male field may help inspire math’s next generation. </p>
<p>In the same way people think of Marie Curie or Jane Goodall as scientific pioneers, Maryam Mirzakhani will go down in history as a trailblazer as well as a mathematical genius.</p><img src="https://counter.theconversation.com/content/81143/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mehrdokht (Medo) Pournader 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>Mirzakhani blazed to the top of her field due to her talent. But who she was and where she came from also make her a role model for those from underrepresented demographics in the world of math.Mehrdokht (Medo) Pournader, Senior lecturer at The University of Melbourne, The University of MelbourneLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/304112014-08-13T02:44:45Z2014-08-13T02:44:45ZMeet the winners of the Fields medal – the ‘Nobel prize of maths’<figure><img src="https://images.theconversation.com/files/56344/original/ht2dfgcq-1407895853.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Maryam Mirzakhani – the first woman to win the Fields medal – was recognised for contributions to understanding the symmetry of curved surfaces.</span> <span class="attribution"><span class="source">Stanford University</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>The four winners of the 2014 <a href="http://www.mathunion.org/general/prizes/2014/prize-citations/">Fields medals</a> – the most prestigious prizes for mathematics – were announced today, including the first female and first Latin American recipients of the 78-year-old prize.</p>
<p>The awarding of these and several other International Mathematical Union (<a href="http://www.mathunion.org/">IMU</a>) <a href="http://www.mathunion.org/general/prizes">prizes</a> – typically by the head of state of the host country – is the climax of the opening ceremony of the quadrennial <a href="http://www.icm2014.org/">International Congress of Mathematicians</a>, held this year in Seoul. </p>
<p>Unlike the <a href="http://www.nobelprize.org/">Nobel prizes</a> and many other modern science prizes, the financial value of the Fields medal (officially known as the International Medal for Outstanding Discoveries in Mathematics) is relatively trivial – roughly US$15,000 – but its prestige is life-changing. </p>
<p>The money comes from the 1932 bequest of Canadian mathematician <a href="http://www.fields.utoronto.ca/aboutus/jcfields/">J C Fields</a> who ran the 1924 Mathematics Congress in Toronto. Indeed, the Canadian mint produces the medals. A friend of mine once transported them to the congress in his luggage.</p>
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<span class="attribution"><span class="source">Wikimedia Commons</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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<p>According to the IMU, as was requested by Fields the man, the award intends:</p>
<blockquote>
<p>to recognise outstanding mathematical achievement for existing work and for the promise of future achievement.</p>
</blockquote>
<p>Since 2003, the <a href="http://www.abelprize.no/nyheter/vis.html?tid=61204">Abel prize</a> has been awarded on the same scale as a Nobel prize, but the Fields medal is the prize mathematicians pay most attention to. The 2006 medallist, Australian <a href="http://en.wikipedia.org/wiki/Terence_Tao">Terry Tao</a>, is considered special even among this august group.</p>
<p>The list of <a href="http://www.mathunion.org/index.php?id=prizewinners">previous winners</a> charts the past century’s mathematical achievements, and the four new winners make exciting additions. They include both the first woman and the first Latin American to win the prize. They also include a number theorist who many (in the know) had tipped to win.</p>
<p>As described in the <a href="http://www.businessinsider.com.au/2014-fields-medal-winners-2014-8">press</a>, the winners in alphabetic order are:</p>
<h2>Artur Avila</h2>
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<span class="caption">Artur Avila.</span>
<span class="attribution"><span class="source">Wikimedia Commons</span></span>
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<p><a href="http://w3.impa.br/%7Eavila/">Artur Avila</a>, a Brazilian mathematician, has contributed to a number of fields. Some of his most notable research is in the study of <a href="https://theconversation.com/explainer-what-is-chaos-theory-10620">chaos theory</a> and dynamical systems. </p>
<p>These areas seek to understand the behaviour of systems that evolve over time in which very small changes in initial conditions can lead to wildly varying outcomes, such as weather patterns. This is typified in the classic example of a butterfly’s wings flapping leading to a change in weather hundreds of kilometres away.</p>
<p>One of Avila’s major contributions to this field of study was in clarifying that a certain broad class of dynamical systems fall into one of two categories. They either evolve eventually into a stable state, or fall into a chaotic stochastic state, in which their behaviour can be described probabilistically.</p>
<h2>Manjul Bhargava</h2>
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<figcaption>
<span class="caption">Manjul Bhargava.</span>
<span class="attribution"><a class="source" href="http://www.mathunion.org/general/prizes/2014/">IMU</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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<p>Canadian-American <a href="https://www.math.princeton.edu/directory/manjul-bhargava">Manjul Bhargava</a>’s research is focused on number theory and algebra. One of the basic subjects in algebraic number theory is the behaviour of polynomials with integer coefficients, such as 3x<sup>2</sup> + 4xy - 5y<sup>2.</sup></p>
<p><a href="http://www-history.mcs.st-and.ac.uk/Biographies/Gauss.html">Carl Friedrich Gauss</a>, one of the greatest mathematicians of the late 18th and early 19th centuries, developed a powerful tool for analysing polynomials such as the one above, where the variables are all raised to at most the second power.</p>
<p>Bhargava, by intensely studying Gauss’ work and adding to it an impressive level of geometric and algebraic insight, was able to extend Gauss’ tool to higher-degree polynomials in which we raise the variables to higher powers than two. This work vastly expands the ability of number theorists to study these fundamental mathematical objects.</p>
<h2>Martin Hairer</h2>
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<figcaption>
<span class="caption">Martin Hairer,</span>
<span class="attribution"><span class="source">Wikimedia Commons</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p><a href="http://www.hairer.org/">Martin Hairer</a> – originally from Austria and currently professor at the University of Warwick – researches stochastic partial differential equations. Differential equations show up throughout mathematics, physics and engineering. They describe processes that change over time, such as the movement of a shell shot from a cannon, or the price of a stock or bond.</p>
<p>Differential equations come in a variety of flavours. Ordinary differential equations are equations that involve only one variable. The motion of a cannonball, for example, can be modelled with a simple ordinary differential equation in which the only variable is the time since the cannon was fired.</p>
<p>Partial differential equations involve processes that depend on multiple variables. In many physical settings, both time and the current position of an object are needed to determine the future trajectory of the object. These describe a much wider variety of processes in the world and are generally much harder to work with than one-variable ordinary equations.</p>
<p>Differential equations can also be either deterministic or stochastic:</p>
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<span class="attribution"><span class="source">Wikimedia Commons</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span>
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<ul>
<li>a cannonball’s movement, or the movement of a <a href="https://theconversation.com/explainer-how-do-satellites-orbit-the-earth-28859">satellite orbiting earth</a>, are <strong>deterministic</strong>: outside of measurement error, once we’ve solved the equation, we have no doubt about where the cannonball or satellite will be at a given point in time.</li>
<li><strong>stochastic</strong> equations have a random element involved. The motion of sugar grains stirred in a cup of coffee, or a stock’s price at a given moment in time, are both best described by models that have an element of noise or randomness. </li>
</ul>
<p>Stochastic partial differential equations — equations that have multiple input variables and random noise elements — have traditionally been very difficult for mathematicians to work with. Hairer developed a new theoretical framework that makes these equations far more tractable, opening the door to being able to solve a number of equations with both large amounts of mathematical interest in their own right and with powerful applications in the sciences and engineering.</p>
<h2>Maryam Mirzakhani</h2>
<p>Iran-born Stanford professor <a href="http://www.newscientist.com/article/dn26044-iranian-woman-wins-maths-top-prize-the-fields-medal.html#.U-qfyPQW1AI">Maryam Mirzakhani</a>’s work focuses on the geometry of <a href="http://mathworld.wolfram.com/RiemannSurface.html">Riemann surfaces</a>. </p>
<p>Riemann surfaces are a classic type of <a href="http://www-history.mcs.st-and.ac.uk/HistTopics/Non-Euclidean_geometry.html">non-Euclidean geometry</a>: while a Riemann surface still has two dimensions such as a plane, and we can still define lines, angles and curves on the surface, the way that the measurement of angles and distances will come out can be very different from what happens on a normal Euclidean plane.</p>
<p>A basic example of this is the <a href="http://plus.maths.org/content/maths-minute-riemann-sphere">Riemann sphere</a>: a version of a sphere in which we still have a notion of lines and angles, but where strange things can happen, such as triangles with three 90-degree angles.</p>
<p>Riemann surfaces can get far more complicated than the Riemann sphere. One of the major research areas in the study of these surfaces is how one Riemann surface can be smoothly deformed or smooshed into another surface. </p>
<p>These deformations themselves have their own strange geometries, called “moduli spaces”, and Mirzakhani has contributed several interesting results in understanding these mysterious spaces. </p>
<p>The <a href="http://www.mathunion.org/general/prizes/2014/prize-citations/">official citations</a> for all the 2014 IMU prizes are available on the IMU site.</p>
<h2>And is it the Nobel?</h2>
<p>The folk history of why there is no Nobel prize in mathematics is amusing and largely bogus. The real history is even more amusing – if you’re interested, a New York Times <a href="http://www.nytimes.com/2014/08/10/opinion/sunday/how-math-got-its-nobel-.html?_r=0">article</a> from last week describes it beautifully.</p>
<p>Helpfully, Scientific American has even published <a href="http://blogs.scientificamerican.com/roots-of-unity/2014/08/11/how-to-talk-about-the-fields-medal-at-your-next-cocktail-party/">cocktail party talking points</a>.</p>
<p>Nobel or not, these four prizes celebrate the enormous depth, range and scope of modern mathematics. Both Fields and Nobel would be proud.</p><img src="https://counter.theconversation.com/content/30411/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from the ARC.</span></em></p>The four winners of the 2014 Fields medals – the most prestigious prizes for mathematics – were announced today, including the first female and first Latin American recipients of the 78-year-old prize…Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleLicensed as Creative Commons – attribution, no derivatives.