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enWhat is pharmaceutical statistics?
https://plus.maths.org/content/pharmaceutical-statistics
<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/pills_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>In this data-driven age, studying statistics at A-level or beyond opens up a wide range of career options. One interesting area for statisticians to work in, which we recently had the chance to find out more about, is the pharmaceutical industry.</p>
<h3>How are new medical treatments tested?</h3>
<p>Before a new drug comes on the market it needs to be tested on people who have the condition the drug has been developed for. Medical scientists will already have an idea that the drug might work from tests on biological cells and on animals, but they still need to see if it also works in humans, measure just how effective it is, and what kind of side effects it might have. </p>
<div class="rightimage" style="max-width: 300px;"><img src="/latestnews/jan-apr10/rct/pills.jpg" alt="Pill bottles" width="300" height="253" /><p>In a medical trial one group of people is given the new treatment and another a placebo or an existing treatment.</p></div>
<p>At first sight, conducting such a test seems easy: give the treatment to a group of people with the condition and see what happens. In reality though, things aren't that simple. One problem is the famous placebo effect, also known as the sugar pill effect: if you have a medical condition, then simply thinking that you’re being treated for it might make you feel better, even if the treatment doesn’t actually work.</p>
<p>Another problem is that even if your patients do appear to get better, this might just be a fluke, or down to some other factors you’re not even aware of. For example, if the patients all live in the same area, and have recently been getting out and about a lot because the weather in that area has been nice, then they might have got better because of all the exercise and fresh air. The treatment may have nothing to do with it.</p>
<p>To get around these problems medical treatments are tested on two groups of people: one group will receive the treatment and the other will receive either a placebo (a "sugar pill" that doesn't work) or an existing treatment you want to compare the new treatment to. People are allocated to these groups randomly. That's the best way of making sure they don’t all share some particular characteristic (such as living in the same area) that can influence the result. A trial which works in this way is called a randomised controlled trial.</p>
<h3>What do medical statisticians do?</h3>
<div class="rightimage" style="max-width: 157px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phamastats/2017_05_15_mci_psi_conference_day_2_168.jpg" alt="" width="157" height="254" />
<p>Mary Elliott-Davey.</p>
</div>
<p>Statisticians are a critical part of every stage of the trial process: from the initial design of a randomised controlled trial to analysing the results and communicating them to colleagues. "My job begins before the trial starts," explains biostatistics manager Mary Elliott-Davey. "I will work with colleagues from other departments to design the trial. For example, how many patients do we need in each group? How long are we going to observe the patients for? How are we going to measure the disease outcome to see if the new medication works better?"</p>
<!-- <div class="rightimage" style="max-width: 300px;"><img src="/latestnews/jan-apr10/rct/blood.jpg" alt="blood pressure measurement" width="300" /><p>How do we test whether a new drug for reducing blood pressure really works?</p></div>
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<p>Even only deciding how many people should be part of a study requires many of the statistical tools you learn about at school. The number of people needed depends on how confident you would like to be that you draw the correct conclusion from the outcome of your trial: the more confident you'd like to be, the more people you need. Measuring this confidence involves significance levels and confidence intervals. The number of people you need also depends on how variable the quantity (or quantities) that will be measured in the trial (for example blood pressure) is in the population you are interested in: the more it varies naturally from person to person, the more people you need in your trial. Working out this variation requires you to understand probability distributions and their measures of variability, such as standard deviation. There are standard formulae that tell you how many people you need depending on these, and other, factors. (You can find out more about this in <a href="/content/evaluating-medical-treatment-how-do-you-know-it-works">this</a> <em>Plus</em> article, or explore the idea yourself with <a href="https://nrich.maths.org/13762">this activity</a> on our sister site NRICH.) </p>
<p>"Once the study has finished I will then analyse the data that was collected," explains Elliott-Davey. "There is a possibility that the results observed were just down to chance, and maybe the new medicine isn't any better. My job is to assess this chance and help the team determine if our new medicine does work better. I also explore the characteristics of the patients that were in the study. For example, what was the average age of patients in the study? How severe was their disease? Did they have any other [diseases]?"</p>
<div class="leftimage" style="max-width: 157px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phamastats/pic2.png" alt="Rhian Jacob" width="157" height="254" />
<p>Rhian Jacob.</p>
</div>
<p>To assess whether the results of a study were down to chance or indicate that a new treatment really is effective, you need to be clear about what questions you are asking of the data. You need to test a clearly formulated hypothesis (the drug reduces blood pressure by x amount) against the possibility that the new treatment is not effective, or no more effective than others. A statistical technique called hypothesis testing delivers the framework to do just that It also involves probability distributions and significance levels. (You can learn more in <a href="/content/evaluating-medical-treatment-how-do-you-know-it-works">this</a> article on <em>Plus</em> or <a href="https://nrich.maths.org/13722">this</a> article on our sister site NRICH, and try out the statistical techniques yourself in <a href="https://nrich.maths.org/13764">this</a> collection of NRICH resources.)</p>
<p>Applying sophisticated mathematics in real-life contexts is never just about the maths, however. It also involves a lot of teamwork. "A medical statistician partners with other non-statistical experts, such as doctors, pharmacologists, and logistics/operations personnel," says statistical scientist Rhian Jacob. "They all have different expertise and generally don't understand statistics. Being able to communicate complex, statistical concepts in simple terms is essential." The ability to visualise complex data sets that potentially involve many variables is another essential tool in this context. </p>
<p>Teamwork is one of the things Jacob likes most about her job: "Being a single member of a very large, global team where everyone has different opinions and expertise but ultimately have the same common goal: to deliver safe and effective medicines to patients."</p>
<h3>Changing lives</h3>
<p>A pharmaceutical statistician usually works on several projects at once. They might investigate the design of a new trial one day, and then analyse the results from the latest study, or explain them to external doctors, the next. They might also travel to attend training sessions, meetings or conferences. "The work is varied, challenging, has large scope for progression and can require international travel," says Jacob. "Most companies are open to new ideas and welcome curiosity; it's nice that contribution from you personally is recognised and valued early on."</p>
<div class="rightimage" style="max-width: 157px;"><img src="/content/sites/plus.maths.org/files/articles/2018/phamastats/jlf_plus_photo.jpg" alt="James Lay-Flurrie." width="157" height="254" />
<p>James Lay-Flurrie.</p>
</div>
<p>Statistics leader James Lay-Flurrie points to the flexibility of career paths in the industry and the potential for exciting discovery. "Many companies will encourage you to take career decisions that play to your strengths. For example, if you have a real interest in being on the cutting edge of new statistical methods then it's possible to go down the statistical scientist route, acting in a consultant role across many projects."</p>
<p>"The great thing about the industry is that statisticians are always striving to develop cutting edge methods and trial designs to ensure medicines can be approved more quickly and at less expense, helping these medicines reach patients as quickly and cheaply as possible," says Lay-Flurrie. "Despite often working for rival companies there is a great statistical community within the industry and willingness to share and develop new ideas." Lay-Flurrie, Jacob and Elliott-Davey are all members of <a href="https://www.psiweb.org/">PSI</a>, an organisation comprising members from many different companies in all areas of drug development, dedicated to leading and promoting best practice and industry initiatives for statisticians.</p>
<p>
But what is perhaps the most satisfying aspect of the job is how the results are put to use. "I love how statistics can be used to improve people's lives," says Elliott-Davey. "It is exciting to know that the study I am working on may be for a medicine that could treat cancer or prevent Alzheimer's."</p>
<p>"Every person will be affected by serious diseases, either directly or indirectly, at some point in their lives. To be able to say that my work might help even just a little bit is what gets me out of bed in the morning."</p>
<p><em>You can find out more about statistics careers in the pharmaceutical industry on the <a href="https://www.psiweb.org/careers-homepage">PSI careers page</a>.</em></p>
<hr>
<h3>About this article</h3>
<p>We are grateful to <a href="https://www.psiweb.org/">PSI</a> for their support in producing this article.</p>
<p>You can try out some of the statistical techniques mentioned here in <a href="https://nrich.maths.org/13764">this</a> collection of resources on our sister site NRICH.</p>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>.</p>
</div></div></div>Thu, 20 Sep 2018 10:40:36 +0000Marianne7102 at https://plus.maths.org/contenthttps://plus.maths.org/content/pharmaceutical-statistics#commentsMaths and politics
https://plus.maths.org/content/politics-and-transcendental-numbers
<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/1966_stamp_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">Christopher Hollings</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 International Congresses of Mathematicians (ICMs) take place every four years at different locations around the globe. They are the largest regular gatherings of mathematicians from all nations (the 2014 congress in Seoul attracted over 5,000 delegates from 122 countries), and provide a forum for the award of one of the premier prizes in mathematics: the <a href="/content/tags/fields-medal">Fields medal</a>. However, as much as the assembled mathematicians may like to pretend that these gatherings transcend politics, they have always been coloured by world events.</p>
<div class="rightimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2018/ICM_history/pre-icm_chicago_1893.jpg" alt="A picture from the zeroeth ICM in Chicago, 1897" width="400" height="252" />
<p>A picture from the zeroeth ICM in Chicago, 1893.</p>
</div>
<!-- Image in public domain -->
<p>The ICMs have their origin at the end of the nineteenth century, at a time when improved communications and transport connections, as well as a period of relative peace over much of Europe, led to a great expansion in international scientific activities (where "international" usually meant just Europe and North America). A very small conference, often termed the "zeroeth ICM", took place in Chicago in 1893, but the first truly international gathering of mathematicians was that held in Zurich in 1897, attended by around 240 delegates from 16 countries – rather modest when compared to more recent ICMs, but evidently successful enough that further congresses were planned.</p>
<p>The 1897 congress was followed in 1900 by a second, held this time in Paris, alongside the many other conferences and exhibitions that were being staged there to mark the new century. A noteworthy feature of the Paris ICM was a lecture given by the respected German mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Hilbert.html">David Hilbert</a>, in which he outlined a series of problems that he thought ought to be tackled by mathematicians in the coming decades. Hilbert's problems went on to shape a great deal of twentieth-century mathematical research; just three remain entirely unresolved (see <a href="/content/transcendental-numbers-and-politics">here</a> for an example of one of the problems).</p>
<h3>Rising tension</h3>
<!-- <p>Amongst Hilbert's list of problems was one concerning so-called <em>transcendental numbers</em>. These are defined in contrast to <em>algebraic numbers</em>: we say that a number is algebraic if it is the solution of a polynomial equation all whose coefficients are integers, that is to say, an equation consisting of a sum of powers of the unknown <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> which can be multiplied by whole numbers. Examples are <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0002.png" alt="$3x^2 + 2x +1 = 0$" style="vertical-align:-1px;
width:122px;
height:15px" class="math gen" /> or <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0003.png" alt="$5x^5 + 6x^3 + 7x + 8 = 0$" style="vertical-align:-1px;
width:167px;
height:15px" class="math gen" />. Thus, for example, the number <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0004.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> is algebraic because it is the solution of the equation <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0005.png" alt="$x – 2 = 0$" style="vertical-align:0px;
width:81px;
height:12px" class="math gen" />. In a similar way, any whole number or any fraction (a <em>rational number</em>) is also algebraic: <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0006.png" alt="$1/2$" style="vertical-align:-4px;
width:23px;
height:16px" class="math gen" /> , for instance, is the solution of <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0007.png" alt="$2x – 1 = 0$" style="vertical-align:0px;
width:89px;
height:12px" class="math gen" />. Another example of an algebraic number is <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0008.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" />: this is a solution of the equation <img src="/MI/30f6daacce325d0934646b52384a9952/images/img-0009.png" alt="$x^2 – 2 = 0$" style="vertical-align:0px;
width:88px;
height:14px" class="math gen" />.</p>
<p> Any number that is not algebraic is called transcendental. Since every rational number is algebraic, it follows that every transcendental number is necessarily irrational (that is, not rational); but not every irrational number is transcendental: take <img src="/MI/fe34a52857184efc7ae3bdb4f4cca715/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> for example (see <a href="/content/maths-minute-square-root-2-irrational"><em>Maths in a minute: The square root of 2 is irrational</em></a>).</p>
<p>Probably the most famous transcendental number is <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> (another is <img src="/MI/ca429a11c2d5253b1d43cba0a0da4053/images/img-0001.png" alt="$e$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" />; see <a href="/content/where-does-e-come-and-what-does-it-do-0"><em>Where does e come from and what does it do?</em></a>). For centuries, mathematicians had had the creeping suspicion that <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> was somehow unlike most of the other numbers that they encountered. It was certainly known that <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> is irrational, but its weirdness as a number seemed to go beyond this. Despite the fact that the notions of algebraic and transcendental numbers had first been defined by <a href="https://www.biography.com/people/leonhard-euler-21342391">Leonhard Euler</a> in the 18th century, it was not until 1882 that the German mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Lindemann.html">Ferdinand von Lindemann</a> demonstrated that <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> is in fact transcendental. As well as proving once and for all that it is not possible to square the circle (see <a href="/content/os/issue21/xfile/index"><em>Mathematical mysteries: Transcendental meditation</em></a>, this also explained why <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> seemed so unlike other numbers: because we can't write down equations of which they are solutions, transcendental numbers are harder to "get hold of" than algebraic ones. In essence, an equation for a number provides us with a finite process by which we can construct that number; in the case of transcendental numbers, we have no such process.</p>
<p>
At the end of the nineteenth century, <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> was one of only very few examples of transcendental numbers that were known. And yet mathematicians knew that there are in fact very many more transcendental numbers than algebraic: the transcendental numbers are <em>uncountably infinite</em>, whereas the algebraic numbers are <em>countable</em> (see <em><a href="/content/maths-minute-counting-numbers">Maths in a minute: Counting numbers</a></em>). More examples of transcendental numbers were needed. Indeed, what would be more useful would be a new method for constructing examples of transcendental numbers, and this is what is embodied in Hilbert's seventh problem.</p>
<p>Suppose that <img src="/MI/f5857f97af42875631e968807dc250ed/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is an algebraic number that is neither 0 nor 1 (to avoid trivial cases), and suppose that <img src="/MI/f5857f97af42875631e968807dc250ed/images/img-0002.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" /> is an irrational algebraic number. Following on from a similar conjecture made by Euler (where <img src="/MI/f5857f97af42875631e968807dc250ed/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> was assumed merely to be rational), Hilbert conjectured that in this situation <img src="/MI/f5857f97af42875631e968807dc250ed/images/img-0003.png" alt="$ab$" style="vertical-align:0px;
width:16px;
height:11px" class="math gen" /> is necessarily transcendental. His seventh problem was to find a rigorous proof of this. If it could be proved to be true, then <img src="/MI/f5857f97af42875631e968807dc250ed/images/img-0004.png" alt="$2\sqrt {2}$" style="vertical-align:-2px;
width:31px;
height:17px" class="math gen" />, for example, would be transcendental, despite the fact that neither <img src="/MI/f5857f97af42875631e968807dc250ed/images/img-0005.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> nor <img src="/MI/f5857f97af42875631e968807dc250ed/images/img-0006.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> is itself transcendental. Mathematicians would thus at last have a means of constructing further transcendental numbers from better-known building blocks (i.e., algebraic numbers) – but they would have to wait until the 1930s.
</p> -->
<p>Following the Paris congress, a four-yearly pattern was established for the ICMs, and so the next one was held in Heidelberg in 1904. It is here that we begin to see clearly some of the political tensions that would shape the later conduct of the congresses: whereas a large number of German mathematicians had attended the Paris ICM, the Heidelberg congress saw comparatively few French delegates – defeat in the Franco-Prussian war of 1870-1 was still a rather raw memory for some. Overall, however, the 1904 congress had the largest attendance yet (almost 400). At the end of the nineteenth century, Germany had been a popular place for people to study mathematics, and it seems that many mathematicians from abroad had relished the excuse to go back.</p>
<p>With three successful congresses having been held, the ICMs had become well established, and further meetings were held in Rome in 1908, and then in Cambridge in 1912. <!-- At the latter, it was agreed that the 1916 congress would take place in Stockholm. This was due largely to the lobbying of <a href=http://www-history.mcs.st-andrews.ac.uk/Biographies/Mittag-Leffler.html">Gösta Mittag-Leffler</a>, a Swedish mathematician, entrepreneur, and canny political operator. He had worked for many years not only to promote the study of mathematics in Scandinavia, but also to gain recognition for Scandinavian mathematics around the world – an ICM in his home city of Stockholm would have enabled him to further his aims. However, it proved impossible to hold an ICM in the war-torn Europe of 1916, even in neutral Sweden, so Mittag-Leffler instead hosted a smaller congress of Scandinavian mathematicians, and clung to the hope that, once the war had ended, the sequence of ICMs would resume with a meeting in Stockholm. --> At the latter, it was agreed that the next congress would take place in Stockholm, however it proved impossible to hold an ICM in the war-torn Europe of 1916. </p>
<h3>Between the wars</h3>
<p>
After the end of the First World War, the mathematicians of Western Europe realised that something ought to be done to help to rebuild their discipline and its international networks. To this end, an International Mathematical Union (IMU) was formed, one of whose immediate tasks was to re-establish the ICMs by staging one in 1920. </p>
<div class="leftimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2018/ICM_history/icm32.jpg" alt="The 1932 ICM in Zurich" width="400" height="271" />
<p>The 1932 ICM in Zurich.</p>
</div>
<!-- Image in public domain -->
<p>However, <!-- contrary to the hopes of Mittag-Leffler and others,--> Stockholm was not to be the venue. The leading members of the IMU, many of whom hailed from the Western European countries that had been particularly devastated by the war – France and Belgium, for example – chose instead to make a bold statement by selecting Strasbourg: a French city that had been incorporated into Germany following the Franco-Prussian War, and that had just been returned to France by the Treaty of Versailles. In fact, more than this, and despite strong protests from such figures as the English mathematician <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Hardy.html">G. H. Hardy</a>, all mathematicians from Germany and her wartime allies were barred from attending the congress – it was a founding principle of the IMU that Germany should never again be allowed to participate in international mathematical activities.</p>
<p>The exclusion of German mathematicians extended also to the 1924 congress in Toronto – indeed, the congress had originally been planned for New York, but the <a href="https://www.ams.org/home/page">American Mathematical Society</a>, which would have played host, failed in its efforts to persuade the IMU to lift the ban and so withdrew its support. The congress was only able to go ahead when <a href="https://en.wikipedia.org/wiki/John_Charles_Fields">J. C. Fields</a>, who would later institute the medal that bears his name, offered to host the event at the University of Toronto. By 1928, however, the pressure on the IMU had become too great, and it was forced to admit German mathematicians to the congress that took place in Bologna that year. From this time on, the ICMs began to take on an existence independent of the IMU, as the influence of the latter waned.</p>
<p>Despite (or, in some cases, because of) the re-opening of relations with German mathematicians, tensions remained in the international mathematical community. Naturally, there were people who believed that the Germans should not have been re-admitted to the ICMs. Moreover, some German mathematicians felt resentment at their earlier exclusion and so boycotted the 1928 congress. In a bid to bring people back together and re-establish ties, the ICM returned in 1932 to Zurich – a deliberately neutral choice. Similar reasoning resulted in Oslo being chosen as the venue for the 1936 ICM, a congress that is significant for the award of the first Fields Medals: to the Finn <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Ahlfors.html">Lars Ahlfors</a> and to the American <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Douglas.html">Jesse Douglas</a>.</p>
<h3>Darkening times</h3>
<!-- <p>One of the plenary speakers at the Oslo ICM was to have been the Russian mathematician <a href=">A. O. Gel'fond</a>. He had been invited precisely because he had solved Hilbert's seventh problem. Spurred on by the proof in 1930 by his fellow Russian <a href="https://en.wikipedia.org/wiki/Rodion_Kuzmin">Rodion Kuzmin</a> that 22 is indeed transcendental, Gel'fond had considered the more general case. He had published a full solution in 1934, and was to speak on this at the ICM, but, in the end, was prevented from attending.</p> -->
<div class="rightimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2018/ICM_history/hitler_stalin.jpg" alt="Hitler and Stalin" width="200" height="582" />
<p>Political agendas dominated the 1936 ICM in Oslo. (Picture of Hitler: <a href="https://commons.wikimedia.org/wiki/File:Adolf_Hitler_Berghof-1936.jpg">German Federal Archives</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/de/deed.en">CC BY-SA 3.0 DE</a>.)</p>
</div>
<!-- Image of Stalin in public domain -->
<p>From the start, the Oslo ICM was a political melting pot of different agendas. The Norwegian hosts, for example, seized upon the opportunity to promote Scandinavian mathematics on an international stage. The wider European political situation also had its effects, with delegations of mathematicians expected from Hitler's Germany, Mussolini's Italy, and Stalin's USSR – though in the end only the first of these groups appeared. The Italians, for example, boycotted the congress in protest at Norway's condemnation of the Italian invasion of Abyssinia.</p>
<p>The goal of the Nazi-led German contingent was clear: to showcase the best of "Aryan mathematics". The German delegation was instructed to be on its best behaviour, to refrain from making racial comments, and to proudly use the German language at all opportunities. The tension that had existed between French and German mathematicians at earlier congresses was present here too, and was expressed through a rivalry over who could "claim" Norway's mathematical heroes <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Abel.html">Niels Henrik Abel</a> and <a href="http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html">Marius Sophus Lie</a>, each of whom had spent time both in France and in (what became) Germany.</p>
<p>The Soviet delegation, on the other hand, was conspicuous by its absence. Like the Germans, Russian mathematicians had had a difficult relationship with the ICMs. Prior to the First World War, they had regularly attended in significant numbers, but had been rather less visible during the 1920s, following the October revolution (1917) and subsequent Russian civil war (1917-1922). As the decade progressed, they began to reappear, but their attendance was sometimes made difficult by the fact that the USSR lacked official recognition by some world governments.</p>
<p>By the end of the 1920s, Stalin was firmly in power in the USSR, and one of the things that he sought to do as he cemented his position was to exercise greater control over the Soviet scientific community. The ability of Soviet academics to travel to foreign conferences was gradually curtailed: whilst, for example, Soviet mathematicians managed an attendance of almost forty at the 1928 ICM, this dropped dramatically to just three in 1932.</p>
<p>At the Oslo congress, around eleven Soviets were expected to attend but when the congress convened in July 1936, it was announced that none of the Soviet delegates had in fact appeared. In an atmosphere in which scientists' foreign connections were increasingly regarded with suspicion by the Soviet authorities, each of the delegates expected from the USSR had been denied permission to travel. </p>
<h3>The cold war</h3>
<p>Just like the planned ICM of 1916, the next one after Oslo, to be held in the USA in 1940, did not take place. In fact, it was not until 1950 that the ICMs resumed with a congress in Cambridge, Massachusetts. No Soviet delegates, nor any from communist Eastern Europe, attended, although several had been invited. The American organisers were at great pains to point out that this had nothing to do with the actions of the US government. Indeed, shortly before the congress, they had received a telegram from the president of the Soviet Academy of Sciences, making the rather lame excuse that Soviet mathematicians were unable to attend due to pressure of work.</p>
<div class="leftimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2018/ICM_history/1966_stamp.jpg" alt="Russian stamp 1966" width="400" height="282" />
<p>A stamp issued to celebrate the 1966 ICM in Moscow.</p>
</div>
<!-- Image of Stalin in public domain -->
<p>During the 1950s, however, we see a similar pattern to that of the 1920s: numbers of Soviet delegates at the ICMs gradually increased in the years following Stalin's death in 1953. <!-- These years saw a thawing of international relations, and so whereas just five Soviet representatives attended the Amsterdam congress of 1954, this went up to around thirty in Edinburgh in 1958, the presence of so many Soviet mathematicians even being remarked upon in an article in <em>The Times</em>. --> The USSR's involvement in the international mathematical community expanded in 1957 when it joined the <a href="https://www.mathunion.org/">International Mathematical Union</a>, which had been re-founded on less prejudiced principles in 1950. <!-- and when Stockholm finally hosted the ICM in 1962, roughly forty Soviet delegates were present. --> The signal that the USSR was again fully engaged in world mathematics came in 1966 when it hosted the ICM that year in Moscow. Nearly 3,000 of the congress' roughly 4,300 delegates came from outside the USSR, the Westerners amongst them seizing the opportunity to visit such a seemingly alien city.</p>
<!-- <p>One mathematician who was able easily to attend the 1966 ICM (since it was held in his home city) was Gel'fond, although he does not appear to have delivered a lecture. As far as we know, this is the only ICM that he ever attended. The congress was certainly attended by a small number of mathematicians who had been in Oslo thirty years earlier, but we can only speculate about the interactions that resulted when they were finally able to meet Gel'fond in Moscow. It is interesting to note that the solution of a generalisation of Hilbert's seventh problem, sometimes termed the <em>Gel'fond conjecture</em>, earned a Fields Medal for the English mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Baker_Alan.html">Alan Baker</a> (1939-2018 ) at the 1970 ICM in Nice.</p> -->
<p>In the decades that followed the Moscow congress, the organisation of the ICMs was certainly not free of difficulties arising from the cold war political climate, or from the policies of the Soviet government. For instance, the second winner of the Fields Medal in Nice in 1970, the Russian mathematician <a href="https://en.wikipedia.org/wiki/Sergei_Novikov_(mathematician)">S. P. Novikov</a>, was denied permission to travel to collect the award, owing to his having publicly criticised the Soviet regime. Moving beyond the USSR, the ICM that was planned for Warsaw in 1982 had to be postponed a year following the declaration of martial law in Poland in December 1981. And, yet again, several Soviet mathematicians were prevented from attending the 1986 congress in Berkeley, California.</p>
<div class="rightimage" style="max-width: 400px;"><img src="/content/sites/plus.maths.org/files/articles/2018/ICM_history/rio_opening.jpg" alt="ICM Rio, 2018" width="400" height="267" />
<p>Modern ICMs attract thousands of delegates and are lavish affairs. This picture was taken at the opening ceremony at the 2018 ICM in Rio de Janeiro.</p>
</div>
<!-- Image in public domain -->
<p>Overall, the International Congresses of Mathematicians provide an excellent means of studying the development of mathematics in the twentieth century: not only can we trace its technical developments and its trends by looking at the choices of plenary speakers, but we can also investigate the ways in which its conduct was affected by events in the wider world, and thereby see that mathematics is indeed a part of global culture.</p>
<p>Since the end of the cold war the ICMs have ventured outside of Europe and North America.
A total of five have now taken place on non-European or North American soil: in Tokyo (1990), Beijing (2002), <a href="/content/category/tags/icm">Hyderabad</a> (2010), <a href="/content/category/tags/icm-2014">Seoul</a> (2014), and <a href="/content/icm-2018">Rio de Janeiro</a> (2018). The Rio ICM, which took place in August this year, was also the first to be staged in the Southern hemisphere. In 2022 the ICM will return to the country whose mathematicians have perhaps suffered most from political tensions as far as their ability to travel is concerned: it will be <a href="/content/icm-2022-bringing-nations-together">hosted by Saint Petersburg, Russia</a>.</p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2018/ICM_history/hollings.jpg" alt="Christopher Hollings" width="200" height="153" />
<p></p>
</div>
<p>Christopher Hollings is a departmental lecturer in the Oxford
Mathematical Institute, where he teaches the history of mathematics,
and a Senior Research Fellow of The Queen's College, Oxford. He
researches various topics from the history of nineteenth and twentieth
century mathematics, including the impact of politics on mathematics,
and the development of abstract algebra.</p>
</div></div></div>Fri, 31 Aug 2018 15:28:21 +0000Marianne7100 at https://plus.maths.org/contenthttps://plus.maths.org/content/politics-and-transcendental-numbers#commentsMaths in a minute: Transcendental numbers (and politics)
https://plus.maths.org/content/transcendental-numbers-and-politics
<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/gelfond_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">Christopher Hollings</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"><h3>What are transcendental numbers?</h3>
<p>Transcendental numbers are defined in contrast to <em>algebraic numbers</em>: we say that a number is algebraic if it is the solution of a <a href="https://www.mathsisfun.com/algebra/polynomials.html">polynomial equation</a> all whose coefficients are integers, that is to say, an equation consisting of a sum of powers of the unknown <img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0001.png" alt="$x$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> which can be multiplied by integers. Examples are <table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0002.png" alt="\[ 3x^2 + 2x +1 = 0 \]" style="width:122px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> or <table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0003.png" alt="\[ 5x^5 + 6x^3 + 7x + 8 = 0. \]" style="width:171px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table> Thus, for example, the number <img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0004.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> is algebraic because it is the solution of the equation <img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0005.png" alt="$x-2 = 0$" style="vertical-align:0px;
width:68px;
height:12px" class="math gen" />. In a similar way, any whole number or any fraction (a <em>rational number</em>) is also algebraic: <img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0006.png" alt="$1/2$" style="vertical-align:-4px;
width:23px;
height:16px" class="math gen" /> , for instance, is the solution of <img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0007.png" alt="$2x-1 = 0$" style="vertical-align:0px;
width:76px;
height:12px" class="math gen" />. Another example of an algebraic number is <img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0008.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" />: this is a solution of the equation <img src="/MI/051a95e33a690cbe190072c01dfdff65/images/img-0009.png" alt="$x^2-2 = 0$" style="vertical-align:0px;
width:75px;
height:14px" class="math gen" />.</p>
<div class="rightimage" style="width: 250px;"><img src="/issue21/xfile/lindemann_small.jpg" alt="Ferdinand von Lindemann" width="250" height="345" /><p>Ferdinand von Lindemann.</p>
</div>
<p> Any number that is not algebraic is called transcendental. Since every rational number is algebraic, it follows that every transcendental number is necessarily irrational (that is, not rational). But not every irrational number is transcendental: take <img src="/MI/fe34a52857184efc7ae3bdb4f4cca715/images/img-0001.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> for example (see <a href="/content/maths-minute-square-root-2-irrational"><em>Maths in a minute: The square root of 2 is irrational</em></a>).</p>
<p>Probably the most famous transcendental number is <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> (another is <img src="/MI/ca429a11c2d5253b1d43cba0a0da4053/images/img-0001.png" alt="$e$" style="vertical-align:0px;
width:7px;
height:7px" class="math gen" />; see <a href="/content/where-does-e-come-and-what-does-it-do-0"><em>Where does e come from and what does it do?</em></a>). Even before <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> was proven to be transcendental, mathematicians had long had the creeping suspicion that it was somehow unlike most of the other numbers that they encountered. It was certainly known that <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> is irrational, but its weirdness as a number seemed to go beyond this.</p>
<p> Despite the fact that the notions of algebraic and transcendental numbers had first been defined by <a href="https://www.biography.com/people/leonhard-euler-21342391">Leonhard Euler</a> in the 18th century, it was not until 1882 that the German mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Lindemann.html">Ferdinand von Lindemann</a> demonstrated that <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> is in fact transcendental. As well as proving once and for all that it is not possible to square the circle (see <a href="/content/os/issue21/xfile/index"><em>Mathematical mysteries: Transcendental meditation</em></a>), this also explained why <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> seemed so unlike other numbers: because we can't write down equations of which they are solutions, transcendental numbers are harder to "get hold of" than algebraic ones. In essence, an equation for a number provides us with a finite process by which we can construct that number; in the case of transcendental numbers, we have no such process.</p>
<h3>Hilbert's seventh problem</h3>
<p>
At the end of the nineteenth century, <img src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" alt="$\pi $" style="vertical-align:-1px;
width:10px;
height:9px" class="math gen" /> was one of only very few examples of transcendental numbers that were known. And yet mathematicians knew that there are in fact very many more transcendental numbers than algebraic: the transcendental numbers are <em>uncountably infinite</em>, whereas the algebraic numbers are <em>countable</em> (see <em><a href="/content/maths-minute-counting-numbers">Maths in a minute: Counting numbers</a></em>). More examples of transcendental numbers were needed. Indeed, what would be more useful would be a new method for constructing examples of transcendental numbers. The respected German mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Hilbert.html">David Hilbert</a> took account of this need when, at the International Congress of Mathematicians (ICM) in 1900, he presented a now famous <a href="http://mathworld.wolfram.com/HilbertsProblems.html">list of problems</a> that he thought ought to be tackled by mathematicians in the coming decades. Finding a method for constructing transcendental numbers was the seventh problem on this list.</p>
<p>Hilbert had a conjecture for how this seventh problem might be solved. Suppose that <img src="/MI/01f740ee71aa13bae87193515a117973/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> is an algebraic number that is neither 0 nor 1 (to avoid trivial cases), and suppose that <img src="/MI/01f740ee71aa13bae87193515a117973/images/img-0002.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:11px" class="math gen" /> is an irrational algebraic number. Following on from a similar conjecture made by Euler (where <img src="/MI/01f740ee71aa13bae87193515a117973/images/img-0001.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:7px" class="math gen" /> was assumed merely to be rational), Hilbert conjectured that in this situation <img src="/MI/01f740ee71aa13bae87193515a117973/images/img-0003.png" alt="$a^ b$" style="vertical-align:0px;
width:15px;
height:14px" class="math gen" /> is necessarily transcendental. His seventh problem was to find a rigorous proof of this. If it could be proved to be true, then <img src="/MI/01f740ee71aa13bae87193515a117973/images/img-0004.png" alt="$2^\sqrt {2}$" style="vertical-align:0px;
width:26px;
height:17px" class="math gen" />, for example, would be transcendental, despite the fact that neither <img src="/MI/01f740ee71aa13bae87193515a117973/images/img-0005.png" alt="$2$" style="vertical-align:0px;
width:8px;
height:12px" class="math gen" /> nor <img src="/MI/01f740ee71aa13bae87193515a117973/images/img-0006.png" alt="$\sqrt {2}$" style="vertical-align:-2px;
width:21px;
height:17px" class="math gen" /> is itself transcendental. Mathematicians would thus at last have a means of constructing further transcendental numbers from better-known building blocks (that is, algebraic numbers).
</p>
<p>The person to finally provide a rigorous proof of the seventh problem was the Russian mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Gelfond.html">A. O. Gel'fond</a>. Spurred on by the proof in 1930 by his fellow Russian <a href="https://en.wikipedia.org/wiki/Rodion_Kuzmin">Rodion Kuzmin</a> that <img src="/MI/c14ba8135f867d140d2aa9ebd84320e3/images/img-0001.png" alt="$2^\sqrt {2}$" style="vertical-align:0px;
width:26px;
height:17px" class="math gen" /> is indeed transcendental, Gel'fond had considered the more general case. He published a full solution in 1934.</p>
<div class="leftimage" style="width: 300px;"><img src="/issue21/xfile/gelfond_small.jpg" alt="Aleksandr Gel'fond" width="300" height="334" /><p>Aleksandr Gel'fond.</p>
</div>
<h3>Why politics?</h3>
<p>A proof of such an important conjecture would normally afford its author centre stage in international mathematics, but Gel'fond became a victim of political tensions. He had been invited to present his proof at the International Congress of Mathematicians in Oslo in 1936. That particular congress, however, proved a melting pot of different agendas. While mathematicians from Nazi-led Germany turned out in force to showcase the best of "Aryan mathematics", the Soviet delegation was conspicuous by its absence. In an atmosphere in which scientists' foreign connections were increasingly regarded with suspicion by the Soviet authorities, each of the delegates expected from the USSR, including Gel'fond, had been denied permission to travel. The world's mathematicians had to be content merely to read Gel'fond's solution of Hilbert's problem (it had been published in French), rather than see it presented at a blackboard by the author. (You can find out more about international maths and politics in <a href="/content/politics-and-transcendental-numbers">this article</a>.)
</p>
<p>Gel'fond was finally able to attend an ICM in 1966, when it was held in his home town of Moscow, although he does not appear to have delivered a lecture. As far as we know, this is the only ICM that he ever attended. The congress was certainly attended by a small number of mathematicians who had been in Oslo thirty years earlier, but we can only speculate about the interactions that resulted when they were finally able to meet Gel'fond in Moscow. It is interesting to note that the solution of a generalisation of Hilbert's seventh problem, sometimes termed the <em>Gel'fond conjecture</em>, earned a Fields Medal, one of the most prestigious prizes in mathematics, for the English mathematician <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Baker_Alan.html">Alan Baker</a> at the 1970 ICM in Nice.</p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" style="max-width: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2018/ICM_history/hollings.jpg" alt="Christopher Hollings" width="200" height="153" />
<p></p>
</div>
<p>Christopher Hollings is a departmental lecturer in the Oxford
Mathematical Institute, where he teaches the history of mathematics,
and a Senior Research Fellow of The Queen's College, Oxford. He
researches various topics from the history of nineteenth and twentieth
century mathematics, including the impact of politics on mathematics,
and the development of abstract algebra.</p>
</div></div></div>Fri, 31 Aug 2018 10:20:40 +0000Marianne7101 at https://plus.maths.org/contenthttps://plus.maths.org/content/transcendental-numbers-and-politics#commentsThe ICM 2018
https://plus.maths.org/content/icm-2018
<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/icm_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>The International Congress of Mathematicians (ICM), which takes place every four years, is the highlight of the mathematical calendar. It's the biggest maths conference of them all, attracting thousands of participants, and also sees the awards of some very prestigious prizes, including the famous Fields medal.</p>
<p>The ICM 2018 took place in Rio de Janeiro at the beginning of August. We were lucky enough to be there, to interview some of the mathematicians who took part and to cover their work. Below are the articles, podcasts and videos we have produced featuring the winners of some of the prizes. To see all of our coverage, click <a href="/content/tags/icm-2018>here</a>.</p>
<p>We are very grateful for the generous support of the <a href="https://www.lms.ac.uk/">London Mathematical Society</a> and the <a href="https://ima.org.uk/">Institute of Mathematics and its Applications</a>.</p>
<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/birkar_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/test-1-0">The Fields medal 2018: Caucher Birkar</a> — Caucher Birkar has been awarded the Fields medal for his contribution to algebraic geometry.</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/%5Buid%5D/%5Bsite-date%5D/af_icon_0.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/test-2-0">The Fields medal 2018: Alessio Figalli</a> — Alessio Figalli has been awarded the Fields medal for his contributions, among other things, to optimal transport theory. You can see a video interview with Figalli <a href="/content/interview-alessio-figalli">here</a> and listen to the interview as a podcast <a href="/content/alessio-figalli-podcast">here</a>.</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/%5Buid%5D/%5Bsite-date%5D/ps_icon.png" alt="" width="100" height="100" /> </div><p><a href="/content/ps">The Fields medal 2018: Peter Scholze</a> — Peter Scholze has received the Fields medal 2018 for transforming arithmetic algebraic geometry.</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/%5Buid%5D/%5Bsite-date%5D/av_icon.png" alt="" width="100" height="100" /> </div><p><a href="/content/AV">
The Fields medal 2018: Akshay Venkatesh</a> —
Akshay Venkatesh has been awarded the Fields medal 2018 for his work exploring the boundaries of number theory. You can see video interviews with Venkatesh <a href="/content/interview-akshay-venkatesh">here</a> and listen to a podcast <a href="/content/node/7092">here</a>.</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/%5Buid%5D/%5Bsite-date%5D/constantinos_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/elusive-equilibria">
The Nevanlinna prize 2018: Constantinos Daskalakis </a> —
The Nevanlinna prize winner Constantinos Daskalakis explains why equilibrium may be unattainable and why it's good to be constructive. You can see a video interview with Daskalakis <a href="/content/nevenlinna-prize-2018-constantinos-daskalakis">here</a> and listen to the interview as a podcast <a href="/content/node/7090">here</a>.</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/%5Buid%5D/%5Bsite-date%5D/nesin_icon.png" alt="" width="100" height="100" /> </div><p><a href="/content/leelavati-prize-2018-ali-nesin">
The Leelavati prize 2018: Ali Nesin</a> —
Ali Nesin has been awarded the 2018 Leelavati prize for creating a mathematical paradise for Turkish students and the world's mathematicians.</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/%5Buid%5D/%5Bsite-date%5D/mk_icon.png" alt="" width="100" height="100" /> </div><p><a href="/content/chern-medal-2018-masaki-kashiwara">
The Chern medal 2018: Masaki Kashiwara</a> —
Masaki Kashiwara wins the Chern medal for his "outstanding and foundational
contributions to algebraic analysis and representation theory sustained over a period of
almost 50 years."</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/donoho_icon.jpg" alt="" width="100" height="100" /> </div><p><a href="/content/gauss-prize-2018-david-donoho">
The Gauss prize 2018: David Donoho</a> —
If you have ever been in an MRI scanner you'll appreciate the work that Donoho is being honoured for with this important prize. In this video Donoho explains what his work is about and why he has an important message to mathematicians. You can also listen to this interview as a <a href="/content/node/7086">podcast</a>.</p></div>
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</div></div></div>Wed, 29 Aug 2018 12:07:47 +0000Marianne7099 at https://plus.maths.org/contenthttps://plus.maths.org/content/icm-2018#commentsIvan Smith: The podcast
https://plus.maths.org/content/node/7089
<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/ivan_icon_0.png" 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 class="rightimage" style="width: 250px;"><img src="/content/sites/plus.maths.org/files/podcast/ICM2018/ivan_frontpage_1.png" alt="" width="250" height="141" />
<p>Ivan Smith.</p>
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<p><a href="https://www.dpmms.cam.ac.uk/people/is200/">Ivan Smith</a> is Professor of Geometry at the University of Cambridge and gave an invited lecture at the <a href="http://www.icm2018.org/portal/en/home">International Congress of Mathematicians 2018</a> (ICM). We talk to him about his work and what he likes about the ICM. </p>
<p><em>You can also watch this interview in a <a href="/content/interview-ivan-smith">video</a>.</em></p>
</div></div></div><div class="field field-name-field-remote-encl field-type-file field-label-inline clearfix clearfix"><div class="field-label">Podcast download link: </div><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="Audio icon" title="audio/mpeg" src="/content/modules/file/icons/audio-x-generic.png" /> <a href="https://plus.maths.org/content/sites/plus.maths.org/files/podcast/ICM2018/ivan.mp3" type="audio/mpeg; length=3722996">ivan.mp3</a></span></div></div></div>Wed, 08 Aug 2018 18:42:02 +0000Marianne7089 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/7089#commentsAn interview with Jack Thorne
https://plus.maths.org/content/interview-jack-thorne
<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/jack_icon.png" 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>
"I like the way coincidence plays a part in the subject." Jack Thorne, from the University of Cambridge, gave an invited lecture at the <a href="http://www.icm2018.org/portal/en/home">International Congress of Mathematicians 2018</a> on number theory. He talks to us about how his work is a little like "faceting a gemstone."
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<p><em>This content was produced in a collaboration with the <a href="https://www.lms.ac.uk">London Mathematical Society</a>.</em></p> </div></div></div>Wed, 08 Aug 2018 18:37:54 +0000Rachel7088 at https://plus.maths.org/contenthttps://plus.maths.org/content/interview-jack-thorne#commentsThe ICM 2022: The podcast
https://plus.maths.org/content/node/7085
<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/peters_icon_1.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 class="rightimage" style="width: 250px;"><img src="/content/sites/plus.maths.org/files/podcast/ICM2018/russians_frontpage_1.png" alt="" width="250" height="141" />
<p>Stanislav Smirnov and Andrei Okounkov.</p>
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<p>The next International Congress of Mathematicians will take place in 2022 Saint Petersburg. We were lucky to catch up with two of its organisers, Stanislav Smirnov and Andrei Okounkov, during a very busy break at the current ICM, to ask them why they wanted to bring the ICM to Russia, what exactly they are planning, and what they have learnt from Rio.</p>
<p><em>You can also watch this interview as a <a href="/content/icm-2022-bringing-nations-together">video</a>.</em></p>
</div></div></div><div class="field field-name-field-remote-encl field-type-file field-label-inline clearfix clearfix"><div class="field-label">Podcast download link: </div><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="Audio icon" title="audio/mpeg" src="/content/modules/file/icons/audio-x-generic.png" /> <a href="https://plus.maths.org/content/sites/plus.maths.org/files/podcast/ICM2018/smirnovokounkov.mp3" type="audio/mpeg; length=9147013">smirnovokounkov.mp3</a></span></div></div></div>Wed, 08 Aug 2018 18:18:40 +0000Marianne7085 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/7085#commentsNalini Joshi: The podcast
https://plus.maths.org/content/node/7084
<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/nalini_icon_1.png" 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><a href="http://wp.maths.usyd.edu.au/nalini/">Nalini Joshi</a> has just been elected Vice President of the <a href="https://www.mathunion.org/">International Mathematical Union</a> (IMU), which organises the International Congress of Mathematicians and awards the Prizes. We talk to her about the work of the IMU, her own work in mathematics, and the <a href="http://www.sciencegenderequity.org.au/">SAGE programme</a> she has helped set up to improve gender equity in STEM subjects in Australia.</p>
<p><em>You can also <a href="/content/interview-nalini-joshi">watch this interview as a video</a>.</em></p>
</div></div></div><div class="field field-name-field-remote-encl field-type-file field-label-inline clearfix clearfix"><div class="field-label">Podcast download link: </div><div class="field-items"><div class="field-item even"><span class="file"><img class="file-icon" alt="Audio icon" title="audio/mpeg" src="/content/modules/file/icons/audio-x-generic.png" /> <a href="https://plus.maths.org/content/sites/plus.maths.org/files/podcast/ICM2018/nalini_podcast.mp3" type="audio/mpeg; length=11055399">nalini_podcast.mp3</a></span></div></div></div>Wed, 08 Aug 2018 18:09:01 +0000Marianne7084 at https://plus.maths.org/contenthttps://plus.maths.org/content/node/7084#commentsAn interview with Clément Mouhot
https://plus.maths.org/content/interview-clement-mouhot
<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/clement_icon.png" 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>
"Mathematics is a science of structures and the main equations of physics are some of the most beautiful structures we can find." Clément Mouhot, from the University of Cambridge, gave an invited lecture at the <a href="http://www.icm2018.org/portal/en/home">International Congress of Mathematicians 2018</a> (ICM). We talk to him about his work on differential equations and mathematical physics.
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<p><em>This content was produced in a collaboration with the <a href="https://www.lms.ac.uk">London Mathematical Society</a>.</em></p> </div></div></div>Wed, 08 Aug 2018 17:45:49 +0000Rachel7083 at https://plus.maths.org/contenthttps://plus.maths.org/content/interview-clement-mouhot#commentsAn interview with Ivan Smith
https://plus.maths.org/content/interview-ivan-smith
<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/ivan_icon.png" 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><a href="https://www.dpmms.cam.ac.uk/people/is200/">Ivan Smith</a> is Professor of Geometry at the University of Cambridge and gave an invited lecture at the <a href="http://www.icm2018.org/portal/en/home">International Congress of Mathematicians 2018</a> (ICM). We talk to him about his work and what he likes about the ICM. </p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/lOqJaHlXS6U?rel=0" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen></iframe>
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<p><em>This content was produced in a collaboration with the <a href="https://www.lms.ac.uk">London Mathematical Society</a>.</em></p> </div></div></div>Wed, 08 Aug 2018 17:40:54 +0000Marianne7082 at https://plus.maths.org/contenthttps://plus.maths.org/content/interview-ivan-smith#comments