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enMaths in a minute: The prime number theorem
https://plus.maths.org/content/maths-minute-prime-number-theorem
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/prime_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 prime numbers are those integers that can only be divided by themselves and 1. The first seven are </p>
<p><img src="/MI/ebb2ebd9364e559eba958bac0a699b3c/images/img-0001.png" alt="$2, 3, 5, 7, 11, 13, 17.$" style="vertical-align:-4px;
width:129px;
height:16px" class="math gen" /></p>
<div class="rightimage" style="max-width: 288px"><img src="/content/sites/plus.maths.org/files/articles/2021/PNT/primes.jpg" alt="Sieve of Eratosthenes illustration" width="288" height="204" />
<p>This is an illustration of the <em>sieve of Eratosthenes</em>, which is designed to catch prime numbers. You can find out more about the sieve <a href="https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes" target="blank">here</a>. Adapted from a figure by <a href="https://en.wikipedia.org/wiki/File:Sieve_of_Eratosthenes_animation.gif" target="blank">SKopp</a>, <a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en" target="blank">CC BY-SA 3.0</a>.</p>
</div>
<p>Every other positive integer can be written as a product of prime numbers in a unique way — for example <img src="/MI/8c0ab8cb00cdd94329fa45498283299b/images/img-0001.png" alt="$30=2\times 3\times 5 $" style="vertical-align:0px;
width:104px;
height:12px" class="math gen" /> — so the prime numbers are like basic building blocks that other integers can be constructed from. This is why people find them interesting.</p>
<p>We have known for thousands of years that there are infinitely many prime numbers (see <a href="/content/maths-minute-how-many-primes" target="blank">here</a> for a proof), but there isn't a simple formula which tells us what they all are. Powerful computer algorithms have enabled us <a href="/content/tags/prime-number-search" target="blank">to find larger and larger primes</a>, but we will never be able to write down all of them.</p>
<p>The prime number theorem tells us something about how the prime numbers are distributed among the other integers. It attempts to answer the question "given a positive integer <img src="/MI/708763d86ca5755dd2c7f3c594d16d38/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" />, how many integers up to and including <img src="/MI/708763d86ca5755dd2c7f3c594d16d38/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" /> are prime numbers"? </p>
<p><p>The prime number theorem doesn’t answer this question precisely, but instead gives an approximation. Loosely speaking, it says that for large integers <img src="/MI/1008c6c596c408bfac13a815fe5e10c8/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" />, the expression </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/1008c6c596c408bfac13a815fe5e10c8/images/img-0002.png" alt="\[ \frac{n}{\ln {(n)}} \]" style="width:40px;
height:35px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> is a good estimate for the number of primes up to and including <img src="/MI/1008c6c596c408bfac13a815fe5e10c8/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" />, and that the estimate gets better as <img src="/MI/1008c6c596c408bfac13a815fe5e10c8/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" /> gets larger. Here <img src="/MI/1008c6c596c408bfac13a815fe5e10c8/images/img-0003.png" alt="$\ln {(n)}$" style="vertical-align:-4px;
width:37px;
height:17px" class="math gen" /> is the natural logarithm of <img src="/MI/1008c6c596c408bfac13a815fe5e10c8/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" />, which you can find on your calculator. </p></p>
<p>As an example, let's take <img src="/MI/f7827bd91265c7625e22479cd72caa52/images/img-0001.png" alt="$n=1,000$" style="vertical-align:-4px;
width:71px;
height:15px" class="math gen" />. The actual number of primes up to and including <img src="/MI/f7827bd91265c7625e22479cd72caa52/images/img-0002.png" alt="$1,000$" style="vertical-align:-4px;
width:38px;
height:15px" class="math gen" /> (which you can look up <a href="https://en.wikipedia.org/wiki/List_of_prime_numbers#The_first_1000_prime_numbers" target="blank">in this list</a>) is <img src="/MI/31bf0721cdbbfe41dbd0d28c0c983172/images/img-0001.png" alt="$168$" style="vertical-align:0px;
width:23px;
height:11px" class="math gen" />. Our estimate is</p>
<p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0001.png" alt="\[ \frac{1,000}{\ln {(1,000)}} = \frac{1,000}{6.9}\approx 145, \]" style="width:189px;
height:38px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> which is a pretty good approximation. </p><p>However, to be precise about what the prime number theorem tells us, we need to say what we mean by "a good estimate". The prime number theorem does <em>not</em> say that for large values of <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0002.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" /> the difference between the true value and our approximation is close to <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0003.png" alt="$0.$" style="vertical-align:0px;
width:11px;
height:11px" class="math gen" /> Instead, it tells us something about the question "what’s the approximation as a percentage of the true value"? </p><p>To go back to our example of <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0004.png" alt="$n=1,000$" style="vertical-align:-4px;
width:71px;
height:15px" class="math gen" />, the true value was <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0005.png" alt="$168$" style="vertical-align:0px;
width:23px;
height:11px" class="math gen" /> and the approximation was <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0006.png" alt="$145$" style="vertical-align:0px;
width:23px;
height:12px" class="math gen" />. Therefore, the approximation constitutes a proportion of </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0007.png" alt="\[ \frac{145}{168}=0.86, \]" style="width:82px;
height:35px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> of the true result, which translates to 86%. Not bad. </p><p>For <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0008.png" alt="$n=100,000$" style="vertical-align:-4px;
width:88px;
height:15px" class="math gen" /> the actual number of primes up to and including <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0009.png" alt="$100,000$" style="vertical-align:-4px;
width:55px;
height:15px" class="math gen" /> is <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0010.png" alt="$9,592$" style="vertical-align:-4px;
width:39px;
height:16px" class="math gen" />, so that’s the true value, and the estimate was </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0011.png" alt="\[ \frac{100,000}{\ln {(100,000)}}=\frac{100,000}{11.5}\approx 8,686. \]" style="width:237px;
height:38px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>Therefore in this case the estimate constitutes a proportion of </p><table id="a0000000005" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0012.png" alt="\[ \frac{8,686}{9,592}=0.9, \]" style="width:89px;
height:38px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> of the true value. This translates to 90%, so here the estimate is better than for <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0004.png" alt="$n=1,000$" style="vertical-align:-4px;
width:71px;
height:15px" class="math gen" />. </p><p>Generally, the prime number theorem tells us that for large <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0002.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" /> the approximation <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0013.png" alt="$n/\log {(n)}$" style="vertical-align:-4px;
width:66px;
height:17px" class="math gen" /> is nearly 100% of the true value. In fact you can get it to be as close to 100% as you like by choosing a large enough <img src="/MI/dde04eed10e07fac26d3f4e5ba5784e8/images/img-0014.png" alt="$n.$" style="vertical-align:0px;
width:13px;
height:8px" class="math gen" /> </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/PNT/primes.png" alt="A tree diagram" width="500" height="363"/>
<p style="max-width: 600px;">The red curve shows the number of primes up to and including <em>n</em>, where <em>n</em> is measured on the horizontal axis. The blue curve gives the value of <em>n</em>/ln(<em>n</em>). The difference between true result and approximation increases as <em>n</em> grows, but the ratio between the two values tends to 1.</p>
</div><!-- Image made by MF-->
<p>To state the prime number theorem in its full mathematical glory using mathematical notation, let’s first write <img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0001.png" alt="$\pi (n)$" style="vertical-align:-4px;
width:31px;
height:17px" class="math gen" /> for the number of primes that are smaller than, or equal to, <img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0002.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" />. The theorem says that </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0003.png" alt="\begin{equation} \lim _{n \to \infty } \frac{(n/\ln {(n)})}{\pi (n)} = 1.\end{equation}" style="width:148px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>1</span>)</span></td>
</tr>
</table><p> If you know a bit of calculus you will know that you can swap the numerator and denominator here, so expression (1) is equivalent to </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0004.png" alt="\begin{equation} \lim _{n \to \infty } \frac{\pi (n)}{(n/\ln {(n)})}= 1.\end{equation}" style="width:148px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>2</span>)</span></td>
</tr>
</table><p>The prime number theorem is usually stated using the second expression (2). It is also sometimes written as </p><table id="a0000000004" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0005.png" alt="\[ \frac{\pi (n)}{n/\ln {(n)}} \sim 1, \]" style="width:97px;
height:39px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> which in spoken language is "<img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0001.png" alt="$\pi (n)$" style="vertical-align:-4px;
width:31px;
height:17px" class="math gen" /> is <em>asymptotic</em> to <img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0006.png" alt="$n/\log {(n)}$" style="vertical-align:-4px;
width:66px;
height:17px" class="math gen" /> as <img src="/MI/7069ef914243c02fb69279220ae0e561/images/img-0002.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" /> tends to infinity." </p>
<hr/>
<p><em>This article was produced as part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI) – you can find all the content from the collaboration <a href="/content/ini">here</a>.
</p><p>
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more. </em></p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" alt="INI logo" width="400" height="51" />
<p style="max-width: 400px;"></p></div></div></div></div>Wed, 25 Aug 2021 14:10:16 +0000Marianne7489 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-prime-number-theorem#commentsCOVID-19 and universities: What do we know?
https://plus.maths.org/content/going-back-uni
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/laws_equations_icon_0.jpeg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-hidden"><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>"This year we will have students going into the third year of their degree who have not had a single complete year of normal face-to-face education," says epidemiologist <a href="https://warwick.ac.uk/fac/sci/lifesci/people/mtildesley/" target="blank">Mike Tildesley</a> from the University of Warwick. It's a stark illustration of just how much the pandemic has disrupted student life, depriving students of what should have been one of the most formative, and fun, periods of their lives. </p>
<div class="rightshoutout"><p>See <a href="/content/tags/covid-19">here</a> for all our coverage of the COVID-19 pandemic.</div>
<p>This year most universities will no doubt do their best to give their students the experience they deserve. But even though there no longer is a government mandate on how to manage COVID-19 in higher education, many are still likely to employ some kind of measures to protect their students and the surrounding communities from COVID-19.</p>
<h3>Better the virus you know</h3>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/UnlockingHE/lecture_hall.jpg" alt="Lecture hall" width="350" height="222" /><p>In the last academic year many lecture theatres remained empty.</p>
</div>
<!-- Image from Adobe Stock -->
<p>In contrast to 2020, when there was very little information to go on, universities will now have a body of research to guide their decisions. Tildesley, together with thirteen other epidemiologists and mathematicians, has recently published a <a href="https://royalsocietypublishing.org/doi/10.1098/rsos.210310" target="blank">comprehensive study</a>, which analyses data from last year's autumn term to see what actually happened at universities in terms of COVID-19, and uses mathematical modelling to see whether interventions could help reduce outbreaks.</p>
<p>Although this year is different, in both good ways (vaccinations) and bad ways (the <a href="/content/variant-concern-what-do-we-know" target="blank">Delta variant</a>), the lessons learnt should on the whole still hold. "Things like vaccination will probably change some of the quantitative aspects of our results, but not necessarily the qualitative aspects," says Tildesley.</p>
<p>The study is a result of a concerted effort by various research groups that started back in the summer of 2020, with a virtual workshop hosted by the <a href="https://www.vkemsuk.org/home" target="blank">Virtual Forum for Knowledge Exchange in the Mathematical Sciences</a> (V-KEMS) and the <a href="https://gateway.newton.ac.uk/" target="blank">Newton Gateway to Mathematics</a>. Two <a href="/content/going-back-uni-during-pandemic" target="blank">follow-up events</a> were run as part of the <a href="https://www.newton.ac.uk/event/idp/" target="blank">Infectious dynamics of pandemics programme</a> at the Isaac Newton Institute in Cambridge, and a working group continued to meet regularly after that. Seven of the study's authors, including Tildesley, are members of the <a href="https://maths.org/juniper/" target="blank">JUNIPER modelling consortium</a>. </p>
<p>The study suggests answers to a number of questions both students and staff of universities will find interesting. You can jump straight to the question you are interested in by clicking on the links below, or just keep reading to see the answers to them all.</p>
<ul><li><a href="/content/going-back-uni#import">How did (and will) university outbreaks originate?
</a></li>
<li><a href="/content/going-back-uni#halls">Was it useful to split halls of residence into smaller households?</a></li><li><a href="/content/going-back-uni#spill">Did university outbreaks spill over into surrounding communities?</a></li><li><a href="content/going-back-uni#stagger">Would it be useful to stagger students' return to universities?</a></li><li><a href="/content/going-back-uni#test">Would routine testing be useful, especially in the face of a more transmissible variant?</a></li>
<li><a href="/content/going-back-uni#thisyear">So what will happen this year?</a></li></ul>
<a name="import"></a>
<h3>How did (and will) university outbreaks originate?</h3>
<p>When the autumn term started last year, the number of COVID-19 cases was on the rise in the UK. Following government advice, most universities offered a mixture of face-to-face and online teaching. Many also offered a testing regime and segmented halls of residence into smaller households.</p>
<p>Despite these restrictions, some universities still saw large outbreaks. You probably remember the newspaper headlines, featuring desperate students stuck in self-isolation and complaining about inadequate food supplies. Luckily these weren't the norm. The recent analysis shows significant variation between institutions, raising the question of how those large outbreaks had originated.</p>
<p>The mass migration at the start of term had always been a prime suspect in this context, so the team behind the recent study estimated the proportion of students likely to have arrived at each university already infected, based on COVID-19 data from the regions students came from. A simple probabilistic model then suggested that this proportion is linked to the probability of the university seeing an outbreak (which was defined as having more than a certain threshold number of cases). In other words, the pattern of observed outbreaks across universities was consistent with the idea that these outbreaks had been seeded by students arriving at the start of term already infected. (If you'd like to look at some of the maths involved in this model, see <a href="/content/going-back-uni#model" target="blank">below</a>.) </p>
<p>While the model is simple and comes with caveats the result suggests a lesson worth keeping in mind for the upcoming autumn term. "The likelihood of seeing campus outbreaks is going to be dependent on the prevalence in the communities students are coming from," says Tildesley. "The size of any outbreak will depend on the transmissibility of the virus — we now have a more transmissible variant than twelve months ago. But we now also have a proportion of students being vaccinated. Those factors will trade off against each other."</p>
<a name="halls"></a>
<h3>Halls of residence</h3>
<p>Once COVID-19 has entered a university, halls of residence
provide an obvious breeding ground. To keep things under control when students arrived last autumn many universities divided their halls into smaller households: students were not supposed to socialise with anyone outside their own household and the whole household would go into isolation once a member showed symptoms or tested positive.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/UnlockingHE/laws_equations.jpg" alt="Apple, books and blackboard" width="350" height="245" /><p>What is the academic year 2021/2022 going to look like?</p>
</div><!-- image from Plus article 'who made the laws of nature?' --><p>At the time these measures were based on intuition: there wasn't much evidence on how the risk of transmission within halls compares to the risk in the community, and whether segmenting halls would make any difference. </p>
<p>To gain some clarity, the team behind the recent study used as an example the data from one particular university, which manages 19 halls in total and had divided each into households of up to 16 members. They used a statistical technique called <em>multi-variate regression</em> (a version of <a href="content/maths-minute-linear-regression" target="blank"><em>linear regression</em></a> which works when there are more than one variable) to figure out to what extent various factors would impact on the proportion of people in the halls that are infected by an incoming infection. (This proportion is called the <em>secondary attack rate</em>.) The factors the researchers considered here included the overall size of the hall, the median size of households in the hall, the proportion of students sharing bathrooms, and the proportion of students on medical courses (as a proxy for students at higher risk due to their course).</p>
<p>The result is perhaps surprising: only the overall size of the hall and the proportion of students sharing bathrooms were associated with the secondary attack rate in the final multi-variate analysis. Although, as ever, there are limitations to the analysis, this suggests that splitting halls into households is of limited value when it comes to reducing the risk of transmission. A lot more effective, the analysis suggests, would be to only partially fill the hall and coordinate things so that the use of shared spaces is limited.</p>
<a name="spill"></a><h3>Spilling over</h3>
<p>
Last year's outbreaks on university campuses condemned many students to self-isolation in small rooms far away from home, with awful consequences on some people's mental health. But with most students being young, fit and healthy, the risk of them becoming seriously ill or dying was low.</p>
<p> The same couldn't be said for the surrounding community, however. One of the worries about university outbreaks was that they would spill over, posing a high risk to vulnerable people living nearby. "One of the things we were particularly interested in was trying to identify any potential links between universities and communities," says Tildesley. "So that's looking for signals that might indicate whether a rise in cases in surrounding communities was in any way linked to a rise in cases on university campuses."</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/UnlockingHE/crw_5118.jpg" alt="CMS, Cambridge" width="350" height="233" /><p>The Centre for Mathematical Sciences at the University of Cambridge (home of <em>Plus</em>). In Cambridge there are many interactions between students and the community.</p>
</div>
<!-- Image from Faculty Site where it appeared without a credit -->
<p>Since students mix and interact with the local community, such links are hard to tease out from the data, but the team decided to use two approaches: one looked at geographical data on confirmed COVID-19 cases in areas with a high density of students and those nearby, while the other took account of people's ages. Those who were 18 to 24 years old served as proxy for students and other age groups represented the rest of the community.</p><p>Careful analysis of these data sets, and information on outbreaks provided by universities, revealed a complex picture. "The conclusion we found here is that we did see evidence of spill-over in some, but not all cases," says Tildesley. "So there is no direct link there that applied in all settings."</p>
<p>The reason for this is probably that universities are unique in the way they sit within their local communities (for example if they are campus universities or not) and that different universities applied different COVID measures. In one case, the researchers suggest that a large outbreak did not spill over because the university in question routinely tested students whether they had symptoms or not. This meant that even those who did not show symptoms could be made to self-isolate, rather than unwittingly spreading their infection further.</p>
<a name="stagger"></a><h3>Staggering returns</h3>
<p>As autumn 2020 progressed into winter Christmas started to loom large. The prospect of students fanning out across the country for their holidays, only to return in January potentially infected, led the UK government to advise universities to stagger their students' return over a five week period. They were also to be offered tests on arrival.</p>
<p>The national lockdown imposed on January 4, 2021, prevented the staggering from being tried out, as most courses were then moved online. Instead of analysing data, the researchers therefore used mathematical models to predict the effect staggering returns might have on outbreaks at universities.</p>
<p> "What we found is that ultimately, over the course of the term, there was no particular impact of staggering on the overall number of cases you would see on campus," says Tildesley.
The work also showed that, while staggering can reduce the time students spend in self-isolation, and the peak number of students self-isolating on any given day, this only happens when the proportion of students testing positive is small. When it is high, then staggering can actually make things worse, as student households end up going into isolation several times.</p>
<p>The researchers used four different mathematical models to arrive at their results. Two of them (an <a href="/content/mathematics-diseases" target="blank">SIR model</a> and an <a href="/content/agm" target="blank">individual-based model</a>) were kept nice and simple, while the other two (a <a href="/content/how-can-maths-fight-pandemic" target="blank">compartmental model</a> and a model based on a network) included a little more complexity. These models simulated the growth or decline of infections in hypothetical populations of students under various assumptions on the exact nature of the staggering and testing policy, mixing patterns between students and the social groups they belong to, the chance a student tests positive, and so on. You can find the details on the assumptions used for each model in the <a href="https://royalsocietypublishing.org/doi/10.1098/rsos.210310" target="blank">paper</a>.</p>
<p>Interestingly, one model also considered students' behaviour — to be precise the extent to which they isolate when they are told to, get themselves tested and have their contacts traced. "The model suggests that adherence to test, trace and isolate is a much stronger driver in reducing transmission than staggering," says Tildesley. You can stagger returns as much as you like, if your students don't stick to test, trace and isolate rules, the effect will be washed out.</p>
<a name="test"></a><h3>Testing, testing, testing</h3>
<div class="rightimage"><img src="/content/sites/plus.maths.org/files/articles/2021/LFTs/adobestock_432022126_web.jpg" width="300" height="200" alt="lateral flow test"/><p style="max-width:300px">The NHS test kit (Image: <a href="https://stock.adobe.com/uk/images/st-albans-hertfordshire-england-5-may-2021-nhs-provide-coronavirus-lateral-flow-or-rapid-antigen-test-kits-for-people-to-self-test-for-covid-19-at-home/432022126">tommoh29</a>)</p></div>
<p>The January 2021 lockdown was imposed because of the emergence of the Alpha variant. This prompted the researchers to model how different testing regimes would fare in the face of more transmissible variants. They were particularly interested in regimes that routinely test students whether they have symptoms or not, and can therefore catch asymptomatic and pre-symptomatic cases. The model they used involved a layered network, with one layer representing students' household contacts and the other representing all their other contacts. The model was then used to gauge the impact on the number of positive cases under five different testing strategies: testing every 3, 7, 10 or 14 days, or no testing at all.</p>
<p>The results show what you might expect: that more frequent testing better controls the number of infections. What's somewhat shocking, though, is that once you're dealing with a more transmissible variant such as the Alpha variant, testing needs to be at its most frequent to prevent a major outbreak — and that means testing every student every three days. The currently dominant <a href="/content/variant-concern-what-do-we-know" target="blank">Delta variant</a> is of course even more transmissible than Alpha.</p>
<a name="thisyear"></a>
<h3>So what will happen this year?</h3>
<p>The biggest difference between this and last year is that now we have the vaccines. But since not all students will be fully vaccinated, and the vaccines aren't 100% effective, outbreaks on campuses are still possible. The results of the recent study still give important guidance on the qualitative nature of these outbreaks, their links to the community, and what measures might work to fight them.</p>
<p>"One of the key things we are trying to establish at the moment is the proportion of students that will be vaccinated, as that will obviously have an impact on what we might expect to see upon students returning come the end of September," says Tildesley." Of course the vaccination status of international students is also important, and could potentially drive what will happen." As mentioned above, the effect of a proportion of students being vaccinated will play off against the more transmissible Delta variant, and the amount of COVID in students' home regions will impact the number of students arriving at university infected. "What happens in September in the community is going to give us an indication of what we expect to see when universities reopen," says Tildesley.</p>
<div class="leftimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2020/vaccine/vaccine_normal.jpg" alt="Vaccine" width="350" height="239" /><p>The effect of vaccines will be payed off against the greater transmissibility of the Delta variant.</p>
</div>
<p>In general, Tildesley thinks that each university should take an approach tailored to its own unique conditions. "Universities are very diverse in the way they operate, so I think they need to take a risk-based approach," he says. "They need to base their decisions on the risk to their students and the potential benefits to their students."</p>
<p>"Clearly we want to prevent large outbreaks on student campuses, but we need to think about students' mental health as well. I know how important it is for students to experience student life and they have just not done that in the last two years. This is why we need to look at the risks associated with returning to face-to-face teaching and relaxation of the rules, and weigh them up against the benefits for students of these things for their own well-being and learning. There has to be some kind of balanced approach moving forward to try and get back to normality."</p><hr/>
<a name="model"></a>
<h3>A bit of maths...</h3>
<p><p>Here’s a way of figuring out whether the pattern of outbreaks at universities is consistent with the idea that they were seeded by students arriving infected at the beginning of term. First stipulate the number of cases that constitute an outbreak. In their paper the researchers used 200 cases, and then repeated their analysis for 400 cases. Next, estimate the number <img src="/MI/1b366650e3294de469516175e975bf9f/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" /> of incoming infected students for each university, and allocate universities into groups according to the value of <img src="/MI/1b366650e3294de469516175e975bf9f/images/img-0001.png" alt="$n$" style="vertical-align:0px;
width:10px;
height:8px" class="math gen" />: the first group contains universities with up to 10 incoming infections, the second with between 10 and 20 incoming infections, and so on. </p><p>From the data for each outbreak it is possible to estimate the probability <img src="/MI/1b366650e3294de469516175e975bf9f/images/img-0002.png" alt="$p$" style="vertical-align:-4px;
width:10px;
height:12px" class="math gen" /> that an incoming infection fails to produce an outbreak. If all the outbreaks were seeded by incoming infections, then the probability <img src="/MI/1b366650e3294de469516175e975bf9f/images/img-0003.png" alt="$P$" style="vertical-align:0px;
width:13px;
height:12px" class="math gen" /> that a university experiences an outbreak would be </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/1b366650e3294de469516175e975bf9f/images/img-0004.png" alt="\[ P=1-p^ n. \]" style="width:84px;
height:16px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> (Here it’s assumed that the probabilities of infections failing to produce an outbreak are independent of each other.) </p><p>Now for each of your groups (which were produced according to the number of incoming infections) calculate the fraction of universities in that group that experienced an outbreak. This then is an estimate of <img src="/MI/1b366650e3294de469516175e975bf9f/images/img-0003.png" alt="$P$" style="vertical-align:0px;
width:13px;
height:12px" class="math gen" /> for universities in that group. If these estimates for <img src="/MI/1b366650e3294de469516175e975bf9f/images/img-0003.png" alt="$P$" style="vertical-align:0px;
width:13px;
height:12px" class="math gen" /> look roughly like the theoretical probability calculated above, then this suggests that the pattern of outbreaks is consistent with the idea that they were seeded by incoming infections. </p></p><p>See the <a href="https://royalsocietypublishing.org/doi/10.1098/rsos.210310" target="blank">paper</a> for more details.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/HE/model.jpg" alt="Plots" width="600" height="344" /><p style="max-width: 600px;">These plots are taken from the <a href="https://royalsocietypublishing.org/doi/10.1098/rsos.210310" target="blank">paper</a>. They show how the theoretical probability calculated above (red curve) compares to the proportion of universities that had an outbreak (blue stars). In the left plot an outbreak was classified as more than 200 cases and in the right plot as more than 400 cases. Reproduced under <a href="https://creativecommons.org/licenses/by/4.0/">CC BY 4.0</a>.</p>
</div>
<hr/>
<h3>About this article</h3>
<div class="rightimage" style="max-width: 150px;"><img src="/content/sites/plus.maths.org/files/articles/2021/HE/mike_t.png" alt="Mike Tildesley" width="150" height="207" /><p>Mike Tildesley</p>
</div>
<p><a href="https://warwick.ac.uk/fac/sci/lifesci/people/mtildesley/" target="blank">Mike Tildesley</a> is Professor in the <a href="https://warwick.ac.uk/fac/cross_fac/zeeman_institute/">Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research</a> at the University of Warwick, a member of <a href="https://maths.org/juniper/" target="blank">JUNIPER</a>, and a member of the Scientific Pandemic Influenza Modelling group (SPI-M) that advises the government on the scientific aspects of the pandemic. You can listen to a podcast featuring Tildesley <a href="/content/mathematical-frontline-mike-tildesley" target="blank">here</a>.
</p>
<p>Tildesley was interviewed by <a href="/content/people/index.html#marianne">Marianne Freiberger</a>, Editor of <em>Plus</em>, in August 2021. She would like to thank <a href="https://warwick.ac.uk/fac/sci/maths/people/staff/dyson/">Louise Dyson</a> and <a href="https://warwick.ac.uk/fac/sci/maths/people/staff/ed_hill/" target-="blank">Ed Hill</a>, co-authors of the study and members of <a href="https://maths.org/juniper/" target="blank">JUNIPER</a>, for their help with this article. </p>
<p><em>This article was produced as part of our collaborations with <a href="https://maths.org/juniper/">JUNIPER</a>, the Joint UNIversity Pandemic and Epidemic Response modelling consortium, and the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI).</em></p>
<p><em> JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing question about the control of COVID-19. You can see more content produced with JUNIPER
<a href="/content/juniper">here</a>.</em></p>
<p><em>The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more. </em></p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2021/Juniper-logos/juniper-light-bg.png" alt="Juniper logo" width="400" height="83" />
<p style="max-width: 400px;"></p></div>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" alt="INI logo" width="400" height="51" />
<p style="max-width: 400px;"></p></div></div></div></div>Wed, 25 Aug 2021 09:10:35 +0000Marianne7498 at https://plus.maths.org/contenthttps://plus.maths.org/content/going-back-uni#commentsCelebrating a new pi record!
https://plus.maths.org/content/celebrating-new-pi-record
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon.jpeg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>This week a team from the University of Graubünden in Switzerland <a href="https://www.fhgr.ch/news/newsdetail/die-fh-graubuenden-kennt-pi-am-genauesten-weltrekord/" target="blank">announced</a> that they have calculated the decimal digits of the number <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/3e5a549093cf5ea72876460ff94ff15f/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to a record 62.8 trillion places. The calculation took 108 days and 9 hours, making it over three times faster than the calculation that gave us the last record, which had calculated the digits of <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/3e5a549093cf5ea72876460ff94ff15f/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to a mere 50 trillion places.</p>
<div class="rightimage"><img alt="Pi" width="350" height="191" class="b-lazy" data-src="/sites/plus.maths.org/files/articles/2012/budd/pi.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /><p>Here are the first few digits of the number π.</p></div>
<p>The number <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/343234984fd472dc9cb0c6c1c1f84569/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> is what you get when you divide a circle's circumference by its diameter and is the same for any circle, no matter how big or small. The number is <em>irrational</em>, which means it can't be written as a fraction. It also means that its decimal expansion is infinitely long and doesn't end in repeating blocks. It would be impossible to write down the entire decimal expansion of <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/343234984fd472dc9cb0c6c1c1f84569/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> (you'd need an infinite amount of time and an infinite piece of paper). All we can ever hope to do is calculate a finite piece of it, giving us an approximation of <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/343234984fd472dc9cb0c6c1c1f84569/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> — the more decimal places, the more accurate the approximation.</p>
<p>Because <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/833b9a509682b939511d363aba9a4c23/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> lies at the heart of any technology involving rotation or waves, it has many uses outside of maths. You need it to construct rotating parts in jet engines, for example, or the rings used in medical imaging devices like CAT or MRI scanners, and for frequency calculations for mobile phone and GPS devices (find out more in <a href="/content/how-add-quickly" target="blank">this article</a>).</p>
<p>Many of these uses of the number <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/9863a77122a42e88254d9bb18d2bf020/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> require you to know its value to a high degree of accuracy, but that's not the reason why the Swiss team, and most other <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/9863a77122a42e88254d9bb18d2bf020/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> record hunters out there, go to the effort they do. The latest record was achieved by a team from the Centre for Data Analytics, Visualization and Simulation (DAViS) at Graubünden, who wanted to test and show off their computing expertise. "Our attempt at the record had several aims," said Heiko Rölke, Head of DAViS. "In the course of the preparation and execution of the calculations we built up a lot of know-how and optimised our processes. This is particularly useful for our research groups with whom we are working on computing intensive projects in data analysis and simulation."</p>
<p>Computers can perform the calculations a lot faster than any human could ever hope to, but at the heart of those calculations lie mathematical formulas you could, in theory at least, work out with pencil and paper. There are various mathematical algorithms that allow you to calculate the value of <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/e826f630347459feb3f1be6ed9729a9e/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to any degree of accuracy if only you execute sufficiently many steps of the algorithm.
</p><p>A relatively simple one comes from the infinitely long sum </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img alt="\begin{equation} 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} + \ldots \[ \]\end{equation}" style="width:176px; height:34px" class="math gen b-lazy" data-src="/MI/eb0121f68ee05a64b90d38c803e3b920/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"><span>(<span>1</span>)</span></td>
</tr>
</table><p>The more terms in this sum you add (or subtract), the closer the result to the value of <img alt="$\pi /4$" style="vertical-align:-4px; width:26px; height:17px" class="math gen b-lazy" data-src="/MI/eb0121f68ee05a64b90d38c803e3b920/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. In fact, by adding sufficiently many terms you can get as close as you like to the value of <img alt="$\pi /4$" style="vertical-align:-4px; width:26px; height:17px" class="math gen b-lazy" data-src="/MI/eb0121f68ee05a64b90d38c803e3b920/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. You then only need to multiply your result by <img alt="$4$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/eb0121f68ee05a64b90d38c803e3b920/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> to get an approximation of <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/eb0121f68ee05a64b90d38c803e3b920/images/img-0004.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. </p>
<p></p><p>The problem with using our expression (1) is that you need to include a large number of terms to get a decent degree of accuracy. If you include 100 terms and multiply the result by <img alt="$4$" style="vertical-align:0px; width:8px; height:11px" class="math gen b-lazy" data-src="/MI/d8402c1407fb143b22b93c82ddcdc6f2/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, you get <img alt="$3.146567747182956$" style="vertical-align:0px; width:136px; height:12px" class="math gen b-lazy" data-src="/MI/d8402c1407fb143b22b93c82ddcdc6f2/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. This is fairly close to <img alt="$\pi =3.141592653589793...$" style="vertical-align:0px; width:181px; height:12px" class="math gen b-lazy" data-src="/MI/d8402c1407fb143b22b93c82ddcdc6f2/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> but it’s not a particularly accurate estimate given the effort involved in adding up 100 terms. </p>
<p>The DAViS team used an algorithm based on a more complex looking sum, which was published in 1988 by the <a href="https://en.wikipedia.org/wiki/Chudnovsky_brothers" target="blank">Chudnovsky brothers</a>. The sum is written as</p><p></p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img alt="\[ \frac{1}{\pi } = 12 \sum _{q=0}^{\infty } \frac{(-1)^ q(6q)!(545140134q+13591409)}{(3q)!(q!)^3(640320)^{3q+3/2}}. \]" style="width:345px; height:49px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>The symbol <img alt="$\sum $" style="vertical-align:-4px; width:16px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> tells us that we are dealing with a sum in which each term has the form that comes after the <img alt="$\sum $" style="vertical-align:-4px; width:16px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> symbol. The <img alt="$q=0$" style="vertical-align:-4px; width:38px; height:15px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> at the bottom of the <img alt="$\sum $" style="vertical-align:-4px; width:16px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> means that for the first term in the sum you substitute <img alt="$q=0$" style="vertical-align:-4px; width:38px; height:15px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0003.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> in the expression after <img alt="$\sum $" style="vertical-align:-4px; width:16px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, for the second term in the sum you substitute <img alt="$q=1$" style="vertical-align:-4px; width:37px; height:15px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0004.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> in the expression after <img alt="$\sum $" style="vertical-align:-4px; width:16px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, for the third term in the sum you substitute <img alt="$q=2$" style="vertical-align:-4px; width:38px; height:15px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0005.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> in the expression after <img alt="$\sum $" style="vertical-align:-4px; width:16px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, and so on. The infinity sign on top of the <img alt="$\sum $" style="vertical-align:-4px; width:16px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> symbol means that you keep substituting values of <img alt="$q$" style="vertical-align:-4px; width:8px; height:12px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0006.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> forever, all the way up to infinity. </p><p>For any <img alt="$q$" style="vertical-align:-4px; width:8px; height:12px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0006.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />, the expression <img alt="$q!$" style="vertical-align:-4px; width:12px; height:16px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0007.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> stands for </p><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img alt="\[ q! = q \times (q-1) \times (q-2) \times (q-3) \times ... \times 2 \times 1. \]" style="width:346px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0008.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p>The more terms of this sum you add up, the closer the result to <img alt="$1/\pi .$" style="vertical-align:-4px; width:28px; height:17px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0009.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> You now only need to divide <img alt="$1$" style="vertical-align:0px; width:6px; height:11px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0010.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> by your result to get an approximation for <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0011.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. </p><p>The Chudnovsky algorithm is the most efficient method that is know for calculating the decimal places of <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/d6595a8b6df06ad61260e3936c9b5ddd/images/img-0011.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. </p>
<p>For more on how to approximate <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/be7478f7f34c6c81207b119ce9a95020/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> using infinite sums, see our article <a href="/content/how-add-quickly" target="blank"><em>How to add up quickly</em></a> by Chris Budd.</p>
<p>For the record, the last ten digits in the known part of the decimal expansion of <img alt="$\pi $" style="vertical-align:0px; width:10px; height:8px" class="math gen b-lazy" data-src="/MI/17a3021d56cdc188024a79ec1d5bccf2/images/img-0001.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> are <img alt="$7817924264$" style="vertical-align:0px; width:80px; height:12px" class="math gen b-lazy" data-src="/MI/17a3021d56cdc188024a79ec1d5bccf2/images/img-0002.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />. Congratulations to the DAViS team!</p></div></div></div>Thu, 19 Aug 2021 16:48:44 +0000Marianne7499 at https://plus.maths.org/contenthttps://plus.maths.org/content/celebrating-new-pi-record#commentsLearning the mathematics of the deep
https://plus.maths.org/content/mathematics-deep-learning
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_52.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>
Many aspects of our lives today are possible thanks to <em>machine learning</em> – where a machine is trained to do a specific, yet complex, job. We no longer think twice about speaking to our digital devices, clicking on recommended products from online stores, or using language translation apps and websites.
</p><p>
Since the 1980s neural networks have been used as the mathematical model for machine learning. Inspired by the structure of our brains, each "neuron" is a simple mathematical calculation, taking numbers as input and producing a single number as an output. Originally the neural networks consisted of just one or two layers of neurons due to the computational complexity of the training process. But since the early 2000s <em>deep neural networks</em> consisting of many layers have been possible, and are now used for tasks that vary from pre-screening job applications to revolutionary approaches in health care.
</p><p>
Deep learning is increasingly important in many areas both outside and inside science. Its usefulness has been proven, but there still are a lot of unanswered questions about the theory of why such deep learning approaches work. And that is why the <a href="https://www.newton.ac.uk/" target="blank">Isaac Newton Institute</a> (INI) in Cambridge is running a research programme called <a href="https://www.newton.ac.uk/event/mdl/" target="blank">Mathematics of deep learning</a> (MDL), which aims to understand the mathematical foundations of deep learning.
</p><p>
To introduce you to the ideas involved, and to understand more about what the MDL programme is all about, we are putting together a collection of articles and podcasts over the next six months. Start by reading our introductions to the area, and stay tuned for more!
</p>
<div style="float: left; 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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/neuron_icon.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> </div>
<p><a href="/content/maths-minute-artificial-neurons"> Maths in a minute: Artificial neurons </a> — When trying to build an artificial intelligence, it makes sense to mimic the human brain. Artificial neurons do just that.</p></div>
<div style="float: left; 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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/cat_icon.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> </div>
<p><a href="/content/maths-minute-machine-learning-and-neural-networks"> Maths in a minute: Machine learning and neural networks</a> — Machine learning makes many daily activities possible, but how does it work? Find out in this brief introduction.</p></div>
<div style="float: left; 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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/mist_icon.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> </div>
<p><a href="/content/maths-minute-gradient-descent-algorithms"> Maths in a minute: Gradient descent algorithms </a> — Whether you're lost on a mountainside, or training a deep neural network, you can rely on the gradient descent algorithm to show you the way!</p></div>
<div style="float: left; 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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/kitty_puppy_icon.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> </div>
<p><a href="/content/what-machine-learning"> What is machine learning? </a> — Find out how a little bit of maths can enable a machine to learn from experience in this more in-depth introduction from the <em>Plus</em> library.</p></div>
<div style="float: left; 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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_16_0.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> </div>
<p><a href="/content/agent-perspective-0"> The agent perspective</a> — In this collection from the <em>Plus</em> library, deep learning pioneer Yoshua Bengio explains why he thinks that true artificial intelligence will only be possible once machines have something babies are born with: the ability to interact with the world, observe what happens, and adapt to the consequences of their actions. </p></div>
<div style="float: left; 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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/integral_icon_0.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> </div>
<p><a href="/content/seeing-traffic-through-new-eyes"> Seeing traffic through new eyes</a> — Traffic is not just annoying, it can also come at a high cost to human and environmental health. This article from the <em>Plus</em> library explores how a form of deep learning is being used to help understand traffic, so that suitable city planning can tame it.
</p></div>
<div style="float: left; 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 alt="" width="100" height="100" class="b-lazy" data-src="/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/xray_icon.png" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" /> </div>
<p><a href="/content/artificial-intelligence-takes-covid-19"> Artificial intelligence takes on COVID-19</a> — In this article from the <em>Plus</em> library, we find out how researchers from the AIX-COVNET project are developing a tool that will utilise deep learning to help diagnose COVID-19.
</p></div>
<p>If you'd like more mathematical detail then you can watch talks from the GRA programme on the <a href="https://www.newton.ac.uk/event/mdl/" target="blank">INI website</a>. To see more <em>Plus</em> articles about machine learning click <a href="/content/tags/machine-learning" target="blank">here</a>.</p>
<p><em>We produced this collection of content as part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI), an international research centre and our neighbour here on the University of Cambridge's maths campus. INI attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more.</em></p>
<div class="centreimage"><img alt="INI logo" width="400" height="51" class="b-lazy" data-src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" src="data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==" />
<p style="max-width: 400px;"></p></div></div></div></div>Tue, 10 Aug 2021 13:44:15 +0000Rachel7497 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematics-deep-learning#commentsMaths in a minute: Gradient descent algorithms
https://plus.maths.org/content/maths-minute-gradient-descent-algorithms
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/mist_icon.jpg" width="98" 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>
You are out on a mountain and the mist has descended. You can't see the path, let alone where it leads, so how on Earth do you find your way down to the safety of the village at the bottom of the mountain?
</p><p>
Never fear! Maths has come to your rescue! You can use a <em>gradient descent algorithm</em> – a mathematical technique for finding the minimum of smooth (i.e. <em>differentiable</em>) functions!
</p><p>
The mist is so thick that you can't see into the distance, so instead, you have to work locally. You can explore the ground very close to where you are standing and figure out which direction is downhill and head that way. That's how a gradient descent algorithm works, though instead of feeling which way is downhill with your feet and body, you can find the downhill gradient mathematically using calculus. (Exactly what the algorithm looks like depends on the problem, you can read some of the details on <a href="https://en.wikipedia.org/wiki/Gradient_descent" target="blank">Wikipedia</a>, which is also where we first came across the lovely misty mountain analogy.)
</p>
<div class="centreimage"><img src="/issue24/features/symmetry/reflection.jpg" alt="Reflections in Loch Katrine" width="400" height="309" /><p>Maths can find the way!</p></div>
<!-- Photo by Charles Trevelyan, used with permission in https://plus.maths.org/content/through-looking-glass -->
<p>
If we are using this algorithm on a mountainside we are working our way over a two-dimensional surface. You can easily imagine this algorithm going wrong and you ending up in some local minimum where every direction is uphill and you are stuck in hole, a pit of despair halfway down the mountain and not the safety of the village at the bottom. In some mathematics problems you might be happy with a local minimum, and the gradient descent algorithm is a reliable way to find one of these. But if what you want is the global minimum, you might have to move to higher-dimensional ground.
</p>
<div class="rightimage" style="max-width:350px"><img src="/content/sites/plus.maths.org/files/articles/2021/miam/gradient_ascent_surface_publicdomain.png" alt="A saddle point"/><p>A saddle point on a surface, the red line tracing the progress of a gradient descent algorithm</p></div>
<p>
On a two-dimensional mountainside you have a limited number of directions to choose from: North, South, East and West and combinations of these. But in higher dimensional spaces there are a lot more directions to try. And thankfully it turns out that you can show mathematically that you'd have to be really unlucky to end up in a place that is a local minimum in every possible direction: even if in many of the directions it looks like you are in a hole (in a local minimum), you can be almost certain that at least one of the many possible directions will lead you out downhill (which means you're at what's called a <em>saddle point</em>, rather than in a hole), and you can carry on moving downhill towards the safety at the bottom. The gradient descent algorithm has a really good chance of leading you to the global minimum.
</p>
<p>
We came across the gradient descent algorithm in the context of machine learning, where a neural network is trained to do a certain task by minimising the value of some <em>optimisation function</em> as you move over a mathematical surface. And in this case the mathematical surface equates to the space of possible configurations of the weights given to the connections between neurons in your neural network. (You can read a brief introduction to machine learning and neural networks <a href="/content/maths-minute-machine-learning-and-neural-networks" target="blank">here</a>.) The gradient descent algorithm is often used in machine learning to find the optimal configuration for the neural network.
</p><p>
The reliability of the gradient descent algorithm in higher dimensions is the solution to something called the <em>curse of dimensionality</em>. We <a href="/content/agent-perspective-0 " target="blank">spoke to Yoshua Bengio</a>, one of the pioneers of something called <em>deep learning</em>, where neural networks are incredibly complex and have lots of layers, leading to very high dimensional vector spaces to explore to find the best mathematical function to perform a machine learning task.
</p><p>
It had been thought that it would be impossible to optimise things for these bigger neural networks. The fear was that the machine learning algorithms, such as the gradient descent algorithm, would always get stuck in some sub-optimal solution – one of those local minima or pits of despair on the side of the mountain – rather than reach the best possible solution. But in fact Bengio and colleagues were able to mathematically prove that you are more likely to find yourself at a saddle point, rather than a local minimum in these higher dimensions, and in fact machine learning performance gets better with bigger neural networks.
</p><p>
So whether you're lost on a mountainside, or training a neural network, you can rely on the gradient descent algorithm to show you the way.
</p>
<p><em>Read more about machine learning and its many applications on <a href="/content/tags/machine-learning" target="blank">Plus</a>.</p>
<hr/>
<p><em>This article was produced as part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI) – you can find all the content from the collaboration <a href="/content/ini">here</a>.
</p><p><em>
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more. </em></p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" alt="INI logo" width="400" height="51" />
<p style="max-width: 400px;"></p></div></div></div></div>Tue, 10 Aug 2021 11:19:37 +0000Rachel7496 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-gradient-descent-algorithms#commentsMaths in a minute: Machine learning and neural networks
https://plus.maths.org/content/maths-minute-machine-learning-and-neural-networks
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/cat_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>
We're not anywhere close to the scifi concept of <em>strong artificial intelligence</em>, where a machine can learn any task and react to almost any situation, indistinguishably from a human. But we are surrounded by examples of <em>weak artificial intelligence</em> – such as the speech recognition in our phones – where a machine is trained to do a specific, yet complex, job.
</p><p>
One of the most significant recent developments in weak artificial intelligence is <em>machine learning</em> – where rather than teaching a machine explicitly how to do a complex task (in the sense of a traditional computer program), instead the machine learns directly from the experience of repeatedly doing the task itself.
</p><p>
Advances in engineering and computer science are key to progress in this area. But the real nuts and bolts of machine learning is done with mathematics. A machine learning algorithm boils down to constructing a mathematical function that takes a certain type of input and reliably gives the desired output.
</p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2018/machibe_learning/kitty_puppy.jpg" alt="Puppy and kitten" width="350" height="234" /><p>Machine learning algorithms can learn how to tell a picture of a cat from a picture of a dog.</p></div>
<!-- image from fotolia -->
<p>
For example, suppose you want a machine that can distinguish between digital images of cats and dogs. An image can be treated as a mathematical object – each pixel is actually represented as a number identifying its colour, and the image as a whole is represented by the long list of the values of the pixels (acting as a <em>vector</em> in some very high-dimensional <em>vector space</em>). The machine then applies a complicated function to this mathematical version of the picture and outputs either "cat" or "dog".
</p>
<p>
The key idea of machine learning is that a human hasn't specified this mathematical cat-or-dog function, instead the machine learnt what mathematical function was best suited to distinguishing between cats and dogs on its own. And it did this by looking at lots and lots of pictures of cats and dogs (the <em>training data</em>), and using an optimisation process to learn what mathematical function worked best.
</p><p>
We are used to functions having <em>parameters</em>: for example the general equation for a line <img src="/MI/1272a28f050209866c3d6060dae55e66/images/img-0001.png" alt="$f(x)=ax+b$" style="vertical-align:-4px;
width:99px;
height:17px" class="math gen" /> has the parameters <img src="/MI/1272a28f050209866c3d6060dae55e66/images/img-0002.png" alt="$a$" style="vertical-align:0px;
width:9px;
height:8px" class="math gen" /> that gives the slope of the line, and <img src="/MI/1272a28f050209866c3d6060dae55e66/images/img-0003.png" alt="$b$" style="vertical-align:0px;
width:7px;
height:12px" class="math gen" /> that tells you where it crosses the vertical axis. Similarly functions resulting from machine learning algorithms have some general structure which is tailored by parameters, but these functions are more complicated and have many, many parameters. </p><p>
At the start of the learning process the parameters are set to some values, and the algorithm processes the training data, returning the answer "cat" or "dog" for each picture. Initially it is likely to get a lot of these answers wrong, and it'll realise this when it compares its answers to those from the training data. The algorithm then starts tweaking the parameters until eventually it has found values for them that return the correct answers with a high probability.
</p><p>
This learning process is generally done by something called a <em>gradient descent algorithm</em> which trains the machine to do a specific task by tweaking the parameters as the learning machine works through lots and lots of training data. (You can read an introduction to gradient descent algorithms <a href="/content/maths-minute-gradient-descent-algorithms" target="blank">here</a>.)
</p><p>
There are various approaches to machine learning but the most common method is using something called a <em>neural network</em>. This is really a way to structure a complicated mathematical function. <a href="/content/maths-minute-artificial-neurons" target="blank">Artificial neurons</a>, which themselves are a mathematical function, are arranged in a series of layers, taking a linear sum of the outputs of the neurons in previous layers as an input. Then a neuron applies a nonlinear function to that input and then passes the output of this on to the neurons in the next layer.
</p><p>
The network is then trained by working through lots of training data, and applying a machine learning algorithm that tweaks the parameters in the linear sums that link one layer to the next. With enough training data the parameters are tuned until the neural network has settled on a complicated mathematical function that carries out its given task, such as correctly distinguishing between cats and dogs.
</p><p>
Machines have successfully taught themselves how to beat us at games like <a href="/content/what-machine-learning" target="blank">chess and Go</a>, rather than us teaching them all our best moves. Machine learning is now part of our everyday life – you use it when you speak to your digital devices, when you click on a recommended product from an online store, or when you use language translation apps and websites. And not only that, machine learning is now playing important roles in <a href="/content/artificial-intelligence-takes-covid-19" target="blank">medicine</a>, <a href="/content/theory-practice" target="blank">particle physics</a>, and <a href="/content/seeing-traffic-through-new-eyes" target="blank">monitoring traffic and analysing tree cover</a>.
</p><p>
<em>Read a more detailed introduction into machine learning with this <a href="/content/rise-machines" target="blank">great collection of articles</a> by Chris Budd (which this article is partly based on). And find out more about the many applications of machine learning on <a href="/content/tags/machine-learning" target="blank">Plus</a>.</em></p>
<hr/>
<p><em>This article was produced as part of our collaboration with the <a href="https://www.newton.ac.uk/">Isaac Newton Institute</a> for Mathematical Sciences (INI) – you can find all the content from the collaboration <a href="/content/ini">here</a>.
</p><p><em>
The INI is an international research centre and our neighbour here on the University of Cambridge's maths campus. It attracts leading mathematical scientists from all over the world, and is open to all. Visit <a href="https://www.newton.ac.uk/">www.newton.ac.uk</a> to find out more. </em></p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/representation/ini_logo_green.jpg" alt="INI logo" width="400" height="51" />
<p style="max-width: 400px;"></p></div></div></div></div>Tue, 10 Aug 2021 10:46:59 +0000Rachel7495 at https://plus.maths.org/contenthttps://plus.maths.org/content/maths-minute-machine-learning-and-neural-networks#commentsOn the mathematical frontline: Ellen Brooks Pollock and Leon Danon
https://plus.maths.org/content/mathematical-frontline-ellen-brooks-pollock-and-leon-danon
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/fig2_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><em>On the mathematical front line</em> is a special series of the <em>Plus</em> podcast featuring epidemiologists whose efforts have been crucial in the fight against the pandemic. They are the people who make sense of the data to estimate things like the <a href="/content/maths-minute-r0-and-herd-immunity" target="blank"><em>R</em> number</a>, and who make the mathematical models that inform (and sometimes do not inform) government policy. </p>
<div class="rightimage" style="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/podcast/2021/Ellen_Leon/ellen_leon.jpg" alt="Ellen Brooks Pollock and Leon Danon." width="306" height="149" />
<p>Ellen Brooks Pollock and Leon Danon.</p>
</div>
<p>In this episode we talk to <a href="https://research-information.bris.ac.uk/en/persons/ellen-brooks-pollock" target="blank">Ellen Brooks Pollock</a> and <a href="https://research-information.bris.ac.uk/en/persons/leon-danon" target="blank">Leon Danon</a>, both from the University of Bristol. The are members of the <a href="https://maths.org/juniper/" target="blank">JUNIPER consortium</a> of modelling groups from across the UK whose research and insights feed into the <a href="https://www.gov.uk/government/groups/scientific-pandemic-influenza-subgroup-on-modelling" target="blank">Scientific Pandemic Influenza Modelling</a> group (otherwise known as SPI-M) and <a href="https://www.gov.uk/government/organisations/scientific-advisory-group-for-emergencies" target="blank">SAGE</a>, the Scientific Advisory Group for Emergencies, both of which advise the UK government on the scientific aspects of the COVID-19 pandemic.
</p><p>
Ellen and Leon are also a couple, who have stuck it out through the lockdowns not just in terms of living arrangements and child care, but also in terms of work. And if you don't live with a partner but instead have benefited from the support bubbles that allowed you to team up with another household, then you have Ellen and Leon to thank for that: as we find out in the podcast, it was <a href="/content/careful-your-christmas-baubbles" target="blank">their work on household bubbling</a> which showed that these support bubbles were safe.</p>
<p><a href="/content/sites/plus.maths.org/files/podcast/2021/Ellen_Leon/ellenleon_final2.mp3"><strong>Listen to the podcast</a></strong>
<p><em>You can listen to the podcast by clicking the link, and you can subscribe to our <a href="https://plus.maths.org/content/podcast-feed/rss.xml">podcast feed</a> in your podcast aggregator of choice, or directly through <a href="https://podcasts.apple.com/us/podcast/plus-podcast-maths-on-the-move/id263456080">Apple Podcasts</a> or <a href="https://open.spotify.com/show/5ZiMqmLdf1iZN3aoMabEgR">Spotify</a>.</em></p>
<hr/>
<p><em>The podcast is part of our <a href="/content/joining-forces-covid19" target="blank">collaboration</a> with <a href="https://maths.org/juniper/" target="blank">JUNIPER</a>, the Joint UNIversity Pandemic and Epidemic Response modelling consortium. JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing questions about the control of COVID-19. You can see more content produced with JUNIPER
<a href="/content/juniper" target="blank">here</a>.</p>
</p><div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2021/Juniper-logos/juniper-light-bg.png" alt="Juniper logo" width="400" height="83" />
<p style="max-width: 400px;"></p></div></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/2021/Ellen_Leon/ellenleon_final2.mp3" type="audio/mpeg; length=16390915">ellenleon_final2.mp3</a></span></div></div></div>Thu, 29 Jul 2021 09:25:11 +0000Marianne7493 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematical-frontline-ellen-brooks-pollock-and-leon-danon#commentsOn the mathematical frontline: Mike Tildesley
https://plus.maths.org/content/mathematical-frontline-mike-tildesley
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_50.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="max-width: 350px;"><img src="/content/sites/plus.maths.org/files/podcast/2021/Tildesley/mike_web.jpg" alt="Mike Tildesley" width="227" height="219" />
<p>Mike Tildesley</p>
</div>
<p>
This is the second episode of our new podcast series <em>On the mathematical frontline</em>, where we talk to the mathematicians working on the COVID-19 pandemic. We explore the maths they do, how they go about it, and what impact their work on the pandemic has had on their lives.</p>
<p>In this episode we talk to <a href="https://warwick.ac.uk/fac/sci/lifesci/people/mtildesley/" target="blank">Mike Tildesley</a>, associate professor in the <a href="https://warwick.ac.uk/fac/cross_fac/zeeman_institute/">Zeeman Institute for Systems Biology and Infectious Disease Epidemiology Research</a> at the University of Warwick, a member of <a href="https://maths.org/juniper/" target="blank">JUNIPER</a>, and a member of the Scientific Pandemic Influenza Modelling group (SPI-M) that advises the government on the scientific aspects of the pandemic.
</p>
<p>Mike tells us about his unusual route into epidemiology, the work he's doing on the pandemic, and about the highs and lows of working on the mathematical frontline.
</p>
<p><a href="content/sites/plus.maths.org/files/podcast/2021/Tildesley/pluspodcast_miketildesley.mp3"><strong>Listen to the podcast</a></strong>
<p><em>You can listen to the podcast by clicking the link, and you can subscribe to our <a href="https://plus.maths.org/content/podcast-feed/rss.xml">podcast feed</a> in your podcast aggregator of choice, or directly through <a href="https://podcasts.apple.com/us/podcast/plus-podcast-maths-on-the-move/id263456080">Apple Podcasts</a> or <a href="https://open.spotify.com/show/5ZiMqmLdf1iZN3aoMabEgR">Spotify</a>.</em></p>
<hr/>
<p><em>The podcast is part of our <a href="/content/joining-forces-covid19" target="blank">collaboration</a> with <a href="https://maths.org/juniper/" target="blank">JUNIPER</a>, the Joint UNIversity Pandemic and Epidemic Response modelling consortium. JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing questions about the control of COVID-19. You can see more content produced with JUNIPER
<a href="/content/juniper" target="blank">here</a>.</p>
</p><div class="centreimage"><img src="/content/sites/plus.maths.org/files/packages/2021/Juniper-logos/juniper-light-bg.png" alt="Juniper logo" width="400" height="83" />
<p style="max-width: 400px;"></p></div></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/2021/Tildesley/pluspodcast_miketildesley.mp3" type="audio/mpeg; length=21578650">pluspodcast_miketildesley.mp3</a></span></div></div></div>Thu, 22 Jul 2021 14:18:28 +0000Rachel7492 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematical-frontline-mike-tildesley#commentsThe mathematical shapes in your brain
https://plus.maths.org/content/mathematical-shapes-your-brain
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/icon_51.jpg" width="100" height="100" alt="" /></div></div></div><div class="field field-name-field-author field-type-text field-label-hidden"><div class="field-items"><div class="field-item even">Rachel Thomas</div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>
This first day of the <a href="https://www.8ecm.si/" target="blank">European Congress of Mathematics</a> (ECM), which took place in Slovenia in June 2021, mathematically captured hearts and minds. Betül Tanbay, who introduced the public lecture by <a href="https://www.epfl.ch/labs/hessbellwald-lab/hessbellwald/" target="blank">Kathryn Hess</a> at the end of that day, noted that the first lecture of the day had been on mathematical modelling of our hearts. And Hess's lecture was a "mathematical mystery tour of the marvellous intricacy of the brain". We enjoyed Hess's tour so much, we wanted to share some of the ideas from her lecture with you!
</p>
<h3>Dimension reduction</h3>
<p>
Our brains are an unimaginably complex network of billions of <a href="https://plus.maths.org/content/maths-minute-artificial-neurons" target="blank"><em>neurons</em></a> connected by trillions of synapses. And the complexity becomes even denser when you consider the neurons come in a variety of shapes and sizes and orientations, and there are other factors such as the layers of blood vessels in a brain to consider. "To avoid being overwhelmed by this complexity, we apply a trick familiar to all mathematicians," says Kathryn Hess, professor of mathematics at <a href="https://en.wikipedia.org/wiki/%C3%89cole_Polytechnique_F%C3%A9d%C3%A9rale_de_Lausanne" target="blank">École Polytechnique Fédérale de Lausanne</a>.</p>
</p><p>
That trick is <em>dimension reduction</em>, finding a simplified representation of the information you are dealing with, and something Hess explains we all instinctively do. Suppose you were lucky enough to be in Slovenia for the ECM in person, and you wanted to travel from your hotel to the main venue. To navigate between these two points you don’t need to know the colour of the buildings or the names of the people who live in the houses. You just need to know when to turn left and right, and the distances between these turns. You can think of it as going from the satellite view of Google maps, which shows you lots of detail, to a street map view, which only shows you the relevant information.
</p>
<div class="centreimage" style="max-width:650px">
<div class="leftimage" style="max-width:300px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/satellite_web.jpg" alt="satellite view"/></div>
<div class="leftimage" style="max-width:300px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/map_web.jpg" alt="streetmap view"/></div>
<br class="brclear">
<p>Two views of the area between Piran and Portorož, Slovenia, where the ECM was physically taking place. The satellite view shows gardens, trees, the style of buildings and the location of swimming pools. The street map view shows only the road layout and the route to walk from a hotel apartment to the main Congress venue.</p>
</div>
<p><br/></p>
<p><br/></p>
<div class="rightimage" style="max-width:300px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/directedgraph.jpg" alt="directed graph"/><p>Networks of neurons can be represented by a <em>directed graph</em> like this.(Image: Kathryn Hess)</p></div>
<p>
Hess uses dimension reduction to represent the tangle of neurons in the brain by a <em>graph</em> or <em>network</em>. Each neuron, despite its shape or size, is represented by a node. And a single edge is used to represent any number of synapse connections between two neurons, even though there can be many synapse connections between them (this redundancy in connections is thought to be important in brain function). Because information passes in a definite direction along a synapse (from the axon of one neuron to the dendrites of another) these edges are <em>directed</em> – the arrows show the direction in which information can pass from one neuron to another. And it is possible to have reciprocal edges, where two neurons are connected to each other by two edges each pointing in opposite directions. This directed network provides a simplified structure of brain function.
</p>
<br class="brclear"/>
<h3>A touch of reconstruction</h3>
<p>
Hess's work on the brain has been a collaboration with the <a href="https://www.epfl.ch/research/domains/bluebrain/" target="blank">Blue Brain Project</a>. "The scale of a model of a human brain is out of reach," says Hess. "With hundreds of billions of neurons the quantity of data is unimaginably vast." Instead, in the quest to reconstruct a mathematical model of a brain the Blue Brain Project are focussing their efforts on the substantially smaller rat brain. But even in a rat brain, Hess says, you are still dealing with a "wild tangle of the network of neurons".
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/tangle_neurons_web.jpg" width"600" height="338" alt="Tangle of neurons"><p style="max-width:500px"> A scientific data visualization of the tangle of different types of neurons in a rat neocortex, by Nicolas Antille, EPFL.</p></div>
<!-- Used with permission -->
<p>
Over the last ten years the Blue Brain project has reconstructed the <em>somatosensory neocortex</em> of a rat brain – this is like a core sample from the surface of the brain deep into the grey matter that forms a microcircuit in the part of a rat's brain that is responsible for the rat's sense of touch. It contains around 30,000 neurons and 8 million connections. This reconstruction process starts by populating the column-shaped space of the microcircuit with a diverse collection of neurons, representing the diversity and number observed in laboratory experiments. These neurons are then connected according to their proximity and information we have about real rat brains.
</p><p>
The algorithm used to reconstruct the microcircuit is <em>stochastic</em>, in that some choices in the algorithm are made probabilistically, and hence the reconstructions produced are different each time it is run. The Blue Brain Project started with the laboratory data from five individual rat brains and an "average" brain (that pooled the data from the individual brains). They then ran the reconstruction process on each of these six datasets seven times, producing seven slightly different reconstructions for each of the individual and the average brain. This produced 42 reconstructed microcircuits in total.
</p><p>
The electrical behaviour of the synapses is then encoded into the reconstructed microcircuits, and the microcircuits can be linked together to form a complete reconstruction of this part of a rat's brain. The ultimate goal is to use reconstructed brains to study neurological disorders, such as Parkinson's disease, and reduce the need for animal testing in areas such as drug development.
</p>
<br class="brclear"/>
<h3>Revealing structure</h3>
<div class="rightimage" style="width: 150px;"><img src="/issue10/features/topology/gluemug.gif" alt="A doughnut mug" width="150" height="150" />
<p>Topology in action</p>
</div>
<!-- image from https://plus.maths.org/content/shape-things-come-part-i --><p>
Topology is the area of mathematics that classifies shapes according to how they can be bent or stretched from one to the other, without cutting or tearing. The famous example is that a donut is topologically the same as a coffee cup – if they were made out of pliable clay you could make an increasingly cup shaped dent in one side of the donut, shrinking the loop to the handle, continuously transforming one to the other. What is preserved in this transformation is the hole. The hole in the middle of the donut becomes the hole in the handle of the cup – the hole can't vanish. Topology is interested in what properties – like the hole – are <em>invariant</em> or unchanging under such transformations.
</p><p>
Topology provides the mathematical language of shape and connectivity, and can also describe the emergence of global structures from local constraints. "Topology is an excellent mathematical filter," says Hess, explaining that it can detect the structures and the connectivity in the reconstructed microcircuit and help discern the role these play in brain function.
</p><p>
Hess described a number of approaches of applying the filter of topology to the reconstructed microcircuit and its 31,000 nodes and 8 million connections. One approach Hess and her colleagues used was considering a small collection of relevant and significant subnetworks within the tangle of neurons. The flow of information is important in neural circuits, so they focussed on small <em>feedforward</em> subnetworks, where the information only travels one way through the network.
</p><p>
Topologically these can be described as <em>directed simplices</em>. Simplices are generalisations of triangles to any dimension: a 0-simplex is a point (a node), a 1-simplex is a line (an edge between two nodes), a 2-simplex is the familiar triangle (3 nodes connected by 3 edges), a 3-simplex is a tetrahedron (4 nodes connected by 6 edges) and so on. If you consider simplices whose edges also have a direction, a feed forward network can be represented by a simplex where one node only has outward edges (the input for the network) and one node only has inward edges (the output of the network). (You can read more about simplices <a href="/content/maths-minute-simplices-atoms-topology" target="blank">here</a>.)
</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/directedsimplices_web.jpg" alt="Directed simplices of different dimensions"><p style="max-width:600px">Directed simplices (Image: Kathryn Hess)</p></div>
<p>
The power of topology comes from how very local information, such as the way these small groups of neurons are connected together, can give you important characteristics of the global network. Hess and her colleagues counted the number of these directed simplices of each dimension in the neural microcircuits, and compared it with the equivalent counts of local structures for other types of comparable networks.
</p>
<div class="rightimage" style="max-width:300px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/countsimplices_web.jpg" alt="Plot showing counts for directed simplices in different but comparable networks"><p>Counts for directed simplices of different dimensions in the Blue Brain neural microcircuits (blue), a random network (green) and a more general biological network (yellow). (Image: Kathryn Hess)</p></div>
<p>
They considered a random network (the green line in the plot) where you start with a similar number of nodes and connect them randomly in such a way the resulting network has a similar number of edges overall, and the nodes themselves have similar numbers of edges, to the neural microcircuit. In this random network there are very few directed simplices, and no directed simplices of dimension 4 or higher. A similar result is found for a more general biological network (the yellow line), that is constructed in a way that doesn't take into account features such as shape or size of the neurons.
</p><p>
The reconstructed neural microcircuit has far higher numbers of directed simplices than the other networks analysed, and contains higher-dimensionsional directed simplices. "The Blue Brain microcircuit has far higher [numbers]," says Hess. "This circuit is far from random." Their work suggests that to enable the necessary coordinated firing behaviour of the neurons, the neurons need to be part of bigger structures such as these higher dimensional simplices.
</p>
<br class="brclear"/>
<h3>Counting holes</h3>
<p>
Thinking back to our donut and coffee cup, the thing they have in common is that they each have one hole. A sphere is not topologically the same as a donut or a coffee cup as it does not have any holes and therefore you can't smoothly transform a sphere to a donut without ripping or cutting it. The number of holes in an object – called its <em>genus</em> – is an important concept in topology as the number of holes is preserved in the smooth transformations. The shape and location of the holes may change but the number of them won't. (You can read more about the importance of holes in topology and one of its applications in maths <a href="/content/very-old-question-very-latest-maths-fields-medal-lecture-manjul-bhargava" target="blank">here</a>.)
</p><p>
The equivalent idea of a hole in a network is a <em>cavity</em>, where simplices overlap on their nodes to form a closed object. You might think of a cavity made of 1-simplices as a window, or <em>2-cavity</em>. And a cavity of eight 2-simplices as a room, or 3-cavity.
</p>
<div class="leftimage" style="max-width:226px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/window.jpg" alt="A window, or 2-cavity, made out of four 1-simplices" width="226" height="205"><p>A window, or 2-cavity, made out of four 1-simplices. (Image: Nicolas Antille, EPFL)</p></div>
<div class="leftimage" style="max-width:208px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/room.jpg" alt="A room, or 3-cavity, made out of eight 2-simplices." width="208" height="205"><p>A room, or 3-cavity, made out of eight 2-simplices. (Image: Nicolas Antille, EPFL)</p></div>
<br class="brclear"/>
<p>
"Cavities in a network are highly non-random, their presence is an indication of structure," says Hess. Hess and her colleagues counted the number of cavities of different dimensions in the 42 different microcircuits reconstructed by the Blue Brain Project. The results of counting the cavities in these different reconstructions illustrate that this topological parameter faithfully reflects the biological reality.
</p>
<div class="rightimage" style="max-width:300px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/cavitycounts_web.jpg" alt="Clustering of topological parameters for biological samples" ><p>Clustering of topological parameters for biological samples (Image: Kathryn Hess)</p></div>
<p>
The plot to the right shows the number of 2-cavities compared to the number of 3-cavities for each of the 42 reconstructed circuits. The colours represent which individual rat data (or pooled "average" rat data) the reconstructions were based on. The counts of cavities clearly cluster together for the reconstonstructions of each of the individual (and averaged) rat brains. This topological parameter is clearly different for the different rat brains, and the parameter is broadly consistent across all of the reconstructions based on each individual.
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<div class="leftimage" style="max-width:300px"><img src="/content/sites/plus.maths.org/files/articles/2021/hess/swoosh_web.jpg" alt="Clustering of topological parameters for biological samples" ><p style="max-width:300px">The "swoosh" plot of the number of active 3-cavities (the vertical axis) against active 1-cavities (the horizontal axis). (Image: Kathryn Hess)</p></div>
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The parameters of the number of directed simplices and cavities also provided a method for analysing the wave of electrical activity as it spreads through the neural network. An edge is <em>active</em> if it corresponds to one neuron triggering another. The researchers could observe the evolution of this activity through time, using biologically meaningful time steps, by counting active simplices and cavities at each step. The plot – which Hess and her colleagues call a <em>swoosh</em> – gives a signature of the activity as it loops around the curve in the plot in a counterclockwise fashion.
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Hess began her talk by saying she had devoted the first decade of her career to the purest of pure maths – <a href="/content/tags/topology" target="blank">topology</a>, <a href="/content/tags/homotopy" target="blank">homotopy theory</a> and <a href="/content/quantifying-occam" target="blank">category theory</a>. Now she is using that same pure mathematics to shed light onto the very stuff that allows us to think up these abstract mathematical ideas. Yet another demonstration of the effectiveness of maths in describing our inner and outer worlds.
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<h3>About this article</h3>
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<a href="https://www.epfl.ch/labs/hessbellwald-lab/hessbellwald/" target="blank">Kathryn Hess</a> is professor of mathematics at École Polytechnique Fédérale de Lausanne. This article is based on her public lecture at the ECM in Slovenia in June 2021. The congress was both in person and virtual, and Rachel Thomas attended virtually from her front room. Thanks to Professor Hess and <a href="http://www.nicolasantille.com" target="blank">Nicolas Antille</a> for allowing us to reproduce their images.
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<p><A href="/content/people/index.html#Rachel">Rachel Thomas</a> is Editor of <em>Plus</em>.</p></div></div></div>Thu, 22 Jul 2021 10:32:49 +0000Rachel7491 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematical-shapes-your-brain#commentsOn the mathematical frontline: The podcast
https://plus.maths.org/content/mathematical-frontline
<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" loading="lazy" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsite-date%5D/juiper_zoom_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>Over the last year and half we have done <a href="/content/covid-19" target="blank">a lot of reporting</a> on the maths of the COVID-19 pandemic. Behind the maths there are of course people — those mathematicians who make the epidemiological models that do (and sometimes do not do not) inform government policy, who are grappling with the unprecedented challenge of coming to grips with a live pandemic unfolding in front of their eyes. </p>
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<p>Our special podcast series, <em>On the mathematical frontline</em>, is about those people. It explores the maths they do, how they go about it, and the impact it has on their personal lives. The epidemiologists we talk to are all members of the <a href="https://maths.org/juniper/" target="blank">JUNIPER consortium</a> of modelling groups from across the UK whose research and insights feed into the <a href="https://www.gov.uk/government/groups/scientific-pandemic-influenza-subgroup-on-modelling" target="blank">Scientific Pandemic Influenza Modelling</a> group (otherwise known as SPI-M) and <a href="https://www.gov.uk/government/organisations/scientific-advisory-group-for-emergencies" target="blank">SAGE</a>, the Scientific Advisory Group for Emergencies, both of which advise the UK government on the scientific aspects of the COVID-19 pandemic.
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<p>The work and personal experiences of these epidemiologists are fascinating and thought-provoking. We hope you enjoy listening!</p>
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<p><a href="/content/mathematical-frontline-julia-gog">Julia Gog</a> — Julia Gog used to see herself as a backroom theoretician, but when the pandemic struck she stepped up to the challenge of a generation. She is now one of the two leads of JUNIPER and indispensable as government advisor.</p></div>
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<p><a href="/content/mathematical-frontline-mike-tildesley">Mike Tildesley</a> — You many have heard Mike Tildesley being interviewed in the media. Find out about his unusual route into epidemiology, the work he's doing on the pandemic, and about the highs and lows of working on the mathematical frontline. </p></div>
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<p><a href="/content/mathematical-frontline-ellen-brooks-pollock-and-leon-danon">Ellen Brooks Pollock and Leon Danon</a> — It was Ellen Brooks Pollock's and Leon Danon's work which showed that it was safe for single people to bubble up with other households during lockdown. But that's only part of the work they did as a professional team and also as a couple.</p></div>
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<p><em>These podcasts are part of our <a href="/content/joining-forces-covid19" target="blank">collaboration</a> with <a href="https://maths.org/juniper/" target="blank">JUNIPER</a>, the Joint UNIversity Pandemic and Epidemic Response modelling consortium. JUNIPER comprises academics from the universities of Cambridge, Warwick, Bristol, Exeter, Oxford, Manchester, and Lancaster, who are using a range of mathematical and statistical techniques to address pressing questions about the control of COVID-19. You can see more content produced with JUNIPER
<a href="/content/juniper" target="blank">here</a>.</p>
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<p style="max-width: 400px;"></p></div></div></div></div>Mon, 19 Jul 2021 10:49:08 +0000Marianne7494 at https://plus.maths.org/contenthttps://plus.maths.org/content/mathematical-frontline#comments