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Mathematical moments: Frank Kelly
https://plus.maths.org/content/mathematicalmomentsfrankkelly
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/kelly_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>We talk to the mathematician <a href="http://www.statslab.cam.ac.uk/~frank/">Frank Kelly</a> about his work developing mathematical models to understand largescale networks, some of his favourite mathematical moments, and how to be wellprepared for ideas coming out of the blue.</p>
<iframe width="560" height="315" src="https://www.youtube.com/embed/QrgNb4UaO4" frameborder="0" allowfullscreen></iframe>
<p>You can find out more about Kelly's work in <em><a href="/content/catchingterroristsmaths">Catching terrorists with maths</a></em>, <em><a href="/content/renewableenergyandtelecommunications">Renewable energy and telecommunications</a></em> and <em><a href="/content/traveltimemapsmdashtransformingourviewtransport">Transforming our view of transport</a></em>.</p></div></div></div>
Fri, 29 Jul 2016 16:21:27 +0000
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Luca lived beneath the sea
https://plus.maths.org/content/lucalivedbeneathocean
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/vent_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><div class="rightimage" style="maxwidth: 350px;"><img src=" /content/sites/plus.maths.org/files/news/2016/luca/vent.jpg" alt="hydrothermal vent" width="350" height="263" />
<p>A hydrothermal vent at the Champagne vent site in the Western Pacific.</p>
</div>
<p>This week scientists announced a fascinating result regarding the
origin of life on Earth. Using come clever genetic detective work,
geneticists at the University of Düsseldorf in Germany found evidence
that Luca (short for <em>last common universal ancestor</em>, that is, the
root of all life) sprang into being at the bottom of the ocean, in
socalled <em><a
href="https://en.wikipedia.org/wiki/Hydrothermal_vent">hydrothermal
vents</a></em>. These are submarine fissures in the Earth's surface
that spew forth water heated up deep inside it. </p>
<p>The team looked at two groups of singlecelled species that are
still alive today: bacteria and socalled <em>archeae</em>. An obvious first step in profiling
Luca is to look for genes that are present in both archeae and
bacteria, and that's what the team did, examining 2000 genomes
comprising around 6 million genes. There's a hitch, however: common
genes don't necessarily come from Luca, but may have transferred from one species to another long after the time Luca was alive. The
team therefore looked for genes for which there's no evidence of such
<em>lateral gene transfer</em>. They ended up with around 350 genes to
consider.</p>
<p>It's those genes that hold the clue to Luca's birth place. They
suggest that Luca lived on gases, including hydrogen. The most
prevalent source for that are submarine hydrothermal vents. What's
more, the distribution of genes identified by the team is also found in
modern microbes still living in these strange ecosystems beneath the
sea, which are so
alien to us.</p>
<p>What's that got to do with maths? <a
href="http://www.molevol.hhu.de/en/profdrwfmartin.html">Bill
Martin</a>, one of the authors of the new study, gave a beautiful
<a href="http://www.bbc.co.uk/programmes/b07lfktd">interview to BBC
Radio 4's <em>Inside Science</em> yesterday</a>. The sentence that
stuck in our mind was "There's a lot of computing involved. We had 1000 processors working
for about 6 months to calculate [our results]."</p>
<div style="float:left; marginright: 20px;"><iframe width="420" height="315" src="https://www.youtube.com/embed/wCDheNQXDA0" frameborder="0" allowfullscreen></iframe>
<p style="width: 400px; fontsize: small; color: purple;">Strange creatures live near hydrothermal vents, including this octopus.</p>
</div>
<p>That's a heck of a lot of computing! And it reminded us of an <a href="/content/evolutionitsrealgravity">article about evolution</a> we recently ran on
<em>Plus</em>. The article mentioned <em>phylogenetic trees</em>,
which depict genetic relationships between species just like a family
tree depicts relations within a family. We made a <a href="/content/mathsfiveminutescountingtreeslife">backoftheenvelope
calculation</a> to figure out the following question: given a number of species
how
many possible phylogenetic trees are there describing
their possible genetic relationships?</p>
<p> The answer gets very big very quickly as
the number of species grows. For ten species
there are over 34 million possible trees. And for 20 species there
are over 8,000,000,000,000,000,000,000 possible trees. That's not that
much less than the <a
href="http://www.physicsoftheuniverse.com/numbers.html">estimated
number of stars in the observable Universe</a>.</p>
<p>We're not sure exactly what kind of calculations Martin in his
colleagues did in their six months of computing time — but our
calculation illustrates the kind of astronomical numbers geneticists
are dealing with. That's why the study of evolution can't do without maths.</p></div></div></div>
Fri, 29 Jul 2016 15:11:48 +0000
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What do you think?
https://plus.maths.org/content/whatdoyouthink
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/people_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>How did you vote in the brexit referendum? Whichever way it was, your decision was probably influenced,
not only by information from the media, but also
by the opinions of the people around you. No person is an
island: the
opinions that prevail in a society
emerge
from lots and lots of
interactions between individual people.</p>
<div class="rightimage"><img src="/content/sites/plus.maths.org/files/articles/2016/opinion/people.jpg" alt="People" width="350" height="234" />
<p style="maxwidth: 350px;">Opinions emerge from interactions between many people.
</p>
</div>
<! Image from fotolia.com >
<p>At the <a href="/content/europeancongressmathematics2016">European Congress of Mathematics</a>, which took place in Berlin
last week, <a href="http://users.sussex.ac.uk/~bd80/">Bertram Düring</a> presented some work which explores this
fact.
"We're interested in modelling and analysing emergent behaviour in
systems that consist of many interacting individuals," he
explains. "For example, the formation of opinions in a society, or the formation of prices in
a financial market. We want to understand how certain patterns emerge
in such a system
and how this whole process works."</p>
<p>Societies (or financial markets) contain far too
many people (market participants) to track their behaviour individually, so at
first sight it's hard to imagine how to attack the problem Düring talks about. But there's a precedent from an entirely different
area. Gases, such as the air in your room, contain far too many molecules for us to track their
individual behaviour, yet we still understand
their bulk behaviour. For example, there is a famous equation named after
<a href="http://wwwhistory.mcs.stand.ac.uk/Biographies/Boltzmann.html">Ludwig Boltzmann</a> which (if you manage to solve it) tells you how many molecules you can expect to find in a given region of space, and travelling with a particular velocity, at a given moment in time. To derive the equation, you start with some basic assumptions of how individual molecules move and what happens when two of them collide.</p>
<p>"People are not particles," says Düring. "So we don't use
statistical physics and apply it to people. What we do is to try and
model the interactions between individuals,
translated into a mathematical language. We don't want to
follow exactly what every individual thinks, but we want to look at society
at an aggregate level. And this where the mathematical tools that were
developed [in statistical physics] come into play. We try to derive an
equation which describes the formation of opinions in the whole of society."</p>
<h3>People and particles</h3>
<div class="leftimage"><img src="/content/sites/plus.maths.org/files/articles/2016/opinion/Bertram.jpg" alt="Bertram Düring" width="336" height="299" />
<p style="maxwidth: 336px;">During explaining Toscani's model at the ECM in Berlin.
</p>
</div>
<p>In 2006 the Italian mathematician <a href="http://wwwdimat.unipv.it/toscani/">Giuseppe Toscani</a> came up with such an
<a href="http://mate.unipv.it/toscani/publi/opinion.pdf">aggregate model of opinion formation</a>. The idea is that when two people meet and
talk about a particular issue or product, they usually come away with
altered opinions. That's a bit like particles in a gas, which, after they
collide, come away
with altered velocities: they change their speed and their direction
of travel.</p>
<p>In analogy to Boltzmann's ideas, Toscani
derived an equation that tells you the distribution of
opinion in a society: how many people hold opinions with a given value
of the opinion variable at a given moment in time. That's the kind of thing
you want to know if you are trying to understand how opinions evolve. </p>
<p>Underlying Toscani's model are some basic assumptions about what happens when two people meet and exchange opinions. But how do you describe this interaction mathematically? It's interesting to have a quick look at Toscani's approach, even if you don't understand the equations (see box) in detail.
Toscani represented
each individual's
opinion about a particular issue or product (say brexit, or a
new type of mobile phone) by a number on a continuous scale from
1 to +1. For example, 1 might mean "I absolutely want out of the
EU", 1 might mean "I definitely want to stay in it", and 0 represents
the undecided. </p>
<div style="maxwidth: 340px; float: right; border: thin solid grey;
background: #CCC CFF; padding: 0.5em; marginleft: 0.5em; fontsize:
75%">
<h3>Change of opinion in Toscani's model</h3>
<p><p>Suppose two people have opinions <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0001.png" alt="$v$" style="verticalalign:0px;
width:8px;
height:7px" class="math gen" /> and <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0002.png" alt="$w$" style="verticalalign:0px;
width:12px;
height:7px" class="math gen" /> before their interaction, where <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0003.png" alt="$1\leq v,w \leq 1.$" style="verticalalign:3px;
width:104px;
height:15px" class="math gen" /> </p><p>Then, in Toscani’s model, their opinions <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0004.png" alt="$v^*$" style="verticalalign:0px;
width:14px;
height:12px" class="math gen" /> and <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0005.png" alt="$w^*$" style="verticalalign:0px;
width:18px;
height:12px" class="math gen" /> after the interaction are </p><table id="a0000000002" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0006.png" alt="\[ v^* = v \gamma P(vw)(vw)+\tilde{\eta } D(v), \]" style="width:275px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><table id="a0000000003" class="equation" width="100%" cellspacing="0" cellpadding="7">
<tr>
<td style="width:40%"> </td>
<td><img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0007.png" alt="\[ w^* = w  \gamma P(wv)(wv)+\eta D(w), \]" style="width:287px;
height:18px" class="math gen" /></td>
<td style="width:40%"> </td>
<td class="eqnnum" style="width:20%"> </td>
</tr>
</table><p> where <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0008.png" alt="$0 < \gamma < 1/2$" style="verticalalign:4px;
width:86px;
height:16px" class="math gen" /> is a constant <em>compromise parameter</em> and the functions <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0009.png" alt="$P$" style="verticalalign:0px;
width:13px;
height:11px" class="math gen" /> and <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0010.png" alt="$D$" style="verticalalign:0px;
width:14px;
height:11px" class="math gen" /> model the relevance of compromise and selfthinking for a given opinion. The quantities <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0011.png" alt="$\tilde{\eta }$" style="verticalalign:3px;
width:9px;
height:15px" class="math gen" /> and <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0012.png" alt="$\eta $" style="verticalalign:3px;
width:9px;
height:10px" class="math gen" /> are independent random variables representing the diffusion of outside information. They both have the same distribution, a mean of <img src="/MI/3c99cb3dbf4352f759fb4667f324fe75/images/img0013.png" alt="$0$" style="verticalalign:0px;
width:8px;
height:12px" class="math gen" /> and a finite variance. </p></p></div>
<p> "Sociology tells us that after [two people interact] there
is often some sort of consensus: our opinions tend to get closer to
each other," explains Düring. Toscani's expression
contains a component that represents this
compromise process. But since personal contact isn't the only thing
that makes our opinions, the expression also contains a component that represents an
individual's <em>selfthinking</em>: the way they might change their
ideas, for example in response to information from the media. The two components aren't the same for all individuals,
rather they reflect the fact that people with extreme
opinions tend to be pigheaded: how open an individual is to changing their mind
depends on the strength of opinion they already have. Those expressions (see the box) form the basis of Toscani's model and lead to a Boltzmannlike equation for the opinion distribution. </p>
<h3>Trends and territories</h3>
<p>The maths behind this gets quite complex, but the assumptions about human behaviour aren't: the model is still simplistic in that respect. An aspect it ignores, for example, is one of my own
personal frustrations, namely that people tend to pay less attention
to my opinions than they do to those of certain other people I
know. But Düring and his colleague <a href="https://www2.warwick.ac.uk/fac/sci/maths/people/staff/wolfram/">MarieTherese Wolfram</a> have <a href="http://rspa.royalsocietypublishing.org/content/471/2182/20150345?hwshib2=authn%3A1469705649%3A20160727%253A2ec9f5a2c7364d0baf2ecb3596c4a64c%3A0%3A0%3A0%3AZJV1c4bjB2vENJcrwiDmvQ%3D%3D#sec10">extended
Toscani's model</a> to take account of just that phenomenon. </p>
<p>
"We can also have additional variables
which model the details of the consensus process, for example how
persuasive individuals are," says Düring. "This could be seen as a characteristic of
opinion leaders: this is a group of people within society who
have a very strong influence on the common people in the dissemination
of opinion."</p>
<p> Similarly, it's possible to add a variable representing
where in a country an individual lives — looking at the results of
the recent referendum, or even the last general election, that's
clearly an important factor too. </p>
<p>Once you have developed a mathematical model, your next task is to
see what it tells you for particular starting assumptions. You can choose the parameters in the model (in Toscani's model that would be things like the compromise parameter <img src="/MI/11b60558ec31bbe855887b9c41a63874/images/img0001.png" alt="$\gamma $" style="verticalalign:4px;
width:9px;
height:12px" class="math gen" />) in a way you think reflects particular reallife situations, and then see what the model predicts.
Düring and Wolfram have found some interesting
results, in particular when it comes to the relationship between
opinion and geographical location. A particularly striking phenomenon
happened in presidential elections in the US
over the
last few decades. Democrat and Republican voters clustered
geographically, and the clustering intensified over the years, leading
to a deep political division between the various counties. In 2008 the
journalist <a href="https://en.wikipedia.org/wiki/Bill_Bishop_(author)">Bill Bishop</a> suggested that the clustering reflects a trend
for US citizens choosing to live near people who are politically
likeminded. A worrying idea, since such a trend would lead to even more
confrontation in US politics. The trend is exemplified by the state of Arizona:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/opinion/arizona.jpg" alt="Arizona election results." width="600" height="167" />
<p style="maxwidth: 300px;">Results of the presidential elections in Arizona in, left to right in 1992, 1996, 2000 and 2004. Republicans are red and Democrats blue. The colour intensity reflects the election outcome in per cent, i.e. dark blue corresponds to Democrats 60–70%, medium blue to Democrats 50–60% and light blue to Democrats 40–50%. Similar colour codes are used for the Republicans. Image adapted from Wikimedia Commons, used under Creative Commons AttributionShare Alike 3.0. </p>
</div>
<p>Düring and Wolfram ran their model including the geographical
variable, assuming that people tend to be influenced by people who are
geographically close to them, and that they feel drawn to counties
that are controlled by the political party they support. They started
their model with the proportion of Democrat and Republican supporters
like that of 1992, and then let it evolve in time. The result is a
political map quite similar to the real one. The exception is the
county of Navajo, but that should come as no surprise. Much of its
area is covered by the Navajo, Hopi and Fort Apache Indian
reservations, to which people wouldn't move as freely as they would to other locations.</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/opinion/arizona2.jpg" alt="Arizona election results." width="600" height="195" />
<p style="maxwidth: 600px;">Distribution of Democrats (left)
and Republicans (right).
</p>
</div>
<p>Models like Düring and Wolfram's could be used to test suggestions
like Bishop's more extensively, but they also find wider
applications.
They could be used in conjunction with polling to predict likely
outcomes of elections and referenda, and also in the worlds of
marketing and finance. "There's a fastgrowing literature in this area
of emergent phenomena," says Düring. "We already have a better understanding
of how consensus works in a society. We can't say that we understand
everything, and probably we will never completely understand human
behaviour. But we're not trying to understand how individuals think, we are
just trying to understand how opinion disseminates through
society."</p>
<p>One important task that lies ahead is to learn from reallife data what the various
parameters in such models should be. This will involve, not
only sophisticated mathematical tools, but also insights from the
social sciences. And the relationship between maths and the social
sciences isn't oneway. "In this century where we are able to store
and acquire a lot of data, I think there's a stronger drive to make
science even more quantitative. Results in the social sciences tend to
be qualitative — when we want to use these results in our models we
have to translate them into a more mathematical and quantitative
language. I think the process could also go the other way, with the
social sciences taking up a more quantitative approach. This is a fairly
new field so I expect there to be some new and unexpected results in
the future."</p>
<hr/>
<h3>About this article</h3>
<p>Düring and Wolfram have published their work in the paper <em><a href="http://rspa.royalsocietypublishing.org/content/471/2182/20150345?hwshib2=authn%3A1469705649%3A20160727%253A2ec9f5a2c7364d0baf2ecb3596c4a64c%3A0%3A0%3A0%3AZJV1c4bjB2vENJcrwiDmvQ%3D%3D#sec2">Opinion dynamics: inhomogeneous Boltzmanntype equations modelling opinion leadership and political segregation</a></em>.</p>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. She interviewed Bertram Düring at the European Congress of Mathematics in Berlin in July 2016.</p></div></div></div>
Fri, 29 Jul 2016 09:14:00 +0000
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Blood, oil and water
https://plus.maths.org/content/bloodoilwater
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/oil_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Marianne Freiberger</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>People often say that mathematics is a language. For mathematician
<a href="https://people.kth.se/~sara7/">Sara Zahedi</a> this was literally the case. The Iranian came
to Sweden at the age of ten, alone, and without speaking a word of
Swedish. It was through the maths she was doing at school that she
made her first Swedish friends. "I made two friends that way," she says. "I found maths easy, and they were
good at it too, so they started to discuss things with me because they
realised that I also understood." </p>
<div class="rightimage" style="maxwidth: 300px;"><img src=" /content/sites/plus.maths.org/files/articles/2016/zahedi/picture.jpg" alt="Sara Zahedi" width="300" height="274" />
<p>Sara Zahedi. </p>
</div>
<p>Since those school days Zahedi has gone far. We met at the
<a href="/content/europeancongressmathematics2016">European Congress of Mathematics</a> in Berlin, where she won one of the <a href="http://www.euromathsoc.eu/prizeseuropeanmathematicalsociety">prestigious
prizes</a> awarded to young mathematicians by the <a href="http://www.euromathsoc.eu">European Mathematical
Society</a>.</p>
<p> Unlike some of the other mathematics presented at the
congress, Zahedi's work has direct uses in the world outside of
maths. She tries to understand, and simulate on computers, the
behaviour of fluids that don't mix together, like oil and water. It's the boundary
between two such fluids, their
<em>interface</em>, that Zahedi is interested in.</p>
<p>An example of the importance of this kind of problem comes from medicine.
"It's something we think people will use in the future: a lab on a
chip," explains Zahedi. "The idea is that instead of sending blood
tests to a lab and waiting for a couple of weeks until you get the
results, you
take a small blood sample and perform lab processes on the chip,
detecting viruses and bacteria within a few minutes." Such chips are
expensive to produce, however, so it helps to simulate their function
on a computer, getting the design right before actually manufacturing them. In these kind of lab tests blood interacts with other chemicals. Interfaces form and chemical processes happen on them. </p>
<p> Oil and water give us another example of why work like
Zahedi's is important. The
<a href="https://en.wikipedia.org/wiki/Deepwater_Horizon_oil_spill">Deepwater Horizon oil spill</a> that happened in the Gulf of
Mexico in 2010 discharged nearly 5 million barrels of oil into the
sea. "People didn't want this oil to reach land, so they added a lot
of <a href="https://en.wikipedia.org/wiki/Surfactant"><em>surfactants</em></a> into the sea to make the oil soluble in water. The
problem with this is that you don't really know what is going to happen, for
example how it is going to effect sea life. Computer simulations could
give you insight into this kind of complex problem too."</p>
<p>To create such simulations, you essentially face two problems. One is how to represent the changing geometry of the interface between two fluids in a computer. Zahedi and her colleagues came up with a way of doing this a few years ago, and it's been implemented in a commercial software package called <a href="https://uk.comsol.com">Comsol</a>. </p>
<div class="leftimage" style="maxwidth: 400px;"><img src=" /content/sites/plus.maths.org/files/articles/2016/zahedi/sara_talk.jpg" alt="Sara Zahedi" width="400" height="396" />
<p>Zahedi giving her prize lecture in Berlin. </p>
</div>
<p>The second problem is how to describe fluid processes that arise on such a changing interface, in which individual droplets can split, merge and generally deform in complex ways. For example, if a surfactant has been added, as in the oil example, how does its concentration change in such a dynamic setting? "As the fluid evolves the interface may get stretched or deformed and the concentration will decrease or increase," explains Zahedi. There are mathematical equations describing how the concentration will change, but the problem is that they are very difficult to solve. "In many cases we don't have a [neat mathematical formula that gives a solution]."</p>
<p> When there's no exact formula, the only way to describe the process on a computer is to find approximate solutions. Existing methods can do this very efficiently, even on complex geometric surfaces, but not on surfaces that change in complicated ways. Zahedi's method, which she has been awarded her prize for, can — and in a mathematically rigorous way to boot. "What is important for us is to be able to say how accurate the approximate solution is, and that's where the mathematics comes in from our point of view. We don't just want to design methods where there's no control over how big the error is." Zahedi's method not only finds approximate solutions, but also gives a way of telling how accurate those solutions are. "We are not quite there yet for the whole complex problem, but at least we can do it for model problems," she says.</p>
<p>The method has not yet been implemented in commercial software, but Zahedi thinks that it will be very soon. Your future medical diagnoses, and perhaps one day even
the welfare of sea life, may depend on the clever piece of mathematics Zahedi has been honoured for.</p>
<hr/>
<h3>About this article</h3>
<p><a href="/content/people/index.html#marianne">Marianne Freiberger</a> is Editor of <em>Plus</em>. She interviewed Sara Zahedi at the European Congress of Mathematics in Berlin in July 2016.</p></div></div></div>
Wed, 27 Jul 2016 09:01:07 +0000
mf344
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https://plus.maths.org/content/bloodoilwater#comments

Some highlights from the ECM 2016
https://plus.maths.org/content/somehighlightsecm2016
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><div class="rightimage" style="maxwidth: 400px;"><img src=" /content/sites/plus.maths.org/files/news/2016/ECM/p1000678.jpg" alt="ECM" width="400" height="267" />
<p>Mathematicians mingle at the ECM at the Technische Universität Berlin. Image from the <a href="http://www.7ecm.de/general/news.html#news_858793aaf1a74af395904b8b89c47d86">ECM website</a>. </p>
</div>
<p>It's day three of the <a href="/content/europeancongressmathematics2016">European Congress of Mathematics</a> in a sunny and hot Berlin, and we've been having fun. The highlights so far include a brilliant talk by <a href="https://people.kth.se/~sara7/">Sara Zahedi</a> for whom mathematics, quite literally, acted as a language. Born in Iran, Zahedi came to Sweden aged ten and alone, not speaking a word of Swedish. It was at school in her maths classes, and through the language of maths, that she made her first friends in a strange country. </p><p>Zahedi works on the dynamics of fluids; in particular she tries to understand what happens at the <em>interface</em> at which two fluids with different properties meet. An example comes from medicine: scientists are working on chips that could act as mobile labs on which blood samples can be tested, for example for viruses or bacteria. Zahedi's work will help scientists simulate how blood and other fluids interact on those chips. That way, they can model the functioning of such chips on computers before actually making them, saving time and money.</p>
<p><a href="https://www.maths.ox.ac.uk/people/james.maynard">James Maynard</a> from the University of Oxford gave another great talk. Maynard has proved that there are infinitely many prime numbers (numbers only divisible by themselves and 1) which don't have the number 7 as a digit. And he didn't do this just for the fun of it. Prime numbers have fascinated mathematicians for millennia, but some very basic questions (for example if there are infinitely many pairs of prime numbers that are 2 apart) still haven't been answered. Maynard's problem about the digit 7 serves as a test bed for other, more fundamental questions: he hopes that the mathematical tools he has developed will generalise. We interviewed Maynard about a different result last year, and you can read the article <a href="/content/findgap">here</a>.</p>
<p>Both Maynard and Zahedi have won a <a href="http://www.euromathsoc.eu/prizeseuropeanmathematicalsociety">prestigious prize</a> awarded by the <a href="http://www.euromathsoc.eu">European
Mathematical Society</a> to young researchers not older than 35 years, in recognition of excellent contributions to mathematics.</p>
<p>Another favourite was a talk by <a href="http://www.dmg.tuwien.ac.at/pottmann/">Helmut Pottmann</a> of the Technical University of Vienna, about the maths of architecture. Pottmann explained that if you want to build beautiful freeform structures in a way that's not only aesthetically pleasing but also feasible and affordable, you are quickly led into a field of maths called <em>differential geometry</em>. And although architects make much use of digital design tools these days, these tools are not yet specialised enough, and Pottmann thinks that mathematicians will have an important role to play in developing better ones.</p>
<p>Watch this space for more detailed articles on the work of Maynard, Zahedi and Pottmann.</p>
<p>For the rest of today we're looking forward to, amongst other things, talks on the history of mathematics, a women in mathematics reception, and an event of the popularisation of maths. We'll report on those tomorrow.</p></div></div></div>
Wed, 20 Jul 2016 12:50:28 +0000
mf344
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The European Congress of Mathematics 2016
https://plus.maths.org/content/europeancongressmathematics2016
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/ECM_icon.png" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><div class="rightimage" style="maxwidth: 350px;"><img src=" /content/sites/plus.maths.org/files/news/2016/ECM/Berlin.jpg" alt="Berlin" width="350" height="200" />
<p></p>
</div>
<p>We've just arrived in Berlin for the <a href="http://www.7ecm.de">European Congress of Mathematics 2016</a>! More than 1300 mathematicians are expected at the Congress, which takes place at the Technical University Berlin. Every four years, this summit gathers
participants from Europe and around the world under the auspices of the <a href="http://www.euromathsoc.eu">European
Mathematical Society</a> (EMS). For one week, the mathematicians will discuss the entire spectrum of contemporary
mathematics. </p>
<p>Unfortunately we weren't able to attend the opening yesterday, but rumour has it that a brass band had mathematicians shimmying to some favourite film and TV tunes — interesting! Over the next two days we're particularly looking forward to meeting some of the winners of the EMS prizes, which are awarded to young researchers not older than 35 years, in recognition of excellent
contributions in mathematics. One of them, Peter Scholze, we already met two years ago at the <a href="/content/category/tags/icm2014">International Congress of Mathematicians</a> in Korea. You can listen to our interview with him <a href="/content/nextgeneration">here</a>.</p> </div></div></div>
Tue, 19 Jul 2016 08:49:04 +0000
mf344
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The Higgs mechanism
https://plus.maths.org/content/higgsmechanism
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/gold_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Juan Maldacena</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p><em>This article is part of the <a href="/content/node/6614">Financial ether package</a>. Click <a href="/content/weakforce">here</a> to see the previous article in the series.</em></p>
<hr/>
<p>As we have seen in the <a href="/content/weakforce">previous article</a>, formulating the theory of the weak force as a
gauge theory leads to a problem: the gauge theory says there should be
massless particles associated to the weak force, and such particles
haven't been observed in nature. </p>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/gold.jpg" width="350" height="234" alt="Gold example"/><p>Gold opens up a new opportunity to speculate.</p>
</div>
<! Image from fotolia >
<p>But it turns out that there is a way around this problem. It is
called the <em>Higgs mechanism</em>, and although it is named after
the physicist <a href="https://en.wikipedia.org/wiki/Peter_Higgs">Peter Higgs</a>, it was actually
proposed by many researchers too (you can find out more <a href="https://en.wikipedia.org/wiki/Higgs_mechanism#History_of_research">here</a>). </p>
<p>Here we will explain it using the economic analogy. So far, we have assumed that you can only carry money
between countries. Now, let us assume that you are also allowed to carry gold.
Gold has a price in each country, which is set by the inhabitants of each country independently
of the others.
A savvy speculator realises that a new opportunity opens up.
You can now buy gold in one country, take it to the next, sell it, and bring back the money to the first country.
As an example, say that the exchange rate between Pesos and Dollars is 4 Pesos = 1 Dollar. And the price of
gold in Argentina is 40 Pesos per ounce and the price in the USA is
5 Dollars per ounce. What would you do? </p>
<p>
Think about it! Do not continue reading until you have the answer. It
is a bit hard, but worth the effort!</p>
<p>You would start with five Dollars in the USA, buy gold there, go to Argentina, sell it for 40 Pesos,
go back to the USA and get 10 Dollars when you cross the bridge back.
This operation then has a gain of a factor of two, or a
100% profit. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/GoldExample2.png" width="600" height="314" alt="Gold example"/><p style="maxwidth: 600px;">Here we see two countries with their respective gold prices. We have also given
the exchange rate between them. For these prices and exchange rate it is possible to earn
money by buying gold in the USA, taking it to Argentina, selling it there and bringing
back the money, exchanging it at the bank, of course. The net gain is a factor of two or
100%. Gold is indicated in yellow and money in green.</p>
</div>
<p>Note that we continue to have the gauge symmetry. If the Argentinean government changes the currency to Australes
your gain would be the same. </p>
<p> Now, we can use this gauge symmetry to choose the currency so that the price of gold is the same in all countries.
Let us call the new units New Pesos and New Dollars. Now the price of gold is 1 New Peso per ounce and 1 New Dollar per
ounce. However, the exchange rates might not be one to one. In fact, they cannot be one to one if
originally there was an opportunity to speculate. For the example above the new exchange rate is
1 New Peso = 2 New Dollars. Note that this is <em>not</em> a "gold standard" that removes all exchange rates.
It is very important that the exchange rates are still present. </p>
<p>In summary, now the new prices and exchange rates are </p>
<p>1 New Peso = 2 New Dollars</br>
1 ounce = 1 New Peso</br>
1 ounce = 1 New Dollar</p>
<p>If you are a speculator, now it is easier to see what to do, isn't it?
With these new currency units obtained by setting the price of gold to 1 one can immediately see that if any exchange rate is different than
one to one, then there is an opportunity to speculate by performing the gold circuit.
The opportunity to speculate remains, and the gain remains a factor of two, or
100% profit. As usual, the net gain does not change when we change currency units. </p>
<p>Now let's go back to looking at a wave in our economic model. Let us first use a gauge transformation to set the price of gold to the be same in all countries. Now any exchange rate that is different than 1:1 will give rise to a speculative opportunity by running the gold circuit along that link, or bridge. It does not matter, whether the wavelength is long or short.</p>
<p>Therefore, the speculator's gain doesn't necessarily decrease with the wavelength. Once we set the price of gold to 1 everywhere, any
exchange rate that is different from one to one leads to an opportunity to speculate. </p>
<p>
In the physics version, this would mean that it now costs some energy to move any exchange rate away from one to one. This cost is present even for long
wavelength configurations. In physics, this leads to a massive particle, to a massive photon. A mechanism of this
sort is happening inside superconductors, and this was indeed an inspiration for the Higgs mechanism. </p>
<div class="leftimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/sps.jpg" width="350" height="272" alt="Gold example"/><p>The massive Z and W bosons were discovered at the the <a href="https://home.cern/about/accelerators/superprotonsynchrotron">Super Proton Synchrotron</a> at CERN in the 1980s. Image © 19762016 <a href="https://home.cern">CERN</a>.</p>
</div>
<! Image from CERN >
<p>We think that a similar mechanism gives
mass to the particles related to
the weak force. At each spacetime point (country) there is some object (the analogue of gold)
that has some orientation in the weak sphere. The details are more complicated because the symmetries correspond to rotations, and we
will not describe it here. Essentially, the presence of this object means that the cost of exciting waves does not go to zero as the wavelength tends to zero. And this in turn means that the particles associated to the waves in the weak force field have a nonzero mass. </p>
<p>If one sets the price of gold to one everywhere, then one is not free to do gauge transformations to further modify exchange rates. For this reason people sometimes say that the gauge symmetry is <em>broken</em>. This is conceptually misleading because the gauge symmetry is still present, if we remember that we need to also change the price of gold. </p>
<p>There is one interesting fact we have neglected to tell you so far. Photons and the corresponding particles for the weak force actually <em>mediate</em> the respective forces. When particles interact through these forces, it's by exchanging mediator particles. For electromagnetism the mediator is the photon, and for the weak force the mediators are the massive <em>W</em><sup>+</sup>, <em>W</em><sup></sup> and <em>Z</em> bosons. All of these have been observed in experiments.</p>
<h3>Quantum mechanics</h3>
<p>The system we have described so far, through the economic analogy, gives rise to
what is normally called a <em>classical field theory</em>. A field is a quantity that is defined at each
point in spacetime. For example, the price of gold is a field; at each point in spacetime it takes
a definite value. Similarly, the exchange rates are also fields. For each point, we have one exchange rate
per spacetime dimension, since the number of neighbours that a country has
is proportional to the dimension of spacetime. </p>
<p>But this is not the whole story. The dynamics of these fields is ruled by the laws of
quantum mechanics. An important feature for these laws is that they are <em>probabilistic</em> — they depend, to some extent, on chance.
One might think that in the vacuum, when there aren't any particles around, all the fields are zero. However, this is not the case, they take
random values.
All we can say is that they are given by a certain probability distribution.
In the economic analogy, we can say that the
values of the exchange rates and the price of gold, are all random. This randomness follows a very
precise law, which is encoded in the precise form of the probability distribution.
We will not give the precise formula here (it can be found in the <a href="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Appendix_2.pdf">appendix</a>), but we simply note
that it is such that configurations of exchange rates and gold prices become less probable when
there are greater opportunities to speculate. We have a precise law for the probability of each configuration, but we
cannot predict with certainty, which of the possible configurations will occur when we look at the system. </p>
<p>Note that all these fluctuations happen at short distances. If we follow a very big circuit, then we pass
through many countries and their exchange rates all average out. In the vacuum, at long distance they
average out to zero so that we recover the classical result where the fields are all zero. </p>
<p> The probability cost that we have to pay when we set the exchange rates to values leading to larger speculative
opportunities is also related to the energy cost we discussed above. They are essentially the same. Higher energy
configurations are less probable.
In nature, the <em>W</em><sup>+</sup>, <em>W</em><sup></sup> and <em>Z</em> bosons that carry the weak force are very massive. They weigh around a hundred times the mass of the proton, which is a lot
for an elementary particle.
Their large mass explains the weakness of the weak force.
It implies that we are very unlikely to produce fluctuations in the "weak exchange rates".
Therefore, a particle that interacts only through the weak force, such as the neutrino, is very difficult to see.
In fact, a few per cent of the energy of the Sun comes out in <em>neutrinos</em> (this shows that the weak force is important for the workings of the Sun.)
However, we are totally oblivious to these neutrinos. They simply pass through us day and night and we do not see them.
You need very big detectors with
very sensitive electronics to catch a very tiny fraction of them. (You can find out more in <a href="/mysteriousneutrinos"><em>Mysterious neutrons</em></a>.)</p>
<p>The mechanism described above does give a mass to the mediators of the weak force, but
it does not explain why there should be a new physical particle, such as the Higgs boson. This is what we will look at next.</p>
<p>Previous article: <a href="/content/weakforce"><em>The weak force and massive particles</em></a> Next article: <a href="/content/higgsboson"><em>The Higgs boson</em></a></p><p>Back to the <a href="/content/node/6614"><em>Financial ether</em> package</a></p>
<hr/>
<div class="rightimage" style="width: 200px;"><img src="/content/sites/plus.maths.org/files/packages/2011/fqxi/fqxi_logo.jpg" width="200" height="42" alt="FQXi logo"/><p>
</div>
<p>These articles contribute to our <a href="/content/stuffhappensphysicsevents">Stuff happens: The physics of events</a> project. </p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" syle="maxwidth: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Maldacena.png" width="200" height="236" alt="EM force"/><p></p>
</div>
<p>Juan Maldacena is the Carl Feinberg professor at the Institute for Advanced Study at Princeton, New Jersey, USA.
He works on quantum field theory and quantum gravity and has proposed a connection between those two (see <a href="/content/illusoryuniverse">this article</a>). </p></p></div></div></div>
Sat, 16 Jul 2016 15:54:59 +0000
mf344
6618 at https://plus.maths.org/content
https://plus.maths.org/content/higgsmechanism#comments

The weak force and massive particles
https://plus.maths.org/content/weakforce
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/Sun_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Juan Maldacena</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p><em>This article is part of the <a href="/content/node/6614">Financial ether package</a>. Click <a href="/economicem">here</a> to see the previous article in the series.</em></p>
<hr/>
<div class="rightimage" style="width: 300px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Sun.jpg" width="300" height="286" alt="Sun"/><p>All the chemical elements around us, except for hydrogen and helium, were
"cooked" in stars like our Sun. Image courtesy <a href="http://nasa.gov"</a>. </p>
</div>
<p>After having explored our <a href="/content/economicem">economic analogy for the force of electromagnetism</a>, let us turn to the weak force.
It is the force
responsible for radioactive
decays. For example, a free neutron (when it's outside an atomic nucleus) decays in about 15 minutes to
a proton, an electron and particle called a <em>neutrino</em>. This is a very slow decay compared to other
processes that happen on microscopic time scales. The weak force is not terribly relevant for our
everyday life. However, despite its weakness, it played an important role in the history of the Universe. More
specifically, in the synthesis of the chemical elements in stars. In fact, all the chemical elements around us, except for hydrogen and helium, were
"cooked" in stars. The weak force played a crucial role in this process.
Closer to home, we can say that the weak force can move mountains!
Weak decays inside the Earth are partly responsible for maintaining its heat, which in turn moves the continents, creating the mountains.</p>
<p>The weak force can also be understood using a gauge theory. In this case, at each point in space
we have the symmetries of a sphere, let us call it the <em>weak sphere</em>. See the figure below.
We do not know whether the
sphere is real or not. What we do know is that when we go from one point in space to another we have to rotate the sphere (just as with electromagnetism we had to rotate the circle). Knowing how to rotate the sphere requires three pieces of information, three "exchange rates": we have to
specify a rotation axis (defined by two quantities) and an angle of rotation around that axis (the third quantity). Instead of carrying money, we are now carrying an object that has some
orientation in the weak sphere. If we start at a country with an object in the weak sphere, as we go to the neighbouring country we have
to reorient the object according to the "weak exchange rates". </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/CirclesSpheresWeak.jpg" width="500" height="312" alt="Weak spheres"/><p style="maxwidth: 500px;">The weak
force has the same symmetries as a configuration where we have a sphere at each point in
spacetime. We do not know whether the circles or spheres really exist as extra dimensions.
What we do know is that the gauge symmetry is the same as if they existed. The circles
or spheres are useful for visualisation but we only think about their symmetry and focus
only on the associated "exchange rates".</p>
</div>
<p>
So, instead of one magnetic field, we have three different types of magnetic fields.
There are equations, similar to the ones for electromagnetism, which govern the
behaviour of these magnetic fields together with the corresponding electric ones. </p>
<p>
When these equations were proposed by <a href="https://de.wikipedia.org/wiki/Chen_Ning_Yang">Chen Ning Yang</a> and <a href="https://de.wikipedia.org/wiki/Robert_L._Mills">Robert L. Mills</a> in 1954 the physicist <a href="https://de.wikipedia.org/wiki/Wolfgang_Pauli">Wolfgang
Pauli</a> strongly objected. Pauli said that the YangMills
theory implied that there should be new particles that don't have any mass, which are not observed in nature.
This was a beautiful theory killed by an ugly fact. </p>
<h3>Why massless particles?</h3>
<p>To understand Pauli's objection, let us first focus on some properties of waves.
In electromagnetism we have electromagnetic waves (such as light, X rays, etc). These waves have a wave length — that's the distances between successive peaks of the wave.
In physical systems one is often interested in the energy needed to excite waves of various
wavelengths. For a wave with a given amplitude this energy cost can depend on the wavelength.
For an electromagnetic wave, the energy cost decreases when we make the wavelength longer and longer. It costs more energy to excite the wave shown on the top than it does to excite the wave shown on the bottom. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/waves.jpg" width="400" height="473" alt="Weak spheres"/><p style="maxwidth: 500px;">These two waves have the same amplitude, but the top one has a shorter wavelength than the bottom one. In electromagnetism it costs more to excite a wave with a shoer wavelength than it does to excite a wave with a long wavelength.</p>
</div>
<! Image made by MF >
<p>
It turns out that an electromagnetic wave can also be thought of as a stream of particles (that's the famous <em>wave particle duality</em> of quantum mechanics — we won't go into detail, you can find out more <a href="/content/schrodinger1">here</a>.) These particles are called <em>photons</em>.
The mass of photons is related to the energy it costs to excite an electromagnetic wave with a very long wavelength.
This is connected to the famous formula <em>E</em> = <em>mc</em><sup>2</sup>. Unfortunately I have not found a short way to explain this, so you will have to trust me on this. </p>
<p>
In our economic analogy we have not talked about energy. Let us simply say that the energy increases as the gain
available to speculators increases. This makes intuitive sense, the more the speculators can earn, the harder it is for the banks!
Therefore configurations with less gain have a lower energy cost. </p>
<p>
How do we represent an electromagnetic wave in our countries analogy? First notice that when you have a square of four countries, it is always possible to adjust their exchange rate so that the rates corresponding to two opposing sides of the square are 1:1. We can use our gauge symmetry to do this without changing anything. The exchange rates of the other two sides of the square simply need to be adjusted accordingly. Here is an example:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/currency_square.png" width="600" height="338" alt="Weak spheres"/><p style="maxwidth: 600px;">We can adjust the exchange rates so that the two horizontal edges of the square each have an exchange rate of 1:1. We need to adjust the other two exchange rates accordingly, but the overall change makes no difference.</p>
</div>
<! Image made by MF >
<p>An electromagnetic wave is represented by a whole sequence of squares as shown in the figure below, which we assume have exchange rates of 1:1 along their horizontal sides. The wave is consists of the oscillating values of the vertical sides. In each of the two example below, the amplitude of the waves, measuring the difference between its troughs and peaks, is the same. However, the top wave has a long wave length (it takes three squares to go from trough to peak) while the bottom one has a short wave length (it takes only one square to go from trough to peak).</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/LongShortWavelength.jpg" width="600" height="590" alt="wavelength"/><p style="maxwidth: 600px;">We can see two configurations of exchange rates, one with a long wavelength and one with a short
wavelength. In each case the wave consists of the fact that the numbers in the middle go up and down
as we move from left to right. Each segment is a bank sitting between two countries. The
countries sit at the line intersections. The number indicates the exchange rate when you
cross the bridge in the direction of the arrow. Notice that the total amplitude of the wave
is the same; the exchange rates go between 1 and 1.3. The gain obtained by following an
elementary square circle, indicated by the green lines, is smaller for the longer wavelength
configuration.</p>
</div>
<p>You can work out for yourself that a speculator following a circuit within a single square (illustrated by the green arrow in the figure) will make a profit that depends on the <em>difference</em> between neighbouring exchange rates. It doesn't depend on the absolute magnitudes of the exchange rates. This is a crucial point: given a wave with a particular amplitude, a longer wavelength means smaller differences between neighbouring exchange rates (that's because the amplitude is divided into more steps). And smaller differences between neighbouring exchange rates mean less gain from following a square circuit.
</p>
<p>We have said above that electromagnetic waves are associated to particles called photons, and that the mass of a photon is related to the energy it takes to excite a wave with a very long wave length. Using our analogy we have seen that this energy cost (the gain) goes to zero as the wavelength gets longer. This, essentially, is why the photon has no mass. </p>
</p>
<p>This argument works for electromagnetism and it also works for the weak force for
the same basic reason. As in electromagnetism, there is a field associated to the weak force, and waves can travel through this field. These waves are related to particles, and the same argument we used for electromagnetism shows that those particles should be massless. At least it is true for the version of the weak force described so far. </p>
<p>Previous article: <a href="/content/economicem"><em>Electromagnetic economics</em></a> Next article: <a href="/content/higgsmechanism"><em>The Higgs mechanism</em></a></p><p>Back to the <a href="/content/node/6614">Financial ether</em> package</a></p>
<hr/>
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</div>
<p>These articles contribute to our <a href="/content/stuffhappensphysicsevents">Stuff happens: The physics of events</a> project. </p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" syle="maxwidth: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Maldacena.png" width="200" height="236" alt="EM force"/><p></p>
</div>
<p>Juan Maldacena is the Carl Feinberg professor at the Institute for Advanced Study at Princeton, New Jersey, USA.
He works on quantum field theory and quantum gravity and has proposed a connection between those two (see <a href="/content/illusoryuniverse">this article</a>). </p></div></div></div>
Sat, 16 Jul 2016 14:59:19 +0000
mf344
6616 at https://plus.maths.org/content
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The financial ether
https://plus.maths.org/content/node/6614
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/clouds_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p>
Current theories to describe the fundamental forces and particles of nature rely on beautiful symmetries known as <em>gauge symmetries</em>. These aren't easy to describe, but the renowned physicist <a href="https://www.ias.edu/scholars/maldacena">Juan Maldacena</a>,
of the Institute of Advanced Study in Princeton, has developed a great analogy: he thinks of space (the ether, so to speak) as a grid of countries and of particles as travellers keen on making money by speculating with currencies. The analogy even explains the existence of the famous Higgs boson — and you don't need to be familiar with exotic <a href="/content/briefhistoryquantumfieldtheory">quantum field theories</a> to understand the picture. This collection of short articles will take you through the analogy. Or if you prefer, you can watch Maldacena explain the analogy in the video <A href="/content/node/6614#video">below</a>. Enjoy!</p>
<ul><li><a href="/content/introem">A brief introduction to electromagnetism</a></li>
<li><a href="/content/itseconomystupid">The economic analogy</a> </li>
<li><a href="/content/economicem">Electromagnetic economics</a> — The economic model applied to electromagnetism</li></ul>
<p>This sums up the economic model for the force of electromagnetism. But there's more! The following articles look at a similar theory for the weak nuclear force, which can be described using the same analogy.</p>
<ul><li><a href="/content/weakforce">The weak force and massive particles</a></li>
<li><a href="/content/higgsmechanism">The Higgs mechanism</a></li>
<li><a href="/content/higgsboson">The Higgs boson</a></li></ul>
<p>For the slightly more advanced here is a <a href="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Appendix_2.pdf">mathematical appendix</a> in pdf format.</p>
<p>(Note that the analogy between foreign exchange and lattice gauge theory was noted in
K. Young, <em>Foreign exchange market as a lattice gauge theory</em>, American Journal of Physics 67, 862 (1999). )</p>
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<p>These articles contribute to our <a href="/content/stuffhappensphysicsevents">Stuff happens: The physics of events</a> project. </p></div></div></div>
Sat, 16 Jul 2016 14:30:04 +0000
mf344
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Electromagnetic economics
https://plus.maths.org/content/economicem
<div class="field fieldnamefieldabsimg fieldtypeimage fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><img class="imgresponsive" src="https://plus.maths.org/content/sites/plus.maths.org/files/abstractpics/%5Buid%5D/%5Bsitedate%5D/exchange_icon.jpg" width="100" height="100" alt="" /></div></div></div><div class="field fieldnamefieldauthor fieldtypetext fieldlabelinlinec clearfix fieldlabelinline"><div class="fieldlabel">By </div><div class="fielditems"><div class="fielditem even">Juan Maldacena</div></div></div><div class="field fieldnamebody fieldtypetextwithsummary fieldlabelhidden"><div class="fielditems"><div class="fielditem even"><p><em>This article is part of the <a href="/content/node/6614">Financial ether package</a>. Click <a href="/content/itseconomystupid">here</a> to see the previous article in the series.</em></p>
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<p>In physics the <a href="/content/itseconomystupid">countries in our economic model</a> are analogous to points, or small regions, in space. A magnetic field comes with <em>magnetic potentials</em> — these are analogous to the exchange rates between currencies. The speculators are called electrons or charged particles. The existence of a magnetic field corresponds to a configuration of exchange rates that enables an electron to "earn money". In the presence of magnetic
fields, the electrons move in circles in order to become "richer".
The amount of gain is
related to the magnetic field.
In fact, the total gain along the circuit is the <em><a href="https://en.wikipedia.org/wiki/Magnetic_flux">flux</a></em> of the magnetic field through the area enclosed
by the circle (put very simply, the flux is the number of magnetic field lines that pass through the area).</p>
<div class="rightimage" style="maxwidth: 350px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/exchange.jpg" width="350" height="225" alt="Bureau de change"/><p>Magnetic potentials are analogous to exchange rates.</p>
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<! Image from fotolia >
<p>If you are familiar with electromagnetism all this will make sense to you. If not, just stick with the analogy: the presence of a magnetic field corresponds to a configuration of potentials (exchange rates) that enable an electron to become richer (with the gain related to the magnetic field in the area) and will therefore set the electrons moving.</p>
<p>
Now imagine that you are a speculator that has debt instead of having money. In that
case you would go around these countries in the opposite direction! Then your debts would be reduced
in the same proportion.
In physics, we have particles called <em>positrons</em>, which are like electrons
but with the opposite charge. In fact, in a magnetic field
positrons circulate in the opposite direction as compared to electrons. </p>
<p>In physics we imagine that this
story about countries and exchange rates is happening at very, very short distances, much
shorter than the ones we can measure today. When we look at any physical system, even empty space, we are looking at
all these countries from very far away, so that they look like a continuum.
See the figure below. When an electron is moving in the vacuum it is
seamlessly moving from a point in spacetime to the next. In the very microscopic description, it would be constantly changing
between the different countries, changing the money it is carrying, and becoming "richer" in the process.
(In physics we do not know whether there is an underlying discrete structure like the countries we have described. However,
when we do computations in gauge theories we often assume a discrete structure like this one and then take the <em>continuum
limit</em> when all the
countries are very close to each other.) </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Continuum.jpg" width="600" height="171" alt="Zooming out"/><p style="maxwidth: 600px;">Here we display the grid of countries at various scales. We are zooming out as we
move to the right. If we zoom out sufficiently we can view the whole grid as a continuum.</p>
</div>
<! Image provided by author >
<p>The <em>gauge symmetry</em> of electromagnetism is a little different from the one in the economic model, which was about dividing or multiplying by a given factor.
The "currency" of electromagnetism isn't described by numbers that lie along the number line, but by points that lie on a circle. Each "country" (point in spacetime) chooses a starting point on its personal circle — that's its base point. The value of any other point <em>P</em> on the circle is given by the angle defined by <em>P</em>, the base point and the centre of the circle. Each country could at any time decide to simply rotate its circle, moving its base point somewhere else. All charged objects sit somewhere along this circle. If the origin is changed, so are their nominal positions. </p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/circle_0.png" width="600" height="325" alt="Circles"/><p style="maxwidth: 600px;">Each point in spacetime "picks" a base point on the circle with respect to which it measures angles. Rotations of the circle are gauge symmetries.</p>
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<! Image made by MF >
<p>The exchange rate between two countries is simply the angle through which you'd have to rotate the first country's circle to move its base point to the location of the second country's base point. </p>
<p>We can picture all this by imagining a little circle for every point (country) in spacetime:</p>
<div class="centreimage"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/CirclesSpheresEM.jpg" width="600" height="363" alt="Circles"/><p style="maxwidth: 600px;">The electromagnetic interaction has the same symmetries as a configuration
where we have a circle at each point of space. Here each space point is where
the black lines intersect. We can think of the circle as an extra dimension.</p>
</div>
<! Image provided by author >
<p>If this little circle really exists, then this would mean that we
need an extra dimension. Unlike our picture above, which is a twodimensional simplification, the space we perceive is threedimensional. If for each point there really is a little circle that does not live in any of the three spatial dimensions, then there needs to be a fourth dimension for it to exist in. However,
we do not know whether this circle is real. We do not know if indeed there is an extra dimension. All we know is that
the symmetry of the theory is similar to the symmetry we would have if there was an extra dimension.
In physics we like to make as few assumptions as possible.
An extra dimension is not a necessary assumption, only the symmetry is. Also the only relevant quantities are the magnetic potentials which
tell us how the position of a particle in the extra circle changes as we go from one point in spacetime to its neighbour. </p>
<div class="leftimage" style="maxwidth: 210px;"><img src="/content/sites/plus.maths.org/files/articles/2015/Maxwell/maxwells_eqns.gif" alt="Maxwell's equations. " width="210" height="180" /><p>These are Maxwell's equations. You can find a nice explanation of them <a href="http://www.irregularwebcomic.net/1420.html">here</a>.</p>
</div>
<p> In electromagnetism the electric and magnetic fields obey some
equations, the socalled<em> <a href="/content/maxwellsequationandpowerunification">Maxwell equations</a></em>.
In the economic model this would be analogous to a requirement on the exchange rates. </p>
<p>We can
intuitively understand this requirement: let us imagine we have a configuration with generic
exchange rates. Speculators start carrying money around.
Suppose
we focus on a particular bridge, where a particular
bank sits. There will be speculators crossing this bridge in both directions. However, if there are more speculators going in one direction than
in the other direction, then the bank might run out of one of the currencies.
For example, consider the bank sitting at a bridge that connects Pesos to Dollars.
If there are more speculators wanting to buy Dollars than there are speculators wanting to buy Pesos, the bank will run out of Dollars. </p>
<p>
If this happened in the real world, the bank adjusts the exchange rate so that
fewer speculators want to buy Dollars.
In fact, if we assume that the number of speculators following a particular circuit is proportional to the gain that they will have along this circle, then
we find that the condition for banks not to run out of either of the currencies, or that the net flow of money across each bridge is zero, is
equivalent to Maxwell's equations.
The mathematically inclined reader can find the derivation in the
<a href="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Appendix_2.pdf">appendix</a>. </p>
<p>
Let's summarise all of this. Particles, such as electrons and positrons, can be thought of as travellers moving through a grid of countries. The electromagnetic field determines the exchange rates between the countries' currencies. The presence of a magnetic field is akin to a configuration of exchange rates that enables electrons to make money, and positrons to reduce their debt, which sets them in motion around a circuit. The total gain they make is related to the magnetic field. The whole thing is characterised by a gauge symmetry: points in spacetime define a "currency" and changing the currency makes no difference.</p>
<p>This is the economic analogy describing the gauge symmetry of the
electromagnetic force. In the next article (link) we will move one to
another of the fundamental forces: the weak force.</p>
<p>Previous article: <a href="/content/itseconomystupid"><em>The economic analogy</em></a> Next article: <a href="/content/weakforce"><em>The weak force and massive particles</em></a></p><p>Back to the <a href="/content/node/6614"><em>Financial ether package</em></a></p>
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<p>These articles contribute to our <a href="/content/stuffhappensphysicsevents">Stuff happens: The physics of events</a> project. </p>
<hr/>
<h3>About the author</h3>
<div class="rightimage" syle="maxwidth: 200px;"><img src="/content/sites/plus.maths.org/files/articles/2016/Maldacena/Maldacena.png" width="200" height="236" alt="EM force"/><p></p>
</div>
<p>Juan Maldacena is the Carl Feinberg professor at the Institute for Advanced Study at Princeton, New Jersey, USA.
He works on quantum field theory and quantum gravity and has proposed a connection between those two (see <a href="/content/illusoryuniverse">this article</a>). </p></p></div></div></div>
Sat, 16 Jul 2016 14:15:44 +0000
mf344
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