Welcome back to the Oxford Mathematics Public Lectures Home Edition.
My name is allegorically and I'm in charge of external relations for the Mathematical Institute as usual.
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It is a great pleasure for me to welcome today, my colleague John Kito with the said Lee and professor of natural philosophy at Oxford University.
As you may know, the Sadlier Professorship is the oldest scientific chair in Oxford and is dedicated to applied mathematics.
John was elected to this chair two years ago after an illustrious career at the University of Bristol.
He has broad research interests in quantum physics,
random matrix theory and its unlikely connexion to number theory through the remains that a function is.
Research is truly fascinating as he draws connexion between feeds that are apparently disconnected,
and I've been always impressed by John's originality and clarity of thoughts.
Continuing in the same vein and exploring unlikely connexions today,
John will take us on a wild ride of extreme events and explain to us how the
heights of unexplored Montaigne gold medals at the Olympics quantum period,
machine learning and prime number are all somewhat connected.
So thank you very much, John, for doing this. Please start now.
Well, let me begin by thanking Alan for the kind invitation to give this lecture.
I'll be speaking about it a collection of problems which superficially may seem to have no connexion,
but which I shall argue and are linked by a mathematical threat.
And it's that mathematical threat that will be the main focus of the talk rather than any
one of the individual problems that I shall speak about and that are connected by it.
I'd like to begin by inviting you to imagine that you're hiking in a mountainous terrain rather like the one shown in this photograph on your hike,
your goals may be to climb one of the peaks that you see shown here,
perhaps to walk along a ridge from that peak to a neighbour to neighbouring peaks down from the peak and then back up to a neighbouring peak.
Or it may be to descend all the way down and find a dip where water has collected
and a lake or a pool has formed in which you can swim or where you can skim stones.
Mathematically speaking, you're visiting what we would call critical points in the height of the Earth's surface,
so critical points might be a local minimum.
That's a point where a step in any direction takes you upwards.
And that's the sort of place where a pool or lake may form as water collects a different type of critical point would be a little maximum.
That's a point where a step in any direction takes you downwards, and the peaks that you visit would be examples of ours.
And finally, there are subtle points two points which are a minimum in one direction and a maximum in the other direction.
And you would encounter these on the ridge between two peaks.
So your hike may take you to these various critical points, but perhaps your goal is more ambitious,
perhaps you're more adventurous and your intention is not to climb any old peak, but in fact to climb the highest peak in this vicinity.
And so you may recognise that to be there Nevis,
which is shown there in the background and much of this talk will be concerned with that kind of problem.
The problem of identifying finding the highest peak in some problem, that's on some surface that looks like the one that I'm showing you here now.
In terms of mountains, you may think this is a little artificial. We know where the highest peaks are.
We have maps that show us that and we even have GPS. But many of the problems I should be speaking about later are not of that kind.
We we don't have an ID in advance of where the highest peak is.
And our job may be to find it and maybe to understand how hard it is likely to be.
And so I'm showing you this more for illustrative purposes,
but I think it conveys the message as another message that I want to convey from this picture, which is that looking at it,
you may form the impression that the height of the Earth's surface in this sense is a
rather irregular and random function of the position on the position where you are.
So knowing your position doesn't necessarily mean that you can automatically deduce what the height of the Earth's surface would be.
This terrain here is seemingly random and or irregular.
But in other ways, you know, the height of the Earth's surface at one point in this terrain isn't automatically tell you
what the height of the Earth's surface would be some distance away or in some other direction.
So that means that we want to model the height of the surface as as that as being random.
And there are many ways to do that. And the simplest one would be to assume that the height of the Earth's surface in
that photograph is described by a normal distribution or a bell shaped curve,
as shown in this picture. So there is an average height that's the centre of the distribution,
the highest point in the graph there and then the probability or likelihood of
finding heights much greater than the mean decay rapidly as a function of height.
And likewise, the probability of finding very low heights decays rapidly as you go away from the main.
Now, I should say you could criticise this as a model for heights in mountain ranges.
That's not my point.
And in fact, the examples I'll be showing you later are ones where we do believe that this is the right way to model wide distribution of heights.
But just for the moment, let's take this to be the distribution heights in the mountain range that I showed you.
The question, then, that you might ask and these will be the ones that I should be focussing on throughout the tour are in a
situation where the terrain can be modelled by a random surface with a normal distribution of heights,
a bell shaped curve of heights.
How effective can we expect message to be for locating the highest maximum is the to the peak yes, peak or equivalently the lowest minimum.
How high should we expect the highest maximum to be? You're given a random service, do you expect exceptionally large hikes to appear?
And how does that depend on the total number of peaks in the in the in the service?
And finally, to what extent do we expect these answers to depend on the dimension of the surface?
The Earth's surface is two dimensional examples.
I'll give you later. At least some of the most important ones will concern services that have a vastly higher number of dimensions.
Thousands, millions or billions. How do we expect random terrains to look like in very high dimensional spaces?
So let me give you a few more examples just to whet your palate.
Here you see the surface of the sea.
And in fact, it turns out that the normal distribution is a very good model for the distribution of heights of the surface of the sea.
And here, as it's painted, the surface is rather irregular, rather random.
So if you were sailing the boat shown in this picture,
you might well wish to know what is the how high is the highest wave you're likely to encounter,
or how deep would be the deepest trough you might descend down into.
In particular, how would that depend on the length of your voyage as you encounter more and more waves?
Do you expect to encounter to find larger and larger highest waves?
Or does the problem not depend too much on how many waves you're likely to encounter on your voyage?
Here's a second example, and this comes from quantum mechanics. Contract is also a wave theory.
In this case, it's a wave theory of how things move.
So the thing I want to the system I want to consider is a point particle moving inside some domain and bouncing off the walls.
Think of it as a billiard ball bouncing around InSight's billiard table.
But in this case, the billiard table is has a cardioid shape. What you see, there is one trajectory of the billiard ball, and it's highly irregular.
The motion inside the cardioid is is chaotic.
What you see on the right is a quantum wave function for this same problem.
So this describes the quantum mechanics of the motion of a Billy Ball inside a cardioid shaped billet table.
You see the peaks of the wave. In fact, what's plotted here is the square of the wave.
So half of those peaks will actually be deep minima reflected back upwards by the act of squaring the wave function.
But physically, that's the right thing to do,
because the square of the wave function in this case gives you the probability of finding the particle at a given place in a given vicinity.
And so you might well wish to know how high is the highest peak of this wave function likely to be?
Are there places where we we might expect to find vastly higher probability of finding the particle than other places?
How much how much just the height of the highest peak in this wave function depend on the total number of peaks that you see there.
As we look at way functions with more and more peaks, do we expect to find places, positions with increasingly high probability?
And how does that depend on the number of peaks? Well, these are the sorts of questions I want to consider and.
To start with, I want to consider a warm up question, which has nothing to do with waves.
Nothing to do with management trends.
It's a rather more elementary question, but I want to argue it captures much of the spirit of the problems that I discussed so far.
So the question I want to discuss is in the Olympics.
Should we expect the number of gold medals won by a country to be proportional to the relative size of that country's population?
So I'm focussing on gold medals here because they do signify extreme ability.
One gets a gold medal for running, for being the very fastest person.
One of having a being an extreme of speed, one gets a gold medal in the javelin for being the person who can throw the furthest teams,
get gold medals in synchronised swimming for being able to swim in the most synchronised way in the competition.
So Gold Medal signified will measure extreme events.
And the question is, does the number of gold medals that a country, when should we expect it to be proportional to the population of that country?
Well, this was picked over. This question was picked over in the press and after the 2012 Olympics and the 2016 Olympics.
And the following was that was the sort of analysis that you found in many, many articles throughout the press, at least the British press,
the when the Great Britain had a population of roughly 65 million and won in the 2016 Olympics, 27 gold medals.
The US population of roughly 320 million, that's about five times the population of Great Britain and one forty six gold medals,
so little under twice the number, China had a population which was roughly 20 times that of Great Britain.
And yet it went about the same number of gold medals. Japan population about twice that of Great Britain.
Yet when half the number of gold medals, the Great Britain did not twice the number.
So the articles where one found this statistical analysis carried out the conclusion drawn in those articles was that Great Britain had done
disproportionately well and that the country had done better than should have been expected on the basis of the size of its population.
So the question I want to address is,
is that is it true that the number of gold medals you expect to win is proportional to the size of the population going back to my questions earlier?
Is it true that the size of extremes, highest peaks, highest waves are proportional to the total number of waves?
Here's another more careful analysis of the Olympics. You find many of these on the web.
I click this one at random and adjusting for population.
These were the most successful countries at the Rio Olympics.
So while the United States won more medals than any other country in the Rio Olympics, the article points out that its population was relatively high.
And so this analysis took the number of medals were for all various countries.
And compared them with the population size taken very carefully from two sources,
so this is a very careful analysis, and the two sources were the United Nations and the CIA World Factbook book,
which I confess I didn't know existed until I did this research to find this website that divided the number of medals by the size of the population.
And they found that the country which did best was Grenada, Grenada, perhaps which only won a single medal, which was a very small population.
And so the number of medals won by the population, it came out top.
And the other countries that did well were the Bahamas, New Zealand and Jamaica.
In fact, and the US didn't do very well at all. Here is the list, or at least the top of the list that you find on this website.
So Grenada or Grenada, I did the best and by some significant margin.
And then the Bahamas, New Zealand, Jamaica, Denmark, Croatia did very well.
Slovenia, Azerbaijan, Georgia, Hungary, Bahrain, Lithuania, Great Britain comes down this list.
It's about three quarters of the way down there. And so it didn't do spectacularly well, according to this analysis,
but did much better than other big countries, certainly much better than the US China.
Japan, etc. But is this a reasonable analysis?
Is this the right? Is it the right way to level the playing field?
To divide by the total size of the population. Well, there are very different answers you can give to this.
Here are two of them two different opposing perspectives, both mathematical.
One mathematical perspective is that Great Britain did do disproportionately well and that you should normalise the
data in order to level the playing field between countries of different sizes by dividing by the population size.
And here's how the argument goes. So my first answer is yes, I'm just let's idealise.
Let's consider an Olympics involving one event and two countries, and let's say the populations of the two countries are A and B.
One of the people in these combined countries has to win the Gold Medal.
And let's assume that there's no bias that the person who wins that one country is not on average athletically more capable than the other country.
And so the likelihood of that person who wins being in one country in the
first country is just a its population divided by the total number of people.
And if you pick somebody at random from the A plus B people in the combined populations,
there's a probability of a over eight plus B that you'll pick someone in the first population and the probability that the winner,
whoever she is, is in the second population in the second country is whereas the fraction of
people of the total combined population in that second country would be over,
eight must be. And so the ratio of the two is a over B ratio that you probabilities.
And so this would suggest that indeed normalising by dividing by the population size is the right thing to do,
the right way to level the playing field. But I would argue that this isn't the most accurate model of how the Olympic Games works.
You don't see the Olympic Games doesn't involve competitions between all members of the populations of the various countries involved.
I've never been involved in the Olympic Games, for example,
so I would argue that a more accurate model of how the Olympic Games works is it's really a competition
between the fastest person and country in the first country against the fastest person in the second,
or the person who can throw the furthest in the first country against the person who can throw the furthest in the second country.
So I would argue a better way to to analyse this would be to ask if you take a population,
a country and how fast is the fastest person likely to be in that country?
Well, how far can the strongest person throw that javelin?
So if people's sporting abilities, speed, strength,
stamina have a normal distribution and are independent of each other out of a population of people,
how fast or strong is the fastest or strongest person likely to be?
And I would argue that this is a better model for the people who enter the Olympics and actually take part in the competition.
So mathematically speaking, we should take end numbers, draw them independently at random from the normal distribution.
So we wait the numbers with the probability that's given by the bell shaped curve or the normal distribution.
And we'll take that distribution to mean new. That's the value at the centre of the distribution and variance.
Sigma squared sigma is the width of the bell shaped curve.
And the question that is, what's the distribution of the largest of these numbers, how large you expect the largest one to be?
And the answer turns out to be the following one.
So it's given by this equation, which I show you in blue, that if you like equations, you'll you'll be able to unpick this very quickly.
But his equations are not quite your thing. Let me unpick it for you.
So the equations series the following to compute what you expect to be the largest of these numbers,
you take the mean centre point off of the bell shaped curve and you add on to that something.
You add on something because of course, the largest is is very likely to be larger than the mean.
The amount you add on is proportional to the width of the bell shaped curve.
That's no surprise if the if you take a normal distribution with a greater width, you expect to find more large values.
And then multiply the width is the square root of two times the log of the number of samples
we've taken from the distribution log event and to work for those who knew their logarithms,
they recognise that this depends barely a tool on and the logarithm grows with and it increases and increases.
But it does so barely at all. It's one of the slowest growing functions you could imagine.
We actually have here the square root of the logarithm, which grows even more slowly.
Some of you may know that a sort of paradigm of rapid growth is the exponential well.
The logarithm is the opposite of that. It's a paradigm of slowest growth, if you like.
So this is the answer, and it's an approximation to the answer, rather.
And it's a beautifully simple formula that tells you how big the extremes are likely to be.
So this same formula tells us how fast the fastest person is likely to be.
But it also tells us how high the highest wave is that we're likely to encounter in our boat.
The height of the highest wave,
if is if we assume a normal distribution of wave heights proportional to the square root of the log of the number of waves that we encounter.
So taking a much longer journey doesn't dramatically affect the height of the highest
wave you're likely to encounter very affected to do likewise in a quantum problem.
The height of the highest quantum amplitude, the highest wave point the in the quantum wave is proportional to the square root
of the log of the total number of peaks that you saw in the picture earlier,
so badly affected at all by the number of peaks.
In fact, if you analyse this little more, you find that typically you expect to get up to this maximum value.
Lots of lots of numbers. And then this is the largest.
And then after this you get nothing. So it's not that you expect to see one outlier dramatically faster than other people.
A wave that's dramatically higher than other people. In fact, you expect to see lots of waves up to about the height of the maximum and then nothing.
Lots of people who can run close to the speed of the fastest person, but then nothing.
And that, of course, of course, with your intuition or your experience at the Olympics,
there are lots of people who can run almost as fast as the fastest person.
That's why race is in the Olympics are exciting. You don't know exactly who's going to win, but the uh, so you get lots of people close the fastest,
not one outlier who's dramatically faster than everybody else.
And so I should say that going back to the Olympics, I'm not arguing that this is how you should model Olympics,
please don't use this as a basis for betting on the next Olympics.
And of course, what this means is the dependence on population is really rather small and dependent on other factors is far more important,
for example, facilities in a country's gross domestic product or the tradition of coaching in that country.
These are far more important effects in the population size.
Let me put in the populations just to illustrate how slowly the square root of the logarithm increases.
She put in the population of China and into the logarithm and divide by the square to the log since the population of Great Britain.
You get an answer this one. So barely any change between China and Great Britain in terms of the dependence on population size.
Now, here's a more accurate answer. This is a better formula, therefore more complicated.
And but let me pick this for you. So this is actually a very accurate formula for the height of the largest or the extremes in a normal distribution.
We have what we had before the mean new centre of the value centre of the distribution.
We have the term that we saw earlier, which increases extremely slowly with an.
And it's like the squirt of log in and then we subtract off a term which depends not just on the logarithm event,
but on the logarithm of the logarithm of. And this term actually decreases and increases.
So we have a term that's about constant, a term that does increase, but extremely slowly.
And a term that decreases, but extremely slowly. And then there are some small fluctuations, which I don't want to describe in this lecture.
Beyond the scope of what I want to discuss now, you don't need to remember this formula or any formula in this talk.
But there's one aspect I would like you to remember, and that's the number here shown in red.
The number one that no, I do wish you to remember because we're going to come back to that a little later.
So we have this very accurate formula, and it's described very accurately and heights of the high sea waves,
heights of the highest quantum waves, et cetera. Now,
you may argue that this analysis I've given you is is too simple because it ignores the
fact that abilities to have people in sporting events aren't independent of each other.
There are dependencies. If we consider the heights of people, once height is low to a good degree dependent on the heights of your parents,
if you have two very tall parents, your height is likely to be taller than average.
If you have two very athletic fast parents, there's a greater likelihood that you two will be athletic and fast.
How can we build this into a model? We get an idealisation of a family tree.
It's very idealised, but it just focuses on the essentials.
And we imagine at the top of the tree, a matriarch, it's the first generation, so to speak.
And we'll imagine just for the sake of simplicity that that matriarch has two offspring.
Each of those offspring has two offspring, each of their offspring have two offspring, etcetera.
Now the number two is not at all relevant here. It's just for illustrative purposes that I'm sharing this.
Nor do we have to assume that everyone has the same number of offspring.
Again, I'm just doing this for illustrative purposes to keep the description simple.
So that's why I'm going from the parent to an offspring,
let's imagine that the offspring acquire some characteristic, but he doesn't acquire a characteristic perfectly.
There's some there's some variance in the degree to which they acquire that characteristic.
So I would assume that as you go from a parent to an offspring, you pick up a factor and attribute drawn from the normal distribution.
The bell shaped curve.
So as you go down the generations each, each person picks up from their parent, an attribute which is drawn randomly from the bell shaped curve.
But let's imagine that your net attribute your net ability is the average of the
attributes of all of your ancestors going back to the matriarchal figure at the top.
So you see at the bottom there, there is a population of people and at the bottom of the Green Line,
there is a person and the attributes they collect is the average of the
attributes acquired through all of the generations going back to the matriarch.
They have a sibling shown at the bottom of the brown line and they to pick up attributes
from all of the generations going back and they'll differ from their sibling.
Only in the one attribute they've acquired from that parent, which is drawn at random and independently from the bell shaped curve.
So this is a way of combining inheritance with randomness.
And the question is, if you look at the population at the bottom, what's the distribution of attributes?
We didn't have that. A beautiful fact about the normal distribution is that if you average lots of numbers from the normal distribution,
they still the answer you get still has a normal distribution. So the people at the bottom in that generation shop bottom there.
The variation in attributes will be described by the normal distribution by the bell shaped curve, but they're no longer independent.
Because, for example, the person at the bottom of the Green Line and a sibling have lots of ancestors in common, lots of attributes in common,
so they will be more similar than people who are more distantly related from that,
and the level of similarity will be determined by their last common ancestor.
What in this case, do we expect to be the highest of those attributes?
Where if we have any people in that population, here's a formula for the for the largest of those end numbers.
And this is a formula that most people have been thinking about this for the last 20 30 years.
So this is relatively recently discovered formula, and the formula looks remarkably like the one I showed you earlier.
In fact, it's almost identical. So, so building in dependence via a family tree dairy effect.
The answer in terms of the size of the maximum height of that of the of the extreme.
The answer is almost identical. It differs only in one way. And that's the thing that was one before is now a three.
But that's in a very small turn. So this dependence is very small, but it is their dependence doesn't matter.
But only at this very low level, this very small turn turn that gets smaller and smaller as an increase.
But it is there, and the difference is that you go from one to three.
Now this three is universal. It doesn't depend on the fact that I assumed two offspring or equal numbers of offspring.
Anytime you have a family tree lying behind your your data, you expect to get a three and not to one.
So that's a description of of some of the.
Mathematics that underpins extreme events.
Now, let me tell you about some of the applications, and the first application I want to describe is to freezing of liquids to form solids.
So any material, if you raise it to a sufficiently high temperature is a liquid that means it's constituent parts,
the atoms and molecules that make up a free to wander around at random and explore various different configurations.
Each configuration carries an energy. And so as you explore various configurations, you're exploring ranges of different energies.
As you lower the temperature, you lower the range of energies that you can explore.
Until you reach the freezing point, when the liquid becomes a solid, where the constituent parts are fixed in space.
And that's because essentially they're stuck at the lowest energy configuration.
So when you're freezing a material, you're finding the lowest energy configuration.
You know, there might be configurations with local minima, but you're finding the very lowest one.
So when a liquid freeze is temperature lowered,
lowered the configuration of atoms and molecules seeks the lowest energy arrangement in many situations.
This lowest energy arrangement is highly symmetrical and highly ordered.
So, for example, if you melt some. Salt.
Common salt and then freeze it again, you know that you form crystals in those crystals are highly awkward and ordered and arranged.
And that's because the lowest energy configuration in that case is very well-defined.
There's a clear winner in the lowest energy configuration, and the system manages to find that every time.
And that's why such systems have a very well-defined freezing temperature and why every time you freeze the system,
you get the same configuration in the solid phase.
There are, however, materials where that's not the case.
And examples would be a glass, the glass that you see perhaps in the window, in your room or the glass that's in my spectacles.
And in these cases, the energy landscape is vastly more complicated, and there isn't a clear,
obvious winning landscape when it comes to being the minimum energy.
In fact, this landscape is so complicated there, rather like the mountain ranges I showed you earlier,
or rather like the surface of the sea or the quantum waves. There are lots of possible different arrangements,
all with more or less the same local minimal energies and finding the right one an obvious winner is difficult.
So when the system, when you lower the temperature, the system explores and finds itself in local minima and these you might get a
different minimum each time you might be stuck in a different minimum each time.
And there's no reason to expect that in the solid phase, the configurations will be the same.
They're certainly not hiding highly ordered. So the question is why do classes have relatively well-defined freezing transitions?
Here's a picture of a cartoon, if you like, of the energy landscape for a glass is computed by Chiara Camerata,
who's an expert on this area of mathematical physics.
And you see that what would I said that there are many possible local minima.
Your system explores this terrain as you lower the temperature and will get stuck in a minimum,
but it may get stuck in a very high line minimum and therefore be have an energy much higher than the potentially lowest one it could reach.
There are many local American. It can it can attain all with different configurations.
Now, this really is a cartoon, because in fact, this isn't two dimensional.
It has many billions of dimensions. And so you have to imagine what this random landscape would look like in an extremely high dimensional space,
not a two dimensional surface of a shown here. Does that help us?
Well, it turns out that in very high dimensional random landscapes, we can get some simplifying features very unexpectedly.
One feature is that the subtle points with higher energies look more like maxima than those with lower energies.
Those with lower energy look more like minima.
So what that means is that the high line shuttle points have many downward directions and not many upward directions.
And this really helps you.
You might be troubled that your your liquid will, as it explores various configurations, end up at a subtle point with a high energy,
but which looks very much like a minimum might get stuck there for a very long time,
and you might freeze in that configuration, but that doesn't happen.
It turns out that subtle points where you may think you would get stuck.
The high energy shuttle points look more like maxima, so you're more likely to fall down in the cascade of energies.
Down to lower energies, right? Turns out that most minima have have low energies and in fact, entries close to the lowest.
Not all of them do. There will be some minima that have high energies, but they're very rare compared to the ones with low energies.
And the lowest energy configuration is only slightly sensitive to the size of the system, as we've already seen.
And so this explains why glasses have a relatively sharp freezing transition.
It's because of this phenomenon that you get lots of things, lots of of local peaks close to the highest peak.
If you're interested in the highest one or the lowest if your lowest dip, if you're interested in the lowest lots,
very close to that and the configuration you end up in may not be the absolute lowest, but his energy is not going to be far away.
And so the freezing temperature is pretty well defined in these systems.
And this explains a puzzle that has been troubling the natural scientists for a very long time.
We can apply this understanding to a different problem that's troubling people currently, and this is the problem of machine learning.
How do you train a machine to recognise or categorise images that is not seen before?
So the idea is you want to show your computer pictures of cats, lots of them and train your computer to recognise what a cat is.
So that if you show it a picture not identical to any of those, it's seen already.
It will still recognise it as a cat. So you put in lots of data into your computer, lots of pictures of lots of cats.
Each picture of a cat contains lots of data.
There are many attributes to a cat, and many things you can measure that would reflect in the data that you put into the into your computer.
So your job here is to take lots of images. Each of them contains lots of information,
and you put it into a computer and try to use this to train your computer to recognise a new image to be a cat as opposed to a dog or a hamster.
How do you do that? Well, you take all the data you've put in,
which now sits in a very high dimensional space because we're putting in a lot of data in each point of data contains a lot of information,
lots of parameters to vary. So this is a very high dimensional space.
We input the data and then we try to find a surface that sits as close as possible to all of that data.
So this surface will necessarily be highly complex and lives in a very high dimensional space.
It has to be complex because it has to fit lots of different looking cats. There are lots of varieties of cat.
So we have these very random surface in a very high dimensional space, and we want it to be as close to all the data that we put in as possible.
So we vary parameters that describe the surface and try to get it to match as closely as possible.
The data that we put in. And we're just as close as possible, Maine,
when it means that the distance between that surface and the data points that we're putting in has to be as small as possible.
That is, we have to find the lowest minimum of distance.
So the problem machine learning is exactly like the problem of freezing of glasses.
We're finding the lowest minimum and is therefore exactly the same as finding the extremes of random surfaces that we discussed earlier.
And you can analyse it in the same way. So we defined the lowest minimum, and this will train the machine in the best possible way,
give it the best possible chance to categorise images that it's not already seen.
So can you find the lowest minimum? Well, this is one of the great challenges of modern computer science.
And every, every, every company interested in in machine learning has teams working on this problem.
It's it's the moon landing problem of of machine learning is how to identify the global minimum,
the lowest minimum of the surface that you fall to approximate the data you've put in.
Which is out, there are algorithms that do that, and they work very well.
In fact, the big surprise in this area is not whether you can find an algorithm we have them.
The big surprise is that they work far better than we might. They might have been expected.
And this was the great puzzle to resolve in the area. So here's an example of the sort of surface you get in the machine learning problems.
This is taken from a paper visualising the lost landscape of neural nets.
Neural nets are a description of these random search of these surfaces.
The one gets in computer science in this area. You see, the surface is highly irregular.
And your job is to find the lowest minimum on the surface, and that's the best approximation to your data.
Here's another illustration of what these surfaces look like.
This is taken from a website lost landscape dot com, where I should say you'll also find films that you can explore,
and you see these surfaces are extraordinarily complex and pockmarked with with local minima, which saddles maxima.
So as you explore the surface, you may well get stuck in the wrong minimum.
You may well get stuck or to settle for a very long time before you find that it really is just a saddle and not not a minimum.
And in these very high dimensional spaces, I remind you, saddles can look very much like like like minima.
If you if this was a surface of a thousand dimensions in 999 dimensions, the saddle might be upwards, but only one direction, maybe downwards.
So it's very easy to be fooled in these high dimensional spaces, but a saddle is really a minimum.
So it looks like this would be an intractable problem, but in fact, it isn't.
And the methods that people have worked extraordinarily well. Why is that?
Well, it turns out that it's the same understanding we've developed all along in these highly high dimensional, highly complex surfaces.
And the structure actually works to your benefit in that the saddles that you see if you're high up look much more like maxim of the minimum.
And so you naturally inclined to roll down the landscape with your algorithm and not get stuck,
not get stuck in in high lying minima because there are relatively few of those.
Instead, you're likely to find a minimum, which is very close to the global minimum.
You may not find the absolute global minimum, but you can quickly get to a minimum that's very close to it.
And that's good enough for all practical purposes. So this is a good example.
I want you to discuss, but let me finish with a final example, which is have a very different flavour and this is the Romans easy to function.
So the remains to function is a mathematical object,
a surface which is designed to understand the prime numbers to the primes of the numbers divisible only by themselves and want.
So here are the examples of the primes up to 100.
And for thousands of years, humankind's been interested in the distribution of these numbers against all of the whole numbers.
Are there any patterns amongst the primes? Are there any ways to predict where the next prime will count,
etc. So people have found a way to analyse the distribution of primes, and this involves the remains to function?
So what's that? Well, if we take a number as.
We do the following to it. We get one. We are doing one over two to the power plus one of the three to the Paris plus one over four to the Paris.
Just one of the height of power, etc. So our faces to this would be one plus one over to square,
which is four plus one of three squared, which is nine.
That's one of the four squared, which is 16, etc. So for each number s, we can put it into this sub and get an answer act for some values of s.
You have to work a little harder, but I don't want to go down that route. We have a way to get an answer for every time you have.
Now, just to make this a little more complicated, it turns out you don't have to have taken no like two or three four s.
You could take a complex number, something that was a combination of a number that we see in the everyday world and the square root of minus one.
So if you understand complex numbers, you'll know what that means.
You put a complex number and threaten the value out that you get will typically be a complex number.
If complex numbers aren't quite your thing. Think of it. That s is really two numbers.
You put two numbers into this device and two numbers come connect.
So how does this help you? Well, there's a remarkable identity,
do you originally to oiler which says that that sum is in fact equal to a product and the product is over all of the prime numbers?
So the Sun one plus whatever of the sort of a three to the S, et cetera,
is identical equal to one over one minus one of a two to the chest and one which would have a three year less tangible match.
One of five to this much more over 70 years will match whatever 11 to the S, et cetera.
And these are all the prime numbers that are appearing in only the prime numbers. So it is realised by Raman that just as modern chefs love to do,
you can take a nice dish and deconstruct it ream and realise that you can
deconstruct this formula that if you understand the behaviour of the to function,
you can deconstruct it to get information about the prime numbers.
So this is how we understand the primes is via the Raymond Z to function deconstructed.
So is the Romans each function look like it's a little hard to plot because of the fact that complex numbers appear,
but one way to representatives is in terms of colours. So here's a plot in two dimensions the two to a point in the coordinates of the points
in this two-dimensional plot are the two input numbers or the one complex number,
if you know what that means.
So you picture X and Y coordinates of a point here, and X and Y correspond to the two input numbers and the output is represented as a colour.
Yeah, it's not the only way to represent the Z to function,
and I'll give you a different one in a minute, but this is the one that we'll focus on for the moment.
So you have colours at each point, but some points are special that points where all the colours meet.
And you see a collection of these down at the bottom of the picture.
They lie on a straight line, a horizontal line at the bottom of the plot.
And even realised that these points where all the colours meet are a very special,
and in fact, these are the points that really determine the distribution of the prime numbers.
This is where the deacon, this is where the deconstruction really acts.
He realised as well that there aren't just these special points on the horizontal line at the bottom.
There are some others that lie away from that horizontal line.
And he he guessed, he hypothesised that those numbers lie on a single straight vertical line.
Now, he wasn't able to prove that he guessed it, and that left that as a challenge for future generations.
And that's the problem that we call the Riemann hypothesis.
It's that these special points were all the colours meet and that aren't the points on the horizontal line at the bottom.
In fact, lie on a single vertical straight line. And we can't prove that yet.
If you do come up with the proof and you're able to get it accepted in a reputable mathematical journal and accepted by the mathematical community,
you win a million dollars because this is one of the problems issued by the Clay Institute, actually one of the clay mathematical millennium problems.
Here's a different way to represent the Romans each function.
And this is a way due to G.H. Hardy, who was a great mathematician in the early part of the 20th century,
and he thought, What does this look like as you look up the vertical line where you expect their special points to lie?
And he found a way to plot that it's called the hardest function, and it's the curve in blue there.
So this curve oscillates is like a wave.
It passes through zero and the zeros are precisely the points in the previous block where all the colours meet going up vertically.
So the zeros of this curve, the point where it intersects the horizontal axis are the Roman zeros,
and these are the points that are identified as points where the colours meet.
What's the Riemann hypothesis in this setting, where it's the statement that this curve,
if you look beyond 10 to the right of 10 on the horizontal axis, that all the maxima will be positive and all the minimum will be negative.
So if you find a maximum of the of this function of this curve that lies below the horizontal axis that disproves the Roman hypothesis.
So that's one of the great challenges about this, about the high desert function,
this curve here show that it has no negative maxima or no positive minima, and that proves the remote hypothesis.
There's another question that people have asked about this curve.
Again, going back about 100 years and this is how big it is to the oscillations get in the heart is that function.
I plotted the hard-edged function in the top curve up to 60 and then down below from fires in 2050.
You see the oscillations continue and they get more rapid and the curve seems to get a little bigger, but not by very much.
The oscillations and Lindelof, about 100 years ago, suggested that perhaps the hardest edge function increases as you go along the horizontal axis.
The size of the oscillations increases, but as slowly as you can possibly imagine, I'm not going to make that.
I'm not. I'm not going to stay that more precisely. But Lindelof made a precise guess that said that the hardy z function does
increase in the size of its oscillations as you go along the horizontal axis.
But as slowly as you can possibly imagine now which people have tried to think about
this work on this for the last hundred years and hard won progress has been made.
But we very far from proving that. So progress is being slow, steady, hard fought, but we're very far from proving that off.
However, recently a different question has been put forward very much in this spirit, which it turns out we can answer very precisely.
And this is the question, not how big does these oscillations get all the way out?
Some long distance along the horizontal axis? What if we look over a short distance to what's the largest value of the heart?
Is that function between tea and tea plus two pi?
It doesn't have to be to be any constant. No, but I'm putting two pi here for frustration.
So how big do you expect? The largest value of the hardy search function to be its largest oscillation in a range of lengths to PI?
Two points about six. Well, as tea increases, you get more and more oscillations at the function in this range.
And so we have more little max maxima the local minima. How big is the highest to lowest, the lowest?
Well, here's the answer.
It turns out to be that this answer is identical to the answer that we saw earlier for the extremes associated with family trees.
There's no three appears, and the terms are exactly matching those that we saw in the formula for the extremes associated with family trees.
Now the history of this is that this formula was guessed by analogy with the family tree problem.
And in the last few years, people have managed to prove it. So we now have very precise information,
astonishingly precise just over the last few years about the extreme values of the Hardees that
function and the and this has come about because we asked the right question in this setting,
a question that does have a very precise answer that we can prove and proving
this relies on showing that there is a family tree like structure to the primes.
And people have been thinking about the primes for a very long time.
But this new understanding has emerged only recently that you can group the primes together in the
way they contribute to the and Z to function in a way that makes them look like a family tree.
And using that fact then allows you to prove this formula. So.
By thinking about extremes, we discover new properties that are primes,
and I should emphasise that it's inconceivable that we would have one would have guessed this formula if
one hadn't been thinking about all the other problems and extremes that I've been describing earlier.
So let me finish with the summary questions relating to the highest,
maximum or lowest minimum connect, many different problems from water waves, quantum mechanics,
the Olympics through to how glasses freeze, machine learning problems,
the efficacy of algorithms and machine learning and through to the remains each function.
And by identifying that thread, we've managed to make progress in recent years that I think wouldn't have been imagined longer ago.
So the statistics of extreme values shows universal behaviour.
You get the same formula appearing time and again, and this commonality is what allows us to make progress on a broad range of these problems.
So it might seem that we've made good progress, and I think we're pleased that we have.
But I want to emphasise that going back to the first story I told you the mountain range,
we're really still in the foothills of this, of this analysis, of this line of research.
It's very clear that there's much more that we don't understand that we do.
And the great challenge now is to take this analysis and make it more precise in each of these separate applications and to get
more and more accurate methods for analysing these problems of at extremes in the various contexts I've described to you today.
Thank you.