yeah so when I was preparing the for
this lecture and you know I thought
since well despite seeing a few familiar
faces I hope that most of the people in
the audience are not professional
mathematicians and you know as a
mathematician you often go to parties
with lots of mathematicians but then
occasionally occasionally you also go to
normal parties and so if you go to a
normal party and somebody asks you what
you do and you say you're mathematician
you know usually there's sort of a split
second of silence for the person
thinking what should I say what should I
say and then they usually say oh I was
really never any good at math when I was
at school but then occasionally the
person thinks for a bit and says you
know what you actually do and so I
thought I'd first try to explain in a
few words what mathematicians actually
do now the first thing is that
mathematicians don't just compute either
means of in school you essentially learn
how to do long division and stuff like
that and you know as you go older you've
learned essentially more sophisticated
versions of doing those long divisions
and so it's not that mathematicians just
come up with really really sophisticated
versions of doing things like this you
know for computing we actually have
computers
these are computers computers was
actually a job it was a relatively low
skilled job you could learn to be a
computer in a few weeks and that was
sort of back in the sixties or something
or in the fifties of this nowadays we
have still have computers but we have
the electronic computers so if we want
to actually occasionally it happens that
we need the outcome of some calculation
so if the calculation is easy which is
do if I had on or scribble on a piece of
paper and if it's hard we ask a computer
to do it because that's what they're
here for now what we actually do it's
much more about
if you want exploring the world of ideas
or the world of what we call
mathematical concepts and so we're
really much more like you know the
people in Plato's cave you know you know
the story about Plato's cave with you
know the idea being that we are a little
bit like down in this cave and there's
this idealized world outside that we
only have glimpses off and what we try
as mathematicians is essentially try to
get glimpses of this idealized world of
the world of ideal objects and so so
what is a mathematical object well
essentially it's just an idea or a
concept that you've described very very
precisely
okay so you've described it so
okay so you've described it so precisely
that there's absolutely no one big
ambiguity left on what it means and
you've described it in a way that rules
out any kind of contradictions okay so
in that sense we're a little bit like
the the opposite of politicians
it also a politician it's usually very
good at using concepts that may have
different meanings for different people
and then use them in such a way that
everyone everybody can hear precisely
what they want to hear so what we do is
exactly the opposite so we try to
describe ideas in such a precise way
that there can be absolutely no scope
for misunderstanding anymore or for any
kind of ambiguity and so then within
this framework if you want what we try
to do is uncover truths about these
objects that are you know as absolute as
possible so in a way mathematics is
probably you know pretty much the only
human endeavor in which you can actually
discover absolute truth now you know you
can start having all sorts of
philosophical discussions about where
the truth exists so in a way the reason
why we can discover absolute truth to
some extent this because you know
mathematics is sort of disconnected from
reality in a certain sense but we'll see
not completely but of course there's
this world of idealized objects and
these objects they are purely objects of
thought right so mathematical objects
are purely objects of thought and a
priori they don't have anything to do
with concrete reality but of course the
way we have very very successfully used
mathematics is to to build models of
reality and so there are many ways of
linking mathematics to realities or one
way of course is physics or the job of a
physicist to some extent a theoretical
physicist is to link mathematics to
reality then there are all sorts of
other models that you can come up with
so we've seen in the previous talk that
when you want to analyze sets of data
you would essentially set up a model and
then you would try to fit the set of
data to this model
then for example computers computers
play a bit of a special role because to
some extent mathematics can describe
what happens in computers but computers
can actually also describe mathematics
or can simulate mathematical object so
computers are probably as close as it
gets for mathematical objects to become
reality but ok so this talk is about the
title of the focus mathematic of
randomness because that's my area of
research so well that's sort of this
small area of mathematics where I'm
competent to say things
so what's probability so in general we
use probabilities when we have some
experiment an experiment doesn't
necessarily mean experiment in a lab
item it can be any kind of situation
where the information that's available
to you is insufficient to determine the
outcome for certain I don't mean we've
seen lots of examples of this in the
previous talk where we have some kind of
partial information of what's going on
but we don't have complete information
of what's going on and so what we do is
we try to assign probabilities to
different outcomes in order to measure
if you want how likely they are I don't
so there are if you want there are two
kind of interpretations for
probabilities so one way is to say well
you know how how likely do I believe an
outcome to be and we've seen that
towards the end of last talk right with
the is Bayesian models so you would try
to encode your belief about how likely
certain outcomes are or if you want
probabilities could have in say when
you're in a situation where there's some
experiment you can repeat which is not
always the case right so many many
experiments have one outcome right the
experiment could be something like the
outcome of an election right that's a
typical example of an experiment and
obviously you can't rerun it 20 times in
order to figure out what was the actual
probability that somebody won right
but when instead in some situation you
can do it and then the probability you
know there's sort of an objective
meaning for the probability which
essentially you just count how often the
different outcomes come out and you look
at the fractions the proportion of time
the different outcomes appear now we
know a few there are a few simple things
we know about probabilities right so
maybe so if you have two positive two
outcomes that are mutually exclusive
like you roll a die okay so you could
have one two three four five and six so
there are two possible outcomes say 1 &
3 they are mutually exclusive right
either you see a 1 or you see a 3 or you
see neither of the two and then the
probability that you know one of them
happens it's just the sum of the two
probabilities that either of them
happens right so the probability that
you just see 1 is 1 over 6 probability
to see 3 is 1 over 6 the probability
that you see 1 or 3 is going to be 2
over 6 5 it's kind of clear and then if
things independent motive probabilities
multiply right so if you throw a die
twice probability that you see a 1 the
first time is 1 over 6 probability you
see a 1 again the second time is 1 over
6 so the probability that you see two
ones in a row is 1 over 36 so there are
some very simple rules that govern these
probabilities and so you see your
problem is people tend to think that you
know that's great because you think
compute the probabilities of everything
that's not true I mean we can't you know
I can't pull out of my hat the
probability that is going to rain
tomorrow something like this so all we
can do is essentially transform
probabilities I don't mean we can build
models that incorporate probabilities
but we have to put something into the
model and so how do you put stuff into
the bottle so one way of doing it is to
actually use data like we've seen in the
previous talk but then there are many
other situations where well there's
either no data around because the
probability
involve things that are say too small to
measure order too many of them or simply
you want to come up with some nice
mathematical model a priori before
having any data and there are
essentially two principles that would
allow you to a priori guess some
probabilities so the first one is
symmetry alright so that we've seen
already with the example of the die if
you roll a die there are six possible
outcomes they're all completely
symmetric I don't mean they're maybe not
symmetric as far as the effect is it
would mean if you bet on the sixth and
for you it's better together six right
but as far as the die doesn't know that
so as far as the dice concern six one
doesn't make any difference so if you
have symmetry so if different outcome
certain outcomes are completely
equivalent well then clearly you
shouldn't sign to them the same
probability alright so the typical
example is when you toss a coin there
are two possible outcomes so each of
them is probability 1/2 because they're
completely equivalent now there's a
second principle and not only is much
more subtle okay and that one you so the
first one you've certainly heard enough
of a nutritive knowledge about it the
second one you probably didn't and
that's universality and the universality
is actually very much sort of driving
principle of modern theoretical physics
and essentially what it says is that if
you have some outcome some random
outcome which is itself you know the
consequence of many many many different
random things that combine together then
very often sort of the distribution of
the outcome or somehow the the way the
randomness of the outcome looks doesn't
actually depend on the details of these
little you know components that make up
this more complicated outcome so one
typical example here is if you take and
data Laplace is the same Laplace as
we've seen already
so one typically example is successive
coin tosses right so you take again -
you just - throw a coin say you throw a
coin a hundred times and you record how
many times you see heads okay
most of the time it's going to be
somewhere around 50 but you know in 90
1552 might be 47 and then you repeat
that many times so say you do a thousand
such runs and then you plot sort of how
many you know the higher the Troxel
horizontally you have you know how many
heads you see and fifty is somewhere on
here in the middle and vertically it
just plot how often you see that in your
thousand runs and you see these
something like this so here we
essentially just asked my laptop to do a
hundred thousand point courses and to
plot what you get and you see these you
know these red crosses now if you do it
ten thousand times you see that they
start to align more and more on these
smooth black curve right if you do it a
hundred thousand times then it aligns
very nicely on this black curve okay so
if you just do a hundred thousand times
this experiment of doing a hundred coin
tosses and recording whether you get 50
heads or 49 or 48 etc you see this very
predictable and very sort of smooth
outcome and this is called the Gaussian
distribution which appears all over the
place in probability theory and the
reason why it appears all over the place
in probability theory is that it's not
anything special about coin tosses you
know instead of tossing coins I could
have thrown dices I could have done
pretty much anything where you have lots
of independent events and they're all
more or less the same thing and you add
them up and then you plot the statistics
of what you get essentially always get
this Gaussian distribution okay so this
is an example of this universality that
you always see this Gaussian
distribution essentially no matter the
details of these things that you add up
so let me go back with about this
symmetry example so here the problem the
thing is you have to be very very
careful I think the previous talk
already
insisted on this very rightfully so
that you have to be careful about having
good mathematical models okay so here is
a nice little paradox that maybe some of
you know what so imagine I have two
envelopes okay
the only information I give you is that
one of them contains twice as much money
as the other one okay but you have no
idea of how much so say I allow you to
look into one of the two right so you
choose pick one
open the envelope have a look at what's
inside okay so you see a certain amount
of money and then I give you a choice
either you keep that amount of money and
then that's it game is over or you
change your mind and you take what's in
the other envelope but then you're
obviously not about to change your mind
again I don't think if the other one has
the lower amount you have to stick with
the lower not crazy so the other
envelope obviously has either twice as
much or it has half as much right that's
all you know so now if we go you know
you'll think oh yes so this is a very
much a symmetric situation right this is
I have absolutely no information about
which one contains more so there should
be basically a probability or half that
there's twice as much and the
probability 1/2 that there's half as
much so on average if I switch well I
have probability 1/2 of getting half as
much I have a forward with you a half of
getting twice as much so an average I'm
going to get four quarters times as much
as if I stick with my first choice right
so I should always switch whatever
happened I opened the envelope I should
I don't even need to look at it I know I
should always switch which is of course
not
you know it's obviously nonsense and so
the reason why you end up with a
nonsensical answer is that you've
implicitly assumed that actually all
possible outcomes are equally likely but
there is actually no mathematical model
that would give equally life you know
that would make all possible outcomes
equally likely in this case okay so in
some sense what you're doing is you
you're basing your analysis on a model
which doesn't actually exist and what
you should do in reality okay so you
should actually make some kind of
educated guess of what you believe would
be you know my probe the probability of
me giving you certain amount of money
all right so so what and that's what you
actually do in reality right because you
would open the envelope and you see a
hundred thousand quid you know what do
you do obviously you take it right and
you're wrong whereas if you open it and
you see you know you see five quid you'd
say well okay I mean I try my luck
because you know okay I don't really
care about losing too quick and so you
you know you would actually have made
the guess that it's pretty unlikely that
I would give you two hundred thousand
quid and right so you would actually
have incorporated this into the model
and so in the first case it was clearly
not a good idea to switch and so usually
wouldn't switch now in this particular
example actually for the mathematicians
in the audience you can actually come up
with genuine mathematical models where
it's still always profitable to switch
but then those would be models where if
you want to start with the expected
amount of money that I would give away
is infinite and that's not very
realistic and so you see in this case of
course using a brown more ill-defined
mathematical model is you know it's just
a toy example but in I mean reality can
have absolutely dramatic consequences
right so you may be you still remember
the story of Sally Clark
whose two small children died of sudden
infant death syndrome and she was
convicted on the basis of expert witness
who said well you know the probability
that a child of this in that age in her
peer groups of upper middle class would
die of sudden infant death syndrome is I
don't remember one in ten thousand volt
one in 20,000 or something like this and
so the probability that two of them in a
row die of sudden infant death syndrome
is one over whatever the product is 100
million or 400 million and therefore she
must be guilty she must have killed one
of them of course the argument is
completely nonsense because there's no
reason to believe that the two events
are independent there might be a genetic
factor for example and of all you know
there might be any number of other
external factors that would link the
tool the two events and in any case even
if it was true that the probability is
something like one in a hundred million
well the UK population is sixty million
right so you have a probability of one
in a hundred million it's not so
unlikely that it actually happens once
in a while somewhere so now I'd like to
go back to this second principle which
was this universality and this goes back
to sort of late eighteenth century
she's the discoverer if something called
Brownian motion so this was first
observed by a guy called young Egon
rules who was observing coal dust
suspended in alcohol and so he you know
he lo he observed is under a microscope
and he saw something like this so these
are the little particles of coal while
the picture would be sort of inverted if
you want so the little white dots are
the coal
and the black background is the alcohol
again you see these particles of coal
dust moving around in a pretty dissolve
negative way and actually about 50 years
later Robert Brown observed pretty much
the same thing except that he wasn't
observing coal dust he was observing
some kind of very small particles
released by falling suspended in water
and well nowadays it's called Brownian
motion I suspect that the fact that
Brown is easier to remember and to
pronounce and even though this probably
has something to do with it
and so okay so they essentially observe
this so in the beginning they thought so
at least Brown who was a botanist at the
very beginning he thought that these
were actually alive I think they were
kind of very tiny animals moving in
water because he was really careful
right so he let the water sit for day is
to make sure that the water itself
doesn't move at all and make sure
there's absolutely no vibration no
external influence so it was clear for
him that it wasn't this motion wasn't
just caused by the water moving around
because you've just transported it from
a room to somewhere else so he thought
this was actually little animals that
were moving in the beginning but then he
pretty quickly convinced himself that
they were not and so it wasn't clear for
a while what would actually cause this
motion so before we go back to the
explanation of this they have been
introduced another historical person so
this is Joseph Fourier so he was a
French scientist as the name suggests he
he was he had kind of an interesting
life so he was a very he's a very well
known mathematician I mean for
mathematicians is a very well-known
mathematician I'm told that for
Egyptology is actually a pretty
well-known Egyptologist because well he
took part in Napoleon's invasion of
Egypt it was actually the director of
the
Egyptian Museum in Cairo so in you know
during that time and he was also a local
politician who is prefect offer is there
one of the French departments for a
while and mathematically so what's
interesting for us here are two big
contributions so the first one is what's
called now the Fourier transform and
what this says is that pretty much any
signal can be decomposed as a
superposition of pure waves okay so if
you want here's an example so you take a
signal like this that's my redline and
what Fourier says is that this redline
you can actually write it as a sum of I
don't know if you can see these light
blue curves you can write it as a sum of
these light blue curves and they you
know you see that they are sort of these
very pure waves they all look the same
they are just rescaling of each other
and he showed that you can actually
always do that
I think this is used a lot for example
in your mp3 players
alright so that's exactly how you encode
sound actually how you compress some
alright so imagine this is your sound
signal and what you do you actually
decompose it as these pure waves with
the hope that there are not too many of
them and then you just have to figure
out you know what's the frequency and
the amplitude of each of these waves and
you just store that and then you use
this little free construct yourself
sound signal from there and this
actually works pretty well because most
you know pleasant sound signals already
look you know pretty close to pure waves
anyway also the for superposition of not
too many pure waves
now the second contribution that for you
is very famous for is what's calls for
years law and that's more important in
physics if you want and in mathematics
and that gives an equation which tells
you how heat
evolves okay so if you take a say a
chunk of metal and you heat it up
somewhere so it's cold say everywhere
except here where it's warm and then you
just let it evolve then of course you
know it's going to spread a bit so the
room bit is going to expand a little bit
but it's going to be cooled down by the
environment so it's going to more or
less expand but go down and well here is
the equation that describes this you
know I don't expect you to necessarily
understand what the equation means I
just meant just thought I'd write down
an equation because it's a math talk
after all so there should be at least
one equation in a math talk it doesn't
matter what the equation is the point is
that he came up with a way of describing
the evolution of heat now why is that
relevant to this problem of Egon whose
and Brown which was to describe the
motion of these little poems well so
here comes n stein and another chap
called Swarovski and they well based on
actually previous work by Lord Rayleigh
so they came up with a theory of
Brownian motions of Brownian motion is
the motion of these little particles and
so the theory came it comes in two bits
so there's a mathematical bit as a
physical bit so the mathematical bit is
about giving a mathematical model
describing it and the physical bit is
about you know giving an actual
explanation of why the model is
reasonable model and so the physical
reason is actually that water is made up
of molecules and so water is made up of
molecules and the temperature is
essentially something like the average
speed at which these molecules move so
even though you have the impression that
your bit of water is completely
addressed what actually happens is that
there are billions and billions of
little water molecules and the old
bounce around at full speed and it's
just on average that it
appears to be addressed and so now if
you put a little particle into your
water if the particle is small enough
then it actually does feel the impact of
the collisions of all these molecules of
water that bounce around at full speed
okay and so occasionally you have maybe
sometimes for a little while there are a
bit more particles bouncing from the
left and then it sort of moves here or
sometimes there are a bit more particles
bouncing from the right so it moves here
and that's what causes this erratic
motion that you saw in the movie okay so
this erratic motion is actually a direct
consequence of the fact that water is
actually made up of molecules and by the
way so this was then used a few years
later by para another french scientist
who used the quantitative predictions of
einstein and smoluchowski in order to
essentially prove conclusively that
indeed the effect that einstein small
shots he described was the one you know
essentially the only good realistic
explanation was the one they gave
because you had quantitative agreement
of the experiment and therefore this is
what really proved they want the
existence of molecules and of atoms so
before that you know it was a very
strong hypothesis so people already were
starting to really believe in it but
Perrins experiments were really the
conclusive experiment that proved if you
wanted matter is made of atoms and
that's what landed him the nobel prize
in the twenties now the mathematical
part of this description that's where
Fourier comes in so the mathematical
part of this description says that what
you can do is you take take one of these
little particles that bounce around okay
so you observe it at one moment it's
here now you close your eyes you open
your eyes
few seconds later it has moved you don't
know where it is because it has some
random motion and so you want to have
some kind of description of where the
particle is okay so you want to be able
to say well it's maybe you know it's
more likely to be pretty close to where
I saw it last time
but maybe it went through certain
distance I want to have some kind of
description of the likelihood to see the
particle at a certain point in space and
what the mathematical description here
says is that this likelihood is governed
by exactly the same equation as the one
governing the diffusion of heat bodies
okay so if you want probability is the
same as heat in that model it obeys
exactly the same laws now here's another
interesting person related to exactly
the same problem or the same
mathematical model so this is bascially
and he was to some extent he was really
the founder of mathematical finance so
he was the first one if you want but
except that he lived in the early 1900's
and so you know there weren't many hedge
funds around so he had actually trouble
getting a job and so some he essentially
you know invented mathematical finance
in his thesis which was published in was
submitted in 1900 and what he did there
is he described an evolution a mechanism
for the evolution of stock prices right
so he was interested in how stock prices
move and so you see the mechanism he
describes is very much the same as the
mechanism that einstein and smoluchowski
described for the four
Brownian motion right because what did
they say so they said water is made of
molecules and these molecules they just
move around at high speed erratically
every now and then they bounce the
particle you're looking at and then they
push it in one direction and that's
completely random they push it in all
directions more or less the same
likelihood in that what causes this
Brownian motion what he says is well you
know market is made up of individual
investors there are many many individual
investors they all behave more or less
erratically every time they sell a stock
or they buy a stock it's likely
influences the price and so all together
well every time somebody sells the stock
the price goes down a little bit
everybody people buy the stuff the price
goes off a little bit and so what you
end up seeing is you know that's erratic
motion of the stock price which is
actually pretty much the same as the one
that brown Indian has observed under the
microscope and the mathematics was
exactly the same so he also came up with
exactly the same conclusion which is
that if you want to describe the
probability of the stocks you know has
takes a certain value after a while well
it's essentially described by this heat
equation which for you invented quite a
long time ago in a completely different
context because this to some extent you
know laid the foundation for you know
more modern mathematical finance and had
also led to a Nobel price actually in
the 90s to black controls so
now in the previous descriptions what
people try to describe is the following
scenario right you see the particle at
the moment you record where it is and
then you want to predict where it's
going to be at a later time and all
you're going to be able to do is predict
some probability and so you want to
predict the shape of you know the
likelihood of where you're from and find
the particular later times and you want
to see how this evolves in time now what
you actually might be interested in is
you know have a look at how the you know
actual shape of the trajectories look
like and so this is where veena comes
into place so he's mathematician of the
first half of the 20th century from MIT
and he was the first to provide somehow
a full mathematical description of this
Brownian motion but really as in you
know as an actual random trajectory and
now the interesting thing here is that
here then also he's also very famous as
one of the earlier researchers in
robotics and cybernetics and you've
certainly heard of one of his PhD
students who was Bose and it's the same
as the noise canceling headphones
missile so the guy who set up the
company for the noise canceling
headphones he was a PhD student of veena
and the algorithm used in the noise
canceling headphones at least of you
know so the first basic version of this
algorithm was actually device also by
that guy so so he gave this mathematical
description of this trajectories of
Brownian motion so here's what it looks
like ok so this is a it's a it's a
random trajectory so because this is
just an instance like I mean since it's
random every time you look at it you see
something different so this is just one
example but it always roughly looks like
this
you can see it sort of you know it does
sort of remind you of the motion of
stock prices right I mean it sort of has
these kind of features because it's a
very very crude moment and then more
recently in the fifties danske he was
you know the first to if you won't
really show that this is universal in
the sense that there are many many
processes that when you look at them on
large scales they look like this
Brownian motion so here is a let me show
you an example of such a process so this
is what we call a random walk so you
have this red dot here so that's a super
toy model of what happens for the motion
of these little particles that Brown was
observing okay so the red dot here is
the little particle and now at every
instant of time the particle can jump
either to the left or to the right or up
or down okay and so it sort of jumps
around occasionally traces back and so
it does you know this kind of erratic
motion but now you saw you start to look
at you know you speed this off so you
look at this from further and further
away so you still see the particles of
moving around like this but now if you
look from further and further away you
know it sort of starts to it looks like
this if you forget about the black bit
if you just focus somehow on the right
dot it does pretty much the same sort of
erratic motion as you know these little
dust particles that brown observed under
the microscope okay well well of course
what you see that you know at the
microscopic level when you when we sort
of zoom back in right here at this level
the mechanism has if you want nothing to
do with water molecules bouncing around
right the thing can actually only go in
four direction
there's a completely artificial model
but what this tells you is that so this
is this idea of universality that as
long as you if you have even if you have
a completely artificial toy model as
long as in code as it encodes the right
features of your system it doesn't
really matter what your toy model is
it's going to behave essentially the
same when you look at it from far away
and so that's why do you know that gives
then a justification if you want for
studying these toy models that seem to
be completely remote from reality
because you you know you believe that
even though your toy model is completely
cooked up it would actually describe
some more realistic situation when you
look at the large-scale behavioral sort
of the both behavior okay so here is
another example which is so now this
Brownian motion business this is now
extremely well understood okay so the
mathematics well you've seen that
somehow my historical survey
ended in the 50s so this is extremely
well understood but there are actually
quite a few somewhat similar situations
that are still now mathematically not
very well understood it's still an
active field of research so I would want
so I wanted to just you know so in the
last 10 minutes wanted to give you an
example of something where now we know
some things but we know much much less
you know we're sort of at the stage that
people were you know somewhere in the
late 19th century in terms of
understanding Brownian motion and so one
of this example is interphase growth
models so the kind of thing that you can
have in mind is a piece of paper burning
okay so you take a piece of paper and
then you set it on fire the problem so
you're careful to set it on fire very
in a very regular way so say for example
you can shoot the laser beam at you I
feel peace of paper now it starts
burning from the bottom at the beginning
it's completely flat but there of course
your flame front is going to start on
the lathe in a little bit right because
some bits are going to maybe burn a
little bit faster some bits are going to
burn a little bit slower so it's going
to actually move in a slightly random
way and so here is an actual experiment
so these guys did the actual experiment
of burning a sheet of paper and
recalling what you see and what you see
here is you know the position of this
flavour flame front at successive times
okay so you see it starts off relatively
flat well it's kind of sloped but you
know just turn the picture a little bit
it's essentially flat here but then you
see it's actually more and more random
looking okay so here you see quite large
fluctuations and here's another
experiment which is somewhat similar it
has this very similar mechanism but here
the experiment is actually some liquid
crystal which comes in two different
phases and there's one phase which is
more stable than the other one but you
prepare it in the unstable one and then
you shoot the laser beam at your liquid
crystal the laser beam activates the
stable phase so it becomes it changes
and it actually has different optical
properties so you really see it of
changing to a different color different
transparency or something and then since
what you've activated is more stable
it's going to invade the unstable face
it's so there's going to be some kind of
front separating the two and again you
just it'll take a high-speed camera or
something and shoot pictures every ten
times a second or something and then you
record these pictures and here so this
is this front and time is color here
okay so different colors correspond to
different times and aligned with the
same color corresponds to you know one
instant of time and here you see also
these front moving and you can make you
know you can make measurements on these
things and you have relatively good
agreement with between the two
situation so you have you know even
though physically it's complete two
different situations well of course in
both cases it's one interface a one
dimensional interface separating two
regions with one of them invading the
other one so at that level they are kind
of the same at the microscopic level
they of course completely different and
you can you know measure certain
features here and you can cook up with
mathematical toy models for things like
that and they exhibit very similar
features so here are simulations of
three mathematical toy models for
interface propagation so let me describe
to you one of these models so that's
what I call the tetris model okay so
that's the tetris model well sort of
clear what the model is right so it's a
it's essentially a random tetris game
where you don't play the game and the
bricks behave in exactly the way that
Tetris bricks behave right and now when
you do this well you know they start
sort of piling up you know you look from
a bit further away you see that there is
a you know there's some sort of
interface right so here you have the
region without breaks here you have a
region with bricks and there is you know
some sort of very wine the interface
between the tool here but now if you you
know if you zoom out more and more and
you run the thing faster and faster you
know in really really far and really
really fast so now you have you know
sort of thousands and thousands of
bricks here and you know now what looked
like a you know sort of very winding
interface here looks actually like a
nice continuous curve right so here now
you see you have the tetris bricks here
you have no tetris bricks here and you
have this kind of interface between the
two
which now looks actually very much like
the kind of pictures that we had on the
previous slide and you can make
measurements on you know this model or
similar models and they agree reasonably
well with the experiments this tetris
brick model is as good as any other at
you know describing the motion of one of
these interfaces and so this is what we
call so this class so now there is so
what physicists call a universality
class is essentially a whole collection
of models that have the same behavior
okay so we've seen before with this
Brownian motion well there are lots of
different models when when you look at
what happens at large scales you see
this Brownian motion which was first
mathematically fully described by wiener
here well again one conjectures that
there is one object at very large scales
that describes these models we have a
name for it because it's sort of easier
to give names to things than to actually
describe them so we call it the KPG
fixed point but one doesn't actually
have any description even of these
objects so one has some partial
description there are some things that
we know there are some things we don't
know and the way we know things is by
this principle of say well you know cook
up a toy model it doesn't really matter
which one it is so you just try to find
one which is as simple as possible so
you can compute things explicitly okay
so then so the idea is try to find one
model which is so simple that there are
certain things that you can compute
explicitly and then try to see what that
one does at very large scales and the
hope is that every other model does the
same thing and so people have been able
to do this for certain models so they
are able to compute certain things okay
so one has some partial understanding of
what happens here are large scales in
particular for example one has a good
idea of how thick you know how big these
fluctuations are
quickly as a function of time so one
knows that they've rolled like time to
the power one-third for example one
knows this exponent and there's so in
some situation you have some models that
are much better understood and here the
much better understood models are the
ones where you have an interface between
two regions but the fluctuations are
symmetric right so here it was
completely asymmetric like a piece of
paper can burn but it cannot unburn
right so it can clearly cause only one
way the tetris model was clearly also it
had the same feature right so tetris
brakes were falling down they pile up
but they never disappeared now there are
some models you can you know you could
imagine so those similar kind of models
where things can also disappear so when
the interface can go up it can also go
down and if this mechanism is completely
symmetric if you want if there's no
difference between up and down those
models are much better understood and
one knows what happens at very large
scales and what has a good description
of what happens and it's actually much
closer related to this Brownian motion
and being a process and so on and so
then you know one could say well maybe I
can cook up models where I have a small
parameter and when the parameter is zero
then it's completely symmetric and if
the parameter is nonzero then it's not
completely symmetric okay then it tends
to move in one direction and so now for
the non zero values of the parameter at
very large scales it should do the same
as this tetris model or all of these
asymmetric models and for the value zero
there it should be described by these
symmetric models as we understand the
behavior okay and so then maybe we could
understand what happens when the
parameter is very small and you look at
sort of very large but not very very
large scales okay and so in that case
there one can do something and this is a
little something I've been
working on for the last few years so
this is if you want what my contribution
comes in but okay so I think I'm going
to wrap it up here so it just want to a
couple of very short concluding remarks
I think there may be two main takeaway
messages so the first one is this
importance of having you know the right
morals okay so it is this fact that you
know as mathematicians you know we can
prove theorems we can tell you absolute
truth but they are always conditional in
the sense that we say well assuming that
this and nothing is true then this and
that has also to be true right but then
you still have to verify that the thing
I assumed is true and we've seen that
very much in the examples of the
previous lecture right so that this is a
very very important feature and you can
you never have completely absolute truth
without any sort of assumptions you
always start from somewhere to go
somewhere else now the other thing we've
seen here is that there's this notion
which arises quite a lot in probability
theory of universality which some extent
for us justifies the study of these toy
models that seem to have at first sight
very little to do with reality right so
you have these very simple toy models
but what this principle of universality
says is that even though they are very
simple and they are clearly not a good
reflection of what happens at small
scales in reality well when you look at
what happens at large scales when you
look at sort of the cumulative effect of
many small events then it doesn't
actually really matter
how you describe things at small scales
okay and of course well one thing they
also wanted to say very much for the
young people in the audience is that
mathematics is still very much of you
know subjects that
still alive and you know it hasn't
stopped in the nineteenth century or
something and there are still many many
things to be discovered I think I stop
here and thank you very much
the I want to talk about the two
envelope gay where you're offering me X
pounds or 2x or half X depending on
which envelope I choose now given that
you're offering me something for nothing
I don't really care
I'll stick with the first one what
happens if I have to pay to play the
game how does what I pay affect my
choice well I suppose that really
depends or that will probably depend on
you right if if if I'm paying X to play
the game then I face a situation where I
either break even not very interesting
or I can double or half my money right
that's pretty unlikely because you
you're not going to have paid typically
you wouldn't have paid X Y if you would
have paid something way off the mark so
you divider pay much more or much less
okay okay I don't think it doesn't make
much of a difference to the point of the
argument here so regarding your toy
models do you have a catalogue of these
toy models or is it a big area of
research in its own right there is a
kind of catalog so physicists have a
pretty good idea of these things and so
there is a there is essentially a
catalogue of these universality classes
if you want and then there are you know
more or less the simplest toy models in
each of these class
very much indeed for your talk you
mentioned black schools by the way I'm a
bit of an interloper I'm not a
mathematician I'm an economist but you
mentioned black holes skulls within the
Nobel Prize Merton was also involved
anybody who's thinking going into
finance should be very wary because a
year after the Nobel Prize was
established the firm that in fact him
scolds the other set up when actually
belly-up I think a lot of the and this
doesn't come through very well except
Keynes talked about trying to make
estimates of what would happen you
shouldn't think be thinking in terms of
risk where you can assign a probability
distribution but trying to guess what
other investors will be doing it's a bit
like trying to judge a beauty contest
where you try to guess what in fact the
other judges will regard as the winner
of the beauty contest so one of the
things that disturbs me a little bit is
there so much emphasis in fact on risk
and not uncertainty and we're completely
riddled with uncertainty and of course
you can't assign the probability
distribution so I thought that might be
mentioning to the younger members of the
audience here I think I'm going into
finance they could burn their fingers
I've seen this sort of I mean it goes
very much with the remark of you know
the importance of having the right
mathematical models and in this
particular case it's not clear what the
right mathematical model is to start
with and it's quite clear that this very
simple mechanism of many many small
events sort of compounding to move the
prices you know it's maybe realistic to
some extent but it certainly doesn't
take into account for things like 9/11
where suddenly all the prices would
change in one go because of one
unpredictable event five I mean because
here clearly the assumption was that you
would just have very very very small
events that have lead to very very small
effects right as soon as you have one
event that leads to a massive effect
just by itself this is this kind of
model is completely out of the window
and I mean what you say is quite right
is that people tend to forget the
underlying assumptions beneath these
models and then sort of you know one has
to be very careful not to apply them
blindly and to just you know trust the
predictions and not question them where
I perceived was basically what's going
on is a dynamical process although be at
random I wonder if God is t has anything
to do with this there it comes into your
research oh good this if you're very
much coming to some part of my research
it doesn't have too much to do with
these kind of models but well it comes
sure I mean for example if you know in
Dawn's case fear and one you want to
somehow say that when you you know if
you have some kind of stationary process
and when you take some additive
functional to it and you look at it at
large scale you want to see Brownian
motion there you go Algirdas city would
enter the games but here it wasn't the
main point these lectures yeah
giving random walk models in general
over n dimensions given a series of time
steps from the starting point how does
the ratio of distances travel from the
starting point vary with dimensionality
that say the number of Acts easily oh it
always moves like a square root of the
oh it doesn't depend as matter it
doesn't matter of the changes that walk
along the line okay you explains how
Fourier's heat equation turns out to be
unexpectedly useful in describing
Brownian motion is there something you
suspect might be unexpectedly useful for
interface models even if you can't prove
it sorry is there something you think
might be unexpectedly useful for
interface models in the same sort of way
even if maybe you can't prove it yet
well okay so for interface models we do
have a sip okay so there is one there is
one object that we have which is okay
there is an object which is called the
kpz equation and so that one it's
supposed to be some sort of semi
anniversary object so it describes these
crossover regimes and well part of what
I've been doing in the last few years
was to actually give a meaning to that
object dissolve you know contribute to
its proper mathematical definition but
it's not so here the full object so the
full object that will describe the
large-scale behavior one doesn't have I
mean at the moment one doesn't have a
full understanding of it one just
doesn't have a full description so I
mean it's a clear conjecture that there
is such an object and there are many
partial results that go in this
direction and so that one would if you
want play the role of Brownian motion in
this context but one doesn't have a you
know one doesn't have a full conjecture
of what the object is one doesn't have a
description how does a computer generate
a random thing I mean isn't everything
it does dictated by a computer program
oh yeah that's a very good question
yes so how does a computer generate
things at random
of course they're a tool because you're
right right so a computer cannot really
generate something random because
everything that it's going to generate
is just going to be the output of some
program and the program whatever you
know if you run it twice in exactly the
same condition it's going to give you
exactly the same output and so it's not
going to be truly random so there are
many ways that people have to go around
this so one thing you can do is to
actually build a physical device that
would generate some random numbers and
attach stuff device to the computer and
ask the computer to get the number from
there and that's if you want to be
absolutely sure
that it's really really random you can
actually do that and you can exploit the
fact that the best description we have
of nature is quantum mechanics which is
random at a very fundamental level and
so you can you can extract this kind of
quantum mechanical randomness and read
it off and use this now of course that's
the sort of expensive way of doing it
the cheap way of doing it which is you
know what I did here and what people
usually do is you you write so there are
standard computer programs that give an
output which is maybe not random but
which has all the same features of a
random sequence and well so there are
various ways in which you can test
whether a sequence is a good a good
candidate for a random sequence but of
course it's not truly random you're
right now what one can do is do a bit of
a mix of the two so for example what you
tend to do is you do a program like this
which in principle which gives something
deterministic which is not random but
very unpredictable and this program you
have to feed it some input right so you
give it some input to start with which
we call a seed and then if you do two
different seeds it's going to spit out
two completely different sequences and
then the way you choose the seed is you
could take for example the time of the
day at the moment you run the program
and so every time you run it is going to
be something different or you can well
in modern operating systems you have
something called an entropy pool where
essentially you you know the computer
all the time every time you do something
you move the mouse or you touch the keys
or something like this it puts this all
together in order to you know create one
number which is essentially as close to
something random as it gets because it
takes into account all of these
essentially random inputs that you give
into the computer with the tetris model
would that still work if you used single
blocks rather than actual
in shapes ah no they would behave in a
different way but so if you use if you
use single blocks then it would behave
quite differently because then you see
one thing about this tetris model see
here what you see is that well the wolf
you don't well here you see it you see
things can kind of crawl sideways like
this right and this is very much a
feature of these models if you just had
single blocks then they would only ever
pile up on top of each other they
couldn't grow sideways right and that
would have a very different behavior
then what you could do is for example if
you only have these square 2 by 2 blocks
right then then it actually behaves in
pretty much the same way so you could do
it with only 2 by 2 blocks like this and
then you can still grow sideways and
that one has essentially the same
behavior look at it here it looks more
or less the same amazing ok I think that
I should cool it to close I just want to
say that personally I as president on
the math Society I go to a lot of
lectures sore society but just once in a
while one feels truly privileged to
listen to the lectures truly privileged
and I think tonight is just one of those
occasions and I think we've done I hope
you feel that too I think we've done a
remarkably well we've had some wonderful
expeditions and I hope everybody and I
include myself has gone away with lots
of new ideas and thoughts so let's thank
the speakers very much
you