thank you very much it's a great honor
to be here and to speak about basically
it will be about results of Cena and his
contributions in mathematical physics
related to mathematics and to physics
and I specifically selected this topic
because I think that Grisha reflected on
our goreck Theory part of Chanel's walk
and domo will probably speak more
dynamical systems so although part of
the results a big part of today's
contributions of dynamical systems can
be considered as mathematical physics as
well I concentrate on other aspects of
it and this between mathematical physics
it was basically about what scene I did
from 70s upwards and Sinai's himself
yesterday said that it was very
important period when mathematicians
started to study physics and go into
physics and take problems from physics
and you know this cat and dogs thing
about physics it's very interesting
issue and of course mathematical physics
came from the same origin and maybe it's
difficult to say Newton was physicist or
mathematician but then separation or
divorce happened and in different types
of different countries relations were
different it looks like in say in
Germany in twenties or in beginning of
the 20th century it was one community
and maybe in Europe in general for
example Pond Korea has written
recommendation letters for and Stein
famously and it's very interesting
letters to read and see Stein give talks
at the seminar of Hilbert and in fact
Hilbert profound so immediately that
equations are not symmetric the way they
supposed to be and even wrote a paper
about that so it was a related community
somehow but made me elite a
mathematician started to talk less with
scissors maybe they had enough of their
own problems to concentrate it to walk
on it so it's a it's an interesting
relationship and say even Soviet Union
again it was not made much contacts
although
we can there's a famous context between
Kolmogorov and landau during the Second
World War
both were in Kazan they emigrate was
evacuated from Moscow and Makarov was
working on turbulence problem which was
actually physics it was not mathematics
equations were not solved to discuss it
it was a very beautiful theoretical
physics work and they were discussing it
that they had different points of view
but this problem was isolated episode
and this collective movement which
happened about many of great Russian
mathematicians of a younger generation
in sixties was a very interesting
phenomena they very distinguished group
moved towards physics and getting
problems from physics and this is sinai'
this is de Bruijn main laws be reason
Nordica four completely different origin
but also moving in the same direction
Manion so I think the source was that it
was a realization that interesting
mathematical problems and they're coming
from physics that this is a source of
natural problems interesting challenging
and that's the direction to go probably
the indication of certain stagnation and
mathematics at that period and that may
be nowadays may be Morpheus is going in
direction in mathematics rather than
mathematicians to physics so it's kind
of a two-way street now and it's very
good it's just indicated both sides need
each other and they need to be in
coordination maybe there is also danger
when they come too close together when
physicists what kind of loose sense of
theories should be confirmed by
experiment or some measurement so
mathematician with the sense of proving
things but the advantages I think they
overcome disadvantages so it was
important movement and I will discuss
part of it so I would list main
contributions of snail of mathematical
physics contribution I so he started to
work on statistical mechanics in late
60s and works through the seventies I
would take with most liquid Librium but
also non equilibrium statistical
mechanics and the main contributors
Pirogov Sinai's theory and realization
theory for hierarchical Dyson's Harar
models this is giantess public play here
of course Pirogov sanitary journalist
para goof that's those were really
fundamental things it was many others
but I just choose the main things it's
really difficult to you know be
selective and so the next one was model
is the same time in 70 sharing operator
with quasi periodic potentials fractal
properties with denaburg
we filmed in a book this was very
pioneering work and I will discuss it
probably during the talk and then sedan
later won't come back to the problem of
trap spectral properties early operator
late in ages when he studied completely
different case there is kind of
dichotomy between continuous spectrum
absolutely continuous spectrum and pure
porn spectrum and of under some
utilization so that is two different
streams of work we continue it in ages
see nice random walk in rhythm and virus
want the most Oh beautiful work on a
random walk in the random environment
new phenomena was studied discovered
there it's easy to discuss and I hope to
have time I'm not sure we have time
render burgers equation what some people
called burger lens it's not just turbid
burial behaviors equation is also it
looks similar to navier-stokes it is not
navier-stokes equation and dynamics is
quite different this is joint work with
Sven and myself and Alec Mazel and
turbulence itself
great proper turbulence random
navier-stokes systems with Wayne and
Jonathan Mattingly blow ups for complex
3d navier-stokes systems used only so
that's kind of at least which I wrote
but it's not everything I mean there is
a lot of things beyond this list but
that's the main thing now I have very
ambitious plan to discuss
statistical mechanics radical pay
attention as ran a walk I don't think I
have time but that's how it stands at
the moment that's my plan and let's see
how it goes so I'm moving in direction
of statistical mechanics now so it
happened in 60s at the end of sixties
when a big large group of mathematician
in Russia and in the West started to
develop new science rigorous statistical
mechanics in close contacts with physics
problems now I think it was realization
that that type a big part of probably
theory was theory of independent of
weakly dependent random variables also
Markov processes and then it was
realization that new type of problems
probabilistic problems and interesting
most interesting problems coming from
statistical mechanics so that was kind
of feeling which appeared simultaneously
in the Western and Russia among this
group of people I think that breezy
namin loss was the first who moved in
this direction and then they kind of
brought to the same direction they kind
of convinced the Bruce and tonight they
moved to the same thing the blossom said
I was very most students of comma Gaurav
so I think that religion was not very
long in statistical mechanics he wrote
maybe one or two papers but the Russian
said I stayed for very long time and
they are not classical papers now the
famous seven arrow statistical mechanics
and 70s in Moscow was chaired by men
lost ablution cineon Molotov say Oh for
here and that was a real interesting
thing because this similar where the
science was basically created one of the
main seminars oh there were no books
apart from rail book on statistical
mechanics rigorous results so this was
created during this seminars and they
were incredible place of concentration
of people at the efforts and the
enjoyment and excitement now later on it
was Zhukov student of Zhukov Gurevich
Perrigo flossman and many others now
this this is not too extensive or
anything I mean definitely many people
should be included there but just
Oscar Lansford Labov its liberal gleamed
rough eyes and manga la voce frolic
Simon Spencer and so on so forth so
that's people who worked in this area
now this is a portrait of Kimagure first
I showed three portraits of Kimagure
because he made medica tribution SEP so
one because he didn't write a single
papers that is comic addicts and this is
a very interesting topic why Kolmogorov
who was found one of the founders of
probability theory and his work was all
about around statistical mechanics
entropy turbulent theory which is part
of statistical physics he never wrote a
paper on statistical mechanics although
two of his students the Bruce and I
became very important in this
and I don't know what the answer this
question actually I was not the first
could post this question I remember it
was posed by Kozlov during the
Centennial Conference of Kimagure in
Moscow it's an interesting question and
I'm not going to discuss a lot just
Arabic that it was not alien idea the
time because Hinton who was also very
prominent in probability theory a lot of
people book about mathematical methods
of statistical mechanics and quantum
field theory maybe it was in 30s I
believe this book so the idea was no
telling but same still cover girl of
didn't work on this right now let's go
to mathematics so this will be crash
course in statistical mechanics now
those who want to take it for credit can
come after the lecture to me and we'll
discuss the exam and other things for
others I'll just try to explain the main
concept and so the basic notion is gives
distribution so we discuss what is
called spin system spirit system on the
right has led these it's very cyclical
object at every point I point of Z D
there is a finite there is a element of
finite set so plus or minus 1 the
typical example this is what is called
agent model because it in goes German
let's say eating note I think they're
different pronunciations so in this case
finite sets consist of plus and minus 1
it can be different finite set this is
the simplest setting your spin takes
finite number of values and so the
Hamiltonian is a basic notion so this is
a commando net forensic model so Sigma
is a configuration of all the spin on a
lattice and this is a formal some it's
not a number it is sum over all this
bracket stand for nearest neighbors so
necessarily related so you take any two
nearest neighbors and you multiply spin
in these two native neighbors they'll
give you a number you take some of them
it's infinite sum that is a formal
expression now in general Hamiltonian
will be more complicated it can be any
more complicated you just have to figure
out some pattern may be three spins
thing like that so try and take some
overall latest position of this button
so it's similar expression but more
general than that let's stay with I
think as a model as as example for
Hamiltonian but think about expression
which is summation over
the latest of certain party have certain
extraction depending on values of spins
alright so next what you can do with
this Hamiltonian you can take what is
called Vivi is a volume you take finite
set on a latest cut certain set and
consider configuration of spin inside
this set and outside outside will be
Sigma bar style inside will be Sigma so
just to configuration we fix actually
but to say to fit configuration of this
Sigma bar because this is a boundary
condition you figure out the condition
then you can define probability
distribution for speeds inside for fixed
configuration of spins outside this is
giving by Gibbs formula so you first
define watches what there has been
energy of interaction of speed this V
should be capital V here this is a
misprint so this energy of interaction
of spins inside the V which themselves
and which spins outside so this is sum
over all nearest neighbors were both
belongs to V my the sum of Sigma I Sigma
bar J where I and J are still near
stables one side V in self is outside so
just print your on ignoring interaction
outside of of V at all but V this
outside you put together so this is a an
energy of interaction and then you can
write the Gibbs formula and this is a
basic formula it's probability
distribution of Sigma V and the
condition of Sigma bar is fixed and it's
giving my exponentiating of this minus
beta should I said that in statistic se
always have minus minus and minus so
minus beta age and one of us that just
normalization factor just to make that
this is probability distribution that
sum of all these probabilities is equal
to one so this factors that is called
partition function and there's a
positive parameter beta which is called
inverse temperature so this beta is 1
over KT T is the temperature this is in
this temperature so it's one parameter
family of problem I should have put here
bit as a parameter but formula would
look a little bit more complicated so
for fixed beta this is a probability
distribution now what we have is a set
of all possible
expression probability just like it it's
purple eyes by V you can choose any
finite set V and you can choose any
boundary condition so you look at this
as a set of prescribed conditional
probability distribution so that's
different from classical probability
theory where you consider further Demers
to distributions and they determine the
field here you determine conditional
distribution and that's the setting of
statistical mechanics for giving set of
conditional distributions the question
is and that is called the blue salon for
the red condition it's the clarinet give
state it's basically definition is a
probability distribution on spin
configurations such that conditional
distribution for all finite subsets we
are giving by the Gibbs formula above
you can express it as the integral
expression in a way but that's the way
so that's the basic definition which
makes statistical mechanics part of
probability theory and then the question
will be to describe these get States and
to see their structure especially when
parameter beta changes from 0 to
infinity because when bit is equal to 0
it's quite simple there is no
interaction so all spin have the say
they independent random variables this
trivial thing there is only one grip
state now what happens with small beta
for large beta that will be the topic of
our discussion so the main problem is to
describe the set of deep States as
parameter beta changes from 0 to
infinity so what is the phase
transitions there are different
definitions what is phase transition
basically singularities of partition
function for me it would be the
following thing they exist more than one
keep state if this set of conditional
probability ocean is consistent with at
least to give States then we have a
phase transition if it's unique keep
state whether it's simplification but
it's good enough for the discussion
today so the set of give States form a
convex set that's a very simple
statement so every convex set has
extrema points so extrema points are
most important if you know extrema
points you know the co set and they're
called pure phases this extreme point so
the task is to describe the extremal
States pure phases now there is a
general
fact which is non-trivial well which is
very simple actually it's a kind of mark
of theory no phase transition one
demerit okay so if dimension of lattice
is one for all beta there's only one
keep state again it's not always true
but for systems I can see that it is
true if finite range system it's correct
and basically that is because the
process is Markov process so that is a
simple statement now the next statement
is less simple it says the following
it's high temperature temperature regime
when beta is small which means the
temperature is high then spin variables
they are not exactly independent as I've
admitted but they are weakly dependent
and there is a perturbation theory which
make use of this and this is called key
resolve or correlation equation this is
non trivial but it tells you that it
allows you to prove that deep state
exists unique and analytically dependent
parameter beta as long as beta is small
not as well understood part of prime of
high should be written different and so
that is well known but it's already non
trivial and then the next thing which I
want to explain and that's very simple
again there's a fun fact that how you
can construct a the existence of gift
States is a very simple fact so you take
increase in volumes VN sequence of
holders will get big and bigger covering
the whole lattice and then any sequence
basically of boundary condition Sigma
and bar and they take any limit point
for this process so you it defies the
probability that you should increase in
set of spins so take limit point not
necessarily even limit now the space is
compact space of measures on this
compact the set of compact we can find
that number of spins so the limit points
always exist that will be get state
question is where the limit exists and
whether you can have different things by
take choosing different boundary
conditions but any limit point of such
process will be a pure phase all right
so we already state what happened with
small beta nothing interesting well
there is unique deep state what happens
for large beta so for large be that
again
the description concentrates and
figurations with small minimal energy
because it's e to the minus beta so when
betas wave takes large you want H to be
very small then it will give dominant
contribution to e to the minus beta H so
this configuration of small or minimal
energy they called ground states so
basically we won't understand what are
the ground States configuration of spins
of minimal energy now what is the
minimal energy you can ask because it's
a form of some the finish is very simple
if you have infinite configuration of
spin it called the ground state we utter
if the following is correct if you
change the spin configuration and finite
number of places then the difference is
the energy is well-defined and it must
be positive so you cannot decrease
energy by changing finite number of
spins you can do it by changing infinite
but not finite number ok so we are
considering either it's important remark
we are considering in translation
variant of periodic Hamiltonian so if we
can see the spin here and there they
interact the same way or there is a
periodic configuration so because of
that way interested in translation
variant or periodic ground States as
well so spin configurations should be
either transition variant or periodic
and freezing mode of course there are
just two ground States all pluses or
minuses it's ferromagnetic Isaac model
easing model so this is two periodic in
fact transition variant configuration
their ground state but for more
complicated cameltoe nyan can be finite
number it can be different number of
ground states now consider as I said
huge volume V but find that volume with
some boundary condition with giving by
one of the ground state C for Isaac
model they will be either plus or minus
and the main question really typical
configuration inside V look like a
ground state typical with respect of
Gibbs mirror or it will do you forget
about three conditions by typical I mean
typical with respect of Gibbs formula
distribution so that's what I'm saying
huge V W just find that thing it's not
going to infinity so you fix boundary
condition and then you see what will be
probability diffusion of spins inside
the W
and you want to say what that will it
feel the boundary condition or when the
increases that the influence will be so
weak and weaker and you will not feel it
anymore so if you feel it it's caught
you have phase transitions long-range
order you are in a good sermon you can
prove interesting phenomena fave
presence of phase transition if you
don't then there is no phase transition
and everything is relatively simple
so does probability diffusion depends on
Sigma synthetically if it doesn't give
state is if it doesn't and give state is
unique if it does there's a phase
transition system exhibits the
long-range order we shall see that there
is a I think in other cases that is it's
the case now that's I consider Isaac
model so I put plus boundary condition
the boundary and then there is a typical
configured some configuration of spins
inside now it's difficult to deal with
kabila Toric's of the spins because
there are too complicated things right
going on how you handle it so the basic
idea goes back to Pyrus is the following
one you introduce geometrical object you
introduce separation boundary between
pluses and minuses this red curves they
called contours so for every
configuration there is set of contours
they separate classes of minuses if you
know boundary condition as set of
contours you can reconstruct the
configuration so you can either speak on
physical ratio speeds inside or
configuration of contours it's up to you
but of course this geometrical object
it's much easier to deal with it and
also notice that whenever you have a
boundary this contour gamma then you
lose an energy because instead of plus
in your - in your Hamiltonian if you
have plus and plus here if you have plus
and plus here - you have - chairman
Kevin Dorian this will give you plus so
you lose two units of your energy
whenever you have a boundary because of
that is very easy to prove the following
formula that this not the probabilities
correlation function so what is the
probability of those configurations
which are consistent with certain gamma
so if you take any gum on the latest on
a dual lattice
how many configurations which are
consistent with
gamma that is a correlation function of
this contour gamma and it's easy to show
that this probability is less or equal
than e to the minus 2 beta gamma look at
this beta this will play important role
here gamma is the length of this contour
oh it will be volume on the higher
dimensions all right now we're
interested what would happen with a skin
at point at the origin when V is large
so what is the probability that the
speed is equal to plus 1 and what is
probability is equal to minus 1 if both
probe uses tends to 1/2 and then
probably you don't have phase transition
but if this probability of having plus
stay above 1/2 bounded away from 1/2
when V increases then there is a phase
transition so that's what you want to
study what is the probability of having
minus here minus 1 so what does it mean
that it is minus 1 it means that there
is at least one contour which separates
boundary from from the origin because
minus should be separated from places
you have pluses everywhere on the
boundary you should have this control ok
the problem is then to estimate the
probability of having a contour
surrounding the origin any contour now
you can say okay I can see the contour
of length N and it's minimal length of
countries for so it will be sum from n
to infinity now you have e to the minus
2 beta n because this is an energy this
is a probability of contra flanks n it
is less than that number times the
number of contours of LAN Xang
surrounding the origin now this is the
gamma datoria number which is easy to
estimate when you do contour you can go
up you can go to the left to the right
three possibilities when you can
continue your country you cannot go the
way you came so that's 3 to the N
possibilities for you to go for every
choice of your lengths of a contour and
also the starting point of contours easy
to see that it's bounded by 1/8 so when
square because it just you cannot start
very far away from the origin and
surround it so it's very easy to
estimate you look this is an estimate
for this probability now of course if
bit erogenous this is a Phi that some
because this term will dominate that
term so if bitties lag larger than la
vary from 3 over 2
then this is a convergence um morovia
tends to 0 and beta goes to infinity so
what it means is that for any alpha less
than 1/2 they exist beta naught such
that this probability will be less than
alpha less than not have uniformly
nearly the same inequality codes in the
limit V goes to infinity for the plus
phase sorry for the plus phase I mean
for the - phase so yes the same
inequality calls for the whole plus
phase when you go to the limit and by
the symmetry simple inequality holds for
the probability of Sigma 0 equal to plus
1 in a minus phase so you can track two
phases plus phase we permitted to have +
is greater than 1/2 minus phase with
probabilities less than 1/2 so at least
two different phases you have phase
transition so that is what the code
Pyrus argument and it was well known and
this would contribute very beautiful
contribution of Paris to this subject
now what is paragraph C my theory
the Russian line theory is a
far-reaching method which allows to
study deep state construct phase
diagrams for general spin system in the
low temperature regime finite number of
spin variables but take a finite number
of wells but general system so you can
see that any periodic Hamiltonian is
fired at range of interaction like I am
a decent model and a finite number of
periodic ground states which is filed a
pair of stability conditions I am NOT
going to say what is possible it is very
natural condition which is very easy to
verify for concrete models so this is
very constructive conditions and do not
buy all the number of different ground
state now you have to consider R minus 1
parameter family of cameltoe nians which
is perturbation of original crematorium
so R is the same ice here so instead of
Comet Orion H naught you have to
consider H mu which is perturbation and
mu is just mu 1 mu R minus 1 and they
also find the trace periodic Hamiltonian
so they can be considered as external
fuels perturbing H naught so it is
assumed that that's very important
assumption that H 1 H R minus 1 resolves
the degeneracy of the ground state so
what it means is that when you choose
when you change mu 1 mu R minus
one for resulting Hamiltonian you can
get any subset of this our ground stage
so these are ground sent only for this
camel Tony but whether it's perturb you
maybe would have all but one and no but
- so any choice or any subset of this
our ground state can be realized for
particular choice of parameter mu 1 mu
or minus 1 so that's condition of on
this camel Tony which means that they
are kind of linearly independent again
easily verifying condition because this
is condition for beta equal to infinity
it's just condition for ground states
and now suppose the condition of the
previous side that so this way then
forbid a large enough the phase diagrams
its small perturbation of ground states
of the ground states phase diagram so
what it means
maybe they exist a Point mu beta such
that this system with this if you fix
beta and take mu and beta system casts
are pure phases the same as ground
states there are our curves which starts
at that point for which you have all
ground states except one and then you
have Co dimension to you have surfaces
to the master where you have all curves
all ground state but two and then the
full stratification so on the up to food
certification the whole thing you can
construct and this will be just for some
nonlinear choice of this may be turned
those curves so let me draw the picture
here so for Isaac models kind of
slightly different because of the
symmetry so for Isaac model the
perturbation would be this external
field some of the spins times mu and
then these two lines gamma 1 and gamma 2
will be just me less than 0 and mu
greater than mu on beta is identical 0
because of symmetry but if you break
symmetry you will not have this
beautiful behavior and so that's the
typical diagram for say three ground
states so on level of ground state that
point mu equal to 0 you have three
ground States here you have just two
second and third here you have two one
and second and here you have two one and
three
in between you have only ground state
which is not one and all to so three
here one here and two there so that is a
picture 4-bit equal to infinity or T
equal to zero now when we perturb it you
can see the large beta new moves to
point nearby my own beta and then you
have this curved gamma 1 gamma 2 gamma 3
which a perturbation of this gamma 1
gamma 2 gamma 3 so here you have two
ground States two phases two phases two
phases here is one phase 1 phase 1 phase
so that is the most you can do for low
temperature regime and that is done by
very complicated control so this is a
far-reaching generalization of contours
and the very complicated technique
probability perturbation technique reply
but you get as good results as you want
so basic it closes the subject and after
that paper basically was good at the
standing of low temperature regime and
high temperature regime and see there
are two general methods basically there
are Gaussian I method and reflection
positivity to study phase transitions
for such systems and P Ragusa 9 theory
was extended in the case of continuous
feed by doubling the gratnik and that
was done later but I said the theory
basically settle thing so I would not
say it was much of developed a little
because the problem basically was solved
I would say that I know the story that I
think we will had the price attached to
this problem which was well known
problem for everybody
and say that he would bring beautiful
beautiful bottle of champagne and he
brought it to the conference in Ripon er
and this champagne was gladly
consumed during this conference by the
participants when the result was
announced this is still a paragraph I'm
sorry it's not a very good picture but
that's what I got on the internet and
this is pair ago from paragraphs in a
theory now I would start about reading a
Pareto see how I can go so sharing a
period is completely different subject
the paper was published more or less at
the same time and I would say that more
in this middle 70s
74 7-5 7-6 all three papers appeared the
wreck of Chanel Theory surely operator
quite absurd Cuttino Specter within a
book and Blake arsenide on the
renovation theory so it was very grand
period and in general dynamical system
as well Feigenbaum was at the same time
his papers on period doubling and
michelle harmonicas global reserve so it
was a very exciting period anyway so we
see that has written two seminal papers
on this subject one and seventy four
jersey the book regarding the existence
of absolute equality respective and
another in 87 dealing with pure porn
spectrum so in the first paper he
considered continuous time continuous
reading operator that classical show you
can produce potential v onyx which was
quasi periodic but in second paper it
was discrete setting and it's slightly
easier to discuss the screen setting so
well consider this operator so it's a
it's discrete laplacian if i add to say
a fan but try to put either kind of
incorporated in this term and this is a
self-adjoint operator which acts on
hilbert space l2 on that so the question
is to tie the spectrum of the superior
is a function of parameter lambda which
is a complan constant constant in front
of the interaction potential v as a
function of theta and the function of
alpha the most interesting case of
alphas being irrational now when office
so sin i can see that the case of large
energies for this equation and this is
equivalent to considering so what means
large energy it means that this term
laplacian dominates the potential
because large energy comes from
laplacian and so it's equivalent in this
setting to can see the small values of
lambda so basically the result is that
for small values of lambda there is an
absolute continuous spectrum maybe not
only the absolute continues but there is
an absolute continuous spectrum and
there are extended eigenfunctions of
continuous spectrum like block wave
function in this problem that was the
main result of sin I denied were
formulated later but the main novelty
was that they introduced the masses of
KM theory to this subject so the method
to prove the
the difficult thing he has small devices
and to do with this problem they
introduce km methods now potential vo
next is a smooth function on a unit
circle the main examples view on X is
equal to cosine 2 pi X this is most at
example and I don't have time and
present some results about it in this
case basically now almost everything is
known about this almost mushy operator
and you see how how far the subject
moved from that time in this case the
petals of course almost watch the
operator so if alpha is irrational
number then this is the periodic
operator not quasi periodic and the
spectrum is absolutely continuous
respective bands separated by gaps it's
well known classical result by floorcare
theory and block block waves it's a wave
to construct this eigenfunctions of
continuous spectrum so should I did
network ever discover governments
addicted area they proved that the case
of rational dear Fantine alpha you have
to consider the typical alpha Devon time
for small lambda they exist a set of
energies e of positive measure
corresponding to block waves of the form
of classic quasi periodic functions time
exponent which satisfy the equation that
Hamiltonian times this acting on this
function so it's eigenfunction of
continuous spectrum this statement
implies the existence website Cuttino
spectrum the main problem is a related
small devices now I like this picture
very much
I call it said I and km this is picture
of 63 so where is Chennai yes and this
is also Chennai and this is Arnold now
this is Moscow State University
so Kimagure office someone there because
he worked there and he also lived there
his apartment was inside this huge
building so its probability close to one
sees inside so where is mother mother
was taking the picture so each picture
taking by mother so it's definitely seen
I were surrounded by km and what I like
in this picture that they look so happy
boss and so it's probably brings a lot
of fun to do mathematics it's for young
people who are taking exam
now let me just say a few words about
shredding a cut cycle would explain what
it means so if we want to solve this
eigenvalue problem write for this
equation take e we don't know e but
let's take any e and let's solve this
equation then we can say what we'll do
is sign plus 1 you can express it cyan
if you look at it basically you take
pair up side plus 1 to N and there is a
matrix if you multiply play and trial
run of this matrix you get new pair so
basically how you solve this
second-order equation you start with
initial data at two points and then
multiply on matrices so if I did not
this mattress by s data which is SL to
our matrix then what we have that try
and train minus 1 is equal to product of
this matrices times pi 0 psi minus 1 so
we have initial data and multiply by
those matrices now what is the structure
of this matrices this matrix is depend
on on point on the circle and basically
my next slide show what we have here so
we have a circle at every point of a
circle there is a matrix and then we
have translation of circle by angle
alpha and we take product of those
matrices alone trajectory of rotation
this structure is called Co cycle when
you take product of matrices the long
transformation T which supposed to be R
goreck mostly so this rotation by angle
it's not very stochastic but it's
ergodic transformation not mixing and so
it's enough to discuss Co cycle and its
statistical properties so basically we
have to take product along this thing
now in fact they just look at it
slightly differently there is a fiber
structure so there is a plane leaving
which kind of parameterize by ad theta
so over every tattoos every fiber which
is two-dimensional plane and this matrix
in fact is a linear operator which
transfer transformation from R to here
to r2 there so when you take product of
the commodities you take digital service
from this one to resulting one why I am
saying that is that then you can ask the
following question suppose I change the
variable at every of this fiber
but I make change of variables depending
on Teta that the dependent change of
variables at every of this plane so I
have family of matrices be depending on
theta which are non degenerate methods
of exchange of variables then instead of
this Co cycle or have conjugated con
cycle and new variables so this new Co
cycle will be bit at times old kasaya
code times this sorry this is not a this
would be s e as time B minus 1 so that
is a conjugated Co cycle the Nuka cycle
which you get and now the task is to
find change of variables such way that
this would be much simpler than that one
what is simple we say Co cycle is
reducible if if one can find a smooth
family of doctor generate medicines in
such a way that this nuchal cycle is a
constant K cycle it does depend on theta
then you're in a good shape and to do
this linearization or a disability you
need to use km theory so K method allows
to prove that in this case a small gamma
lambda the Kasai Co is reducible for
positive measure of energy E and this
implies the existence of absurd
continuous spectrum and that is Dina
Berkshire nice theorem now later
Eliassen
hoc analysis showed that reducibility
holds for almost all e which proves that
that spectrum is purely observed
continues in the case of small lambda so
there is no other part of spectrum there
is no eigenvalues with decreasing result
to eigenfunctions and that's later is
out of a lesson now to prove what about
poor poor spectrum to view that now the
task is different it's like a
statistical mechanics low bidders small
beta lodge beta here small lambda large
lambda so for large lambda you want to
prove that potential dominates they play
more important role than the laplacian
so to prove that for large lab that
predicate pure point spectrum well helps
to construct a basis of eigenfunction in
this space l2 which i'll two functions
in fact they are going to be
exponentially decaying like understand
localization and this was done by sinai
and better expensive and bitter
which independently did the same thing
actually it looks like three years but
in fact it was maybe less years between
the work done soon I proved that for the
Aventine alpha and typical theta the
specter was pure point provided that
lambdas Lodge
moreover corresponding inversion decay
exponentially fast understand
localization address of localization was
in case of random cellular equation so
this most settled part of understand
localization theory and the proof is
very involved and is based on difficult
analysis of resonances so I'm citing
here listed a chemise of Michael Stein
he cites and I'm saying that
localization is a game of resonances and
in fact the method of proof is based on
very careful analysis of reason answer
to this problem I don't have time to go
with it I just maybe one see what does
it mean from Kosaku point of view so
that is about these oscillators may be
stable stable directions which I
discussed so we have this Co cycle with
a product of this matrices along the
trajectory right and we know that if we
applied it to this thing we for large
lambda this Co cycle would have one
unstable unstable point of exponent so
what it means that there is a direction
and unique direction such that if you
apply this matrices this will tend to
zero exponentially fast that's fine so
there is one five zero five minus one
now but you also want this to decay in
other direction so it means that there
is another vector such that you apply
this backward operator
it also comes exponentially fast in
other direction now unfortunately this
two vectors would be different but if
you find e in a special way then for
special values of e they would coincide
and when they coincide then you will
have exponentially decaying
eigenfunction and that's how you can
think about ID well fortunately it
doesn't what proof doesn't go this way
but give you some intuition with my
changing parameter you can maybe some
time attain that this unit vector in
this universe to coincide but again you
have to go to complicated theory of
residences to do so and so again various
it's when they coincide and I would say
that is a huge activity this area right
now and in the last twenty fifteen years
so the lost list of people who worked on
it extensively
Arvelo and beliefs are Borg in demonic
alas and for illegal Stan measurement
laudatory Korean lost Buick slug
Spencer's Simon everybody the results we
discussed before perturbative in concern
was two different asymptotic regimes we
can strong potentials a big progress was
achieved in developing the global theory
studying the whole range of parameters
lambda so not only large land and small
lambda but the whole range for
particular components particular it
became possible to study the transition
from absolute continuous spectra
corresponding to small lambda to pure
point to large number the best studied
is the case of almost maturity for which
almost a complete Specter pictures
understood by now and I'm happy to say
the formation I students and there's a
Thomas K he she worked all her life on
this problem and she was very
instrumental she made the outstanding
contribution to this program and very
recently I said the time traveller
develop a global theory for general
smooth potential not for a cosine which
was done before but for Jo potential
which must have theories to complex to
formulate the results okay now I want to
see them from physics point of view and
we're discussing mathematical physics
what does it mean its transition from
extent that grounds the eigenfunctions
of block spectrum to localized fire
eigenfunctions of understand what it
means is that when they extend it
electrons can go freely through the mail
through the material so this is metallic
state when you have electric electric
conductivity and when you have localized
eigenfunction it's insulated state
electricity is not going through the
material so it's transition from
conducting to insulating state and now
about almost by Chiappetta what is known
so that is for this particular potential
so if lambda and lambda this parameter
in front of the potential between zero
and two respectable purely absolute
continuous for all alpha and theta
even if alpha is rational we know it
spiritual continues now when lambda is
equal to two the spectrum is almost
surely pure singular continues almost
surely means that for all irrational
alpha there is a countable set of Tet
which possibly exceptional conjecture is
there not exceptional and conjecture is
that for all various surely not almost
surely there is a singular continuous
spectrum it's not proved yet so but with
just accept of countable set of phases
that the result is known and then the
one lambda is greater respect it was
pure pore inspect them almost surely now
this is known that almost surely cannot
be made surely because it is known that
the exception to exist that's
exceptional alpha and theta for which
spectrum is not pure point so I'm done
here I don't think I have much time to
discuss the nice random walk and it's
Jeff was confirmed by a chair person but
I would just finish by showing one
picture well it's I would say that
what's and I proved is that in his this
particular case the render work behave
like not squared event like normal
rather one behave but like la Greve
square and it's basically stuck it's not
moving much and that was a particular
case is very J in a very very natural
setting so unfortunately I don't have
time to discuss it in details I want to
point just one picture this I am NOT the
worth of this picture this picture was
prepared when scene I was 75 by group of
his students former students and buyers
colleagues friends and obviously you can
see that this is Moses and this is
tablets it's not a paper it's a tablet
this one where Ten Commandments are
written it looks like six Commandments
rather than ten but still this is coming
from where it's coming where else like
she and I said yesterday from Sinai
right and those are Commandments and you
know famously Moses when he took people
of Israel from Egypt he told them that
they have to have random walks in Sinai
desert for 40 years and they they had
this random walks and desert for 40
years without being able to get to their
to their goal
and I actually met for the first time
versus in a in 74 except I think it was
in May so it's exactly 40 years now and
so it was 40 years of random walks a
very enjoyable random walks and the
guidance of these commandments and
leadership of Shania and it was a very
exciting journey and I was a graduate
student and I was a graduate student and
colleagues in London and friends and I
want this Georgia to continue beyond 40
years thank you