00:00
thank you thank you for Alis p and the organizer for the this invitation so
many thanks to Alise because she has introduced Coit on this morning that
will be the main topic of my talk uh so I will try to explain what
are the geometric structure of thermodynamic introduced by Jean Mar Soo
on how to extend it for machine learning especially for what we call
thermodynamic informal networks on Le group machine learning start with this
code by Vladimir Arnold every mathematician knows that it is
impossible to understand an elementary course in theramic so it's why very
difficult I remember my course on theramic and I have good good not

00:01
notation in theonics but I have the feeling not to understand really uh
structure would like to D dedicated my talk to even sh pass away this week she
was a the first woman elected at the French Academy of Science and she worked
a lot on Geometry for physics with Andre Lovitz and Je which
has the French School of geometry for physics and especially
his book analysis manifold on physics on
I invite you to look at this video at is for his 100th birthday on so his
conference on his proration with Einstein the French Academy of Science
also the Aro of today is sados so we have just sheary of the of the book of

00:02
[Music] s so we have organized Rafael pizo organized in Le a conference Carol
2024 and the book will be edited by Springer on on on on this
conference there is also M Frances dord has organized with
also Rafael Pizano but also Robert Fox from University of
Oxford a meeting a seminar on S Caro and also at e poly technque you have a s
Caro Legacy conference organized by uh by Gil Kristoph gil on at the dinner at M
po technician the family of caro so Olivier and Alex Carno have a good idea to
launch a project is to put uh to place sad Caro in the pon with Lazar Caro his

00:03
father and with his NE few siko the French President uh so uh sadano is not so
popular in France more in UK or so you could find very good video of
Michelle s on S and I just find a song on S generated by
Ai and only recently in the last version in December of the French Academia of
science dictionary they have added the fact that s Caro was the inventor of
thermodynamic and also in I was in Glasgow in November and it was also it
correspond also to the benty of Lord Kelvin that has rediscovered the
invention s Caro and popularize this invention and he was award by the grand
qu by the French President the Republic of the of this period for this
rediscovery of sadik at uh you could find also a

00:04
theramic in Arin Shing girl who I invite you to read the book statistical teric
byin Shing so my talk I will give you an elementary stic structure of
theramic so you have to remember only that the other is mathematic development
but the main idea is the first chapter then I will introduce the first idea by
Conant was to try to use this F form of theramic to introduce an axiomatization
of theramic that is it was in fact a contact structure contact geometry of of
theramic then I will introduce information geometry because I need to
introduce information geometry to make the link with the work with Jean Mari
Soo we has introduced stic foliation model via moment map of information
geometry on what we call Le group thermodynamic and then we give a
definition this is the main topic of my talk that you have a geometric

00:05
definition of Shannon has given axiomatic definition of entropy but Su
has given a geometric definition of entropy and we will see that the entropy
the cmia function in the coant representation uh con so it is a
constant function on the quadent orbit on the stic leave so I will explain here
after during the seminar you have a talk by batist Kino because in the second
part I will address dissipative phenomena on the link with metritic flow
so batist has explained you in the seminar you could see the video on the
on the seminar channel that the metritic flow is compliant wither relation in
theramic and then we will study the transverse tic flation structure and I
will uh I will conclude with some extension with a link with Lille

00:06
complete integrability of information geometry on the
application for theramic inform nor Network okay the structure what is very
important in physics is equivariance as explained by Alise uh and I I invite you
to read this paper it is a white paper done by be clay MIT yel prinston on
symmetry group equivalent architecture for physics is the main basic tool of
know many development on modelization in
physics but also now in machine learning the main tool I will use will be the
quadrant orbit so the coant orbit has been introduced by kilov so at least
make reference to kilov Coston suo two forms so kiril has introduced in this
book Element elements of the theory of representation so it is a representation
of the group uh so this is a Russian school

00:07
which has introduced first this tool and
here you have a testimony of Jack dimier on the work of of kirilov and he say
kirov thesis published in 1962 immediately Ares a great deal of
Interest kov quickly convinced himself and he convinced the mathematical
community that this orbit method should be applicable to groups much more
General than npon groups uh so we will see uh what is what
are the the roots of Li group theramic we have two Roots uh you have the
classical axiomatic roots with Baki and set and also category the so information
geometry has been extended by Russian Nikolai chenso in the framework of
category Theory and J Kul has introduced some structure with Le algebra chology

00:08
which are the basic uh bedrocks of information geometry but you have
another path which is based on stic Geometry with J Maro who has introduced
the tool of SED geometry in in in mechanics and also the tools introduced
by charl rman and George Reb on foliation on web the so fation Fage in
French and webs is tissue in French and so J Soo has introduced leg group
thermodynamic it is in fact an extension of information geometry for Le group on
Ines we have made the link with with information geometry because there is no
information geometry in J jio book and also the link to metritic flow for
dissipation dis phenomena and then you have new tools for Le group machine
learning on theramic in neural network okay so my talk will talk about

00:09
Le group on algebra representation ter theor so you have the notion of the
algebra chology quadrant orbit cycles and it could be extended also for
Quivers obviously simple geometry and geometry and we will introduce the K to
form C kov suo to form on the moment map the main invention by suo so M map is a
geometrization of not theorem it is also linked to integrable system because in
fact information geometry and statistical manifold are linked to
complete Integra integrable system and so there is a
recent connection between information geometry and delone polytop
Theory you see the link with foliation Theory because we will give a foliation
interpretation of theramic and information

00:10
geometry um information geometry itself uh but I will introduce uh this
information geometri with Kul tool so Jean L Kul was the PHD student of HRI
carton but he has developed the theory of elarton on omous domain [Music]
of the gemetry of symmetric homogeneous bonded
domain first very simple example is point caral plane or Point Unit dis and
we will see how to use it for with natural gradient in machine
learning on metritic Flow so here we extend uh the classical model of suo for
dissipative phenomena so it could be a very interesting also topic for machine
learning because for the time being machine learning only use non
dissipative phenomena on what we call gradient of learning is is simple stic
gradient which is not dissipative yes and I will make the link

00:11
with with the Philip on BST Cino work to make the link with
onang relation obviously you can put all of that in the context of calculus of
variation uh on the the Ser group thermodynamic so about dissipative non
dissipative so uh men fena could be reduced to by capturing the Symmetry on
the envar of the problem by a Pon bracket so as explained by Alice with
the pron geometry so you can reduce the classical Amon equation to this Dynamic
with presson bracket but this pron bracket only describe a Dynamics for
nondissipative phenomena so phip Morrison has
introduced during the 80s metritic flow uh where you will had another bracket
but it is not a Pon bracket it is in fact metric flow bracket and so you have

00:12
two bracket now a PR bracket describe a dynamic of non disipa phenomena and
dissipation is given by this bracket uh in in my notation H will be the energy s
will be the entropy and we will see that
this metritic equation as explained also by batist Kino preserves the first
principle so which is the quantitative preservation of energy on the second
principle of theramic which is the qualitative degradation of energy for trpy
production uh yes so what is metritic metritic is a aggregation of two term
metric and stic so in fact you will have the metric term
on the stic term will give you this metritic equation uh with two

00:13
characteristic is why I've made the first time the connection with the
liopic of Soo for this model you should have that the entropy is a Casmir
function of the pl bracket on the energy H is the Casmir
function of the uh metric bracket okay so it's very uh very simple to interpret
in fact you have two foliation you have a stic foliation which are the leaves
which are correspond to the co orbit so the the quadrant orbit will generate
this slective foliation and Soo has observed
that in fact the entropy is a Casmir function on this coant orbit so it means
that the entropy is a casmier function of the stic
leav and when when you move on on this on this
leaf on this foliation you use the plom bracket

00:14
but if you want to describe dissipation dissipative phenomena you have to use
the other term so in Red so you will have a
transverse Dynamic to the stic fation and you have another foliation which is
not a stic foliation but a remanion foliation and also this is very uh
interesting because it correspond to the level set of energy okay so when I when
I understand this structure I I think I have understood the thermodynamic for
the first time you have two fation one foliation which is a SLE foliation level
set of entropy and a transverse fation which is a level set prman fation level
set of energy and you can describe all dissipative phenomena in this
way okay and so in theramic informal Network they use uh they use these two

00:15
con train uh to preser the first principle of theramic and the second
principle of theramic I will come back to that so the only thing that you have
to remember to my talk uh is this definition of entropy entropy is the
invariant casmier function of Le of stic foliation Associated to the quadrant
orbit generated via moment map of symmetry group acting on the system so I
will explain try to explain this definition
but this is a definition of a casmier function s is the casmier function is
for all Amon H the pl bracket sh is equal to zero okay so here s is the
entropy H could be an or the energy and Q is a it we will see Q will be an
element of the Dual of the Le Algebra I I put in index Theta which is related to

00:16
cyes so Alise has presented some model where you
have equivalent action equivalance of the coant operator but in some case the
coant operator is not equivalent and you have an additional term and you have to
modify the pron bracket to take into account what we call a default of
chology of the group and uh this is another this is the
same expression written in another way with the uh with the coant operator in
the Le algebra so the derivative of the entropy
with respect to the the itat will be the pl temperature we will see after but
here you could see that you have only two variable the entropy on the heat and
you have some method to build from this uh equation the casimer function okay so

00:17
the idea yes this is a braet ah the metric bracket metric braet
I will give the definition after okay this is a uh symmetric positive
definite bracket okay because it is linked to a remanion
foliation okay and you have also to make the link with information geometry to
this foliation will be uh will be linked to the fure metric but extended by suo
for Li group thermodynamic and the other way is the Dual the Dual of the fish
metric which is the as of entropy so you have two dual metric given by

00:18
information geometry so this is uh our proposal to replace the channel
definition of entropy which is purely axiomatic by a geometric definition of
entropy of Soo so it's not obvious in Soo you have
this equation not not this one I will see you here after it was okay so the
first time I I find the equation is not in the book of Soo but in a paper of Soo
you have this equation so you have right it is obvious that uh so so we we take
two mons with Char Michelle M to try to demonstrate that it is right that this
equation is is this equation Q is a itat beta is a plank
temperature usually the inverse of temperature Z is an element of the Le
algebra Theta T is a cycle in case of nonology so I try to put the the caros

00:19
cycle on the foliation okay you are familiar with the caros cycle but you
can rewrite in the volume on pressure temperature but you can rewrite the
caros cycle in the parameterization of entropy on temperature and in fact here
you have a constant also e q1 and Q2 and in fact you have you have a
thermodynamic modification at a constant entropy then at a constant energy and
then constant entropy and constant energy okay so is Aio cycle interpreted
in the foliation model okay is it clear or it's completely
obscure uh so it's uh it's very simple in term of stic geometry on Pon geometry

00:20
to interpret thermodynamic like that so in the yes yes I'm not
so excited by the K Liber entropy in fact when I introduce uh I will
introduce you information geometry I will use coal
structur uh the the history of information geometry introducing a cuby
Liber as a as a initial uh equation to introduce a ra Fisher metric is not the
good way in fact I will explain after but we we we we should be able to extend
kby Liber geometry I will explain you uh you have to replace the classical laas
transform to laas transform on stic manifold which is obvious for
mathematicians see their work during the the last century how to define a for

00:21
transform on the group uh so I will give you the the
equation yes yes yes yes yeah we'll explain you here after so here you are
three Theory you have the first Theory which is a so group thermodynamic with
entropy as a kmia function uh and also information
geometry introduced by Jean Kul I will give you the so Jean Kul was also member
of baki group of the second generation and so on the relation with metritic for
dissipation and batist Kino has explained in the seminar the Rel the
link between metritic flow and oer relation to validate that the metritic
flow is a good flow and the last one is re foliation U theory on stic foliation

00:22
which has been introduced by charl erish man on George
red okay so we have launched a seminar which is called norbas parisan seminar
it is a seminar between n and stasbor and also our colleagues from Belgium uh
and we make reference this is small seminar compared to sud Rod seminar but
during the80s they have launched s Rod seminar between Mar and M working on
stic Geometry on Mechanic uh foliation on geometric quantification stic
geometry on contact geometry uh prion foliation uh
Singularity foliation on Aman mechanics you can
find Herman is I make a new edition of this of this

00:23
seminar so on in this seminar we have discovered three three publication
geometry geometry of moment map by molo gamma structure so GMA structure on
liberman foliation by gazer and molo on stic Duality foliation on Geometry of
moment map by Molino and in fact this transverse
structure are link uh what is called also gamma structure uh on also a liberman
foration at the beginning of theramic they try to understand what are the
geometric structur this is here you have josia will Gibs diagram sked by James
cler Maxwell TR to understand what is the geometric structure of of
thermodynamic and also in Caro you have some idea that
also if you have to consider uh when bodies when bodies are in motion

00:24
what are the impact on the thermodynamic and also sh Michelle m g
me this this remark that in the book of Gibs you have this idea that the
generalization of GB States built with a moment map of the product of the one
dimensional group of translation in time and the three dimensional group of
rotation in space or the study of system contained in rotating vessel so this is
the main idea of that I will explain that classical terit is valid only for
one group time translation okay but if you look for
centrifuge you cannot apply classical terminam so the main idea of suo could
we Define a theramic which is covariant under the action of the Gallo Group
which was his first question so this idea to take into account the covariance

00:25
with respect of the GB state with respect to a to a group especially a
translation and rotation group was in the book of
Gibs uh during the the beginning of the the end of the 19th century so K tried
to axiomatize the thermodynamic on using
the fa form so I will not have two times to explain this part but the main idea
uh so it's very difficult to read the paper of of of of K but I invite you to
read the the lecture of v p which has rewrite this model of of of Kor in a
more clear view and it has been only recently interpretated that the model of
car is a contact geometric model of theramic okay so contact geometry is
very interesting but uh so we have you are in dimension 2 n + one but you can

00:26
stify this model to have a 2 n + two Dimension and then to have a stic one so
I will not give too much detail but I give you the reference ofor Frankel the
geometry of physics where he interpret the car model as a contact
geometry recently I have discovered uh I don't know if you know Herbert Kalen if
you want to learn thermodynamic you have to read the book of Kalen
say me that there is only one book in theramic The Book of Ken and after his
book but and I have discovered that you have written two
paper during the 70s a symmetry interpretation of
theramic and theramic as a science of symmetry and he he gives this idea of
Soo but he was not aware of of the S work so symmetry under spal translation

00:27
implies conservation of momentum on rational symmetry implies conservation
of angular momentum it's relevant as the
theramic coordinates underlies the first law of theramic time translation spal
translation and spal rotation symmetries
are interated in CLE class of continuous SpaceTime
symmetries the Symmetry interpretation of theramic immediately suggests that
energy linear momentum and angular momentum should play fully anal roles in
thermodynamic the main idea that we will explain with Soo
uh you have a group acting on your thermodynamic system so you take the
Gallo group is a Time translation space translation uh space rotation and you
want that your G State should be coant under the action of this group okay this
is not the case for classical teric because the classical teric is only
covariant under the action of time translation okay information geometry so
uh the main idea come from from frese in

00:28
fact so I'm very happy of this figure of this picture I found it at Eon normal
Superior is a very young picture of fret at Eon normal Superior on in a paper of
1943 he has introduced what we call the kar bound uh but you have many more in
this paper but uh only written in in French and he say that the the the men
content of this paper is in this lecture at Institute in Winter
1939 was not a very good period to write to write a paper so I will not give you
a detail but I have written deeply this paper and you have the structure of
Fisher metric and also you has discover the the leg transform uh existing for

00:29
density of Maximum entropy so I inite you to read this this paper and uh after
Russian School are axiomatize the foundation of information
geometry and I make reference to Nikolai nikolich
chenso so classically we introduce information geometry by by this way so
you have a density of par of probability parameterized by Theta and the idea of
RA is to find a metric which is invalent
by change of transformation uh parameter parameters transformation so you take
the kback Liber uh Divergence between P Theta and P theta plus d Theta you make
a t expansion the first term cancel and the second term make appear what we call
the fure Matrix which is given here okay and this fure Matrix is used as inverse
as a lower Bond of the Coan matrice of the statistical estimators

00:30
so if you apply that for Goan law it's very simple so the fishal Matrix is
diagonal and if you apply the computation of the Fisher retric you
have this one and if you are familiar if hyperbolic geometry You observe that it
is a metric ofal plane of Point car okay why because the mean the mean is real
the standard deviation is positive so the space of par of Gan density is in
the upper Al plan half plane of Point car so it's obvious that we will recover
the metric of Point car and here in the model of RA is that you take into
account that the variance on the mean is Sigma 2 and the variance estimation of
Sigma is Sigma 2 ided by two you cannot integrate this metric so
this is a famous idea of Point car so to

00:31
complex specify the plane and to use the K transform to transform theal plane of
point to the to the dis of P so you are now a new variable W given by the K
transform you can no longer integrate but Point integrate from the middle to a
point on the radius and then use use the isomorphism of the plary
disc to have the general formula so what
is the main idea is when you have a gion when you have a gion you can represent
it by one point in the p dis okay if you
have a second go you have a second point the ra Fisher distance information
geometry distance will be the geodic the point car unit dis okay and this is a
link with Jean L Kul work because je Kul work on homogenous bonded domain on the

00:32
most simple example is a point car unit dis the question of elarton to jul is to
try to define a metric which is invariant by all the automorphism of this
space uh so if you have familiar with information geometry also we use a
natural gradient so we apply the Fisher Matrix the inverse of the Fisher Matrix
to have a gradient of learning which is invariant by change of parameterization
and uh during the last decades Yan olier has extended this natural gradient to
define a natural L Dynamic mixing the natural gradient on the stochastic
gradient on we give him the GSI base paper award for this for this
work the natural gradient is used I found recently that Google tried to use

00:33
the natural gradient for llm also so you have this paper using this natural
gradient to improve llm on convergence of llm so now stic fation VI onul fish
metric from information geometry for legal teric so I enter in the in the in
the work of Soo so in 2019 has been organized in Paris a congress for the 50
birday of his book structure the system Dynamic structure of dynamical system
and Je B has G testimony on the discovery of this book when he was young
student at e poly Technic and they have put in place a seminar of young students
because they consider that the lectures on mechanics and Aral po technique
during this period was not so good and they have discovered this famous book of
Soo and all was so clear with this book Soo was gradated from Eon normal

00:34
Superior in 1942 uh and in the same uh in the same
uh in 1942 you have also Jack DM which is also very well known foring
algebra and also Rene for from quadratic so so in 19 69 he has written
this book structure the system Dynamic uh uh
where he has introduced uh stic geometry
in mechanics but in chapter four of this book was dedicated to uh statistical
mechanic and I think that nobody has written this book uh during this period
because it was read only by people from geometric
mechanic and uh I have made the link with with je Co we have discovered ures

00:35
given by Jean Kul in China and we have convinced Springer to make the
translation of this book which is Introduction to stic Geometry where you
have the theory of suo which is explained in modern notation because the
book of Soo is very difficult to read it use its own notation for
leg so we can read this sentence uh these formula are Universal uh uh because
the are is involved only the stic manifold the Le group is cycle istic
cycle on the CLE the the itat on the temperature the temperature on the it
and it call that theramic the group The the group
theramic okay so what is it is in fact the SLE theory of it uh so you know for

00:36
theory of it Clos mechanical theory of it or mathematical theory of it but so
has introduced geometric theory of it on Simple Theory of
e a translation has been made in English structure of dynamical system a stic
view of physics so why I read this book uh because in this book uh you have this
property the total mass of an isolated dynamical system is a class of chology
of the default of equalent for the moment map it could be very obscure but
just to say that the mass in classical mechanic is a geometric object due to
the cycle of the default of chology of the gallog okay and in in in in
relativity you have not that because the
the point car group in relativity has no default of chology it's why the mass

00:37
don't play the same role in class iCal meic and in
relativity but I read especially the chapter four where he develops the group
[Music] thonic what he say so you have some
trumpet in in the book of Soo each time you have important theor you have a
trumpet uh no it just say in French I will translate you but that the the
classical teric on the Gibs density is not covariant under the action of the
Gallo group this is a first remark of [Music]
suo and what he do uh so it proves that in fact the pl temperature so the
inverse of temperature inic is an element of the Le algebra of the group
acting on the system on the it is an element of the Dual of the Le
algebra generalize the notion of entropy as a Lo transform of minus logarithm of

00:38
theas transform and has a casmier function on
stic and also he has extended without knowing information geometry he has
introduced a metric for Le group uh because he has defined a statistic of Le
group and so it is an extension of fure metric for Le
group and he also he has made the the the analogy with it call that we will
see that the fure metric would be interpreted as what we call a calorific
capacity in thermodynamic okay so the main formal is
fully covariant with no special coordinates and you have the covariance
of the gity with d Dynamic group he has also developed this theory in another
paper 1974 uh statistical mechanics leg group
on cosmology and the first part is dedic D at to stic model or statistical

00:39
mechanics it is in French so I have translated we have organized at Le
seminar and I have a paper on Springer book of the proceeding of the seminar
where I have translated three paper of Soo in English developing this his
story okay so all is based on all is based on stic Geometry it's a long long
history of stic geometry uh the main idea of of suo is that the main idea
come from lrange because suo was as a first
position it Institute the Tunis and he live in cage and he have nothing to do
with many people from from uh not from from science and he he decided to re to
read the lrange book and he has discovered stic structure in

00:40
lrange and so you have different contributors to stic geometry and if you
are interested I have given at Mami seminar a link to this
history and discovery of this stic structure and obviously suo was the PHD
student of Lovitz which was the main actor during this period at College the
France uh and we will make the link also so with the work of
Po so it was very well known because at the beginning we was invited byman on at
the first Congress in Strasburg which is very well known in
1973 so in the background you have je Mari suo which was invited and here you
have je Kul but you have many famous and the most famous geometer of this of
this so what is the main idea the main idea of solo is very simple so take take

00:41
the Newton equation okay here you have the Newton equation a is a position M
Mass F Force so you have the Newton equation you can rewrite with the
variable the speed the derivative of position by T by the time and you have
another equation so you have two equations that you can Rewrite by the
system of two equations okay on this system of equation in a new space that
call Evolution space so time position speed you have two equations that form a
fum form system and this function system will generate a foliation and what has
observe sio that you can introduce two second order ter Sigma
that is called the L form that is now called K form C form

00:42
uh so he he make only one reference in his book is fr galiso the form exterior
on Mechanic is PhD in chart 1954 so galiso tried to find uh give
this theorem there are only three type of differential form generating the
equation of material Point motion and you have this form a Form B and form C
and suo has used this form B this is always the same idea to be
invariant by the action of the G group okay skip
that uh I will skip that also so I will use this notation uh sometimes it's not
the same notation in the book on papers on Quant orbit so I Define the quadrant

00:43
orbit by a duality bracket uh this notation so you have a genius minus one
here not a problem so as introduced by Ali this morning so you have the adant
operator the adant operator is the action of the group on the Le
algebra and you have by Duality you can define a coant operator will be the
action of the group and the Dual of the lebra okay and what is important as
explained by Alice is the coant of orbit so you take G an element of G and for
all G and element of the Le group you look at the action of the group on the
Dual of the Le algebra and you will generate this coant Orbit on this fation
Associated to the quadrant orbit you can introduce a two form which is called K
two [Music] formo and what is the main idea of Soo

00:44
this is an idea that to have in lrange if you want to make statistic on the
movement of Planet uh the idea is that take a group the Le group acting on on
your system take the coant orbit and Associate it to this coint orbit a stic
structure with this two form the K two form okay so you have one points in the
space of f group and you will put this point on the stic
manifold in this structure this group act homogeneously on this homogeneous
sythetic manifold so obviously if you want to make statistic on the group the
idea of su and lrange is to make statistic on the stic manifold okay when
you have you have some n amount in the group you want to make statistic you
want to Define some mean or variances of for the Le group all is in the coint

00:45
orbit Okay so this is the main advantage of artificial intelligence that we can
make moving Gary on jeul I come back to information geometry
the good structure for introducing information geometry is this one you
have to introduce Fe which is a uh generating function of cumulant okay
minus log of the laas transform with Lambda is a Le measure and if you take
the ation of this function you will find the Fisher
metric at the other relation that the loan transform of this function given by
this one we have to define a duality bracket uh this give you the entropy so
the entropy the shanon entropy will be the loan transform of this function in
fact this relation has been introduced by franois m Jean franois M was a

00:46
engineer from cemin and he has discovered this
coration and we call that the massio potential be careful this is not the
this is not the uh time I give a a lectures want to I want to kill the free
energy the free energy is not the good uh the good uh
potential uh okay so classically people use free energy okay okay so I will
energy will be energy minus temperature by the entropy but in fact you have to
use the mass potential which is this one
you have to divide it by the temperature and so uh yes
right okay that could I could rewrite this way okay and here you have the

00:47
leion transform between this variable which is the pl temperature the energy
the pl the mass potential and the entropy okay so you have to stop to use
uh stop to use a free energy because when you use a free energy you you broke
the structure and you have no longer the Leon
transform this is a mistake of Gibs on DM to have used this kind of potential
on the good potential are the mass potential so if you are interested there
is a very good paper of Ro Balon on the website of the French Academy of Science
about massio potential so you have two system of coordinate which are D to each
other okay what is important is this this structure Le transform on two AAL
coordinate on two potential Mass potential on the
entropy okay so Kul has extended that uh no so you are here you are on on the on

00:48
air uh but for Kul he has toed for on a sharp homogeneous sharp cone sharp
convex cone so Omega is a sharp convex cone but it is exactly the same relation
minus log of the laas transform on this sharp convex cone and the question of
Kul is what is the matric which is invariant by all the
automorphism of this cone yes yes yes yes no no no yes yes
yes and so it has also been studing in Russia by Ernest vinberg and we talk
about this function which is the the co be characteristic function you know the

00:49
characteristic function of probability but the history of characteristic
function is probability is thatare give a lecture on thermodynamic
and as he was gradated from cordine he takes a book of mass and he has
discovered the mass potential and then he gives the idea to P to introduce
characteristic function probability and then give a lecture
atbon on probability and introduced characteristic function with the idea of
mass thermodynamic and so Kul Prov that the only metric which is invariant on
this sharp convex cone which is invariant under the action of the
automorphism of the cone are given by the a of the log of this function so it
is exactly the fure metric information geometry okay and suo is exactly the

00:50
structure we have not changed the structures we have a new potential but
now the lapas transform is no longer on a sharp convex cone but m is a stic
manifold on U here is a the moment map invented by suo so it's a map from the
stic manifold to the coant orbit is in the DU algebra and you have know theion
transform but know the two variable Q will be the e
will be an element of the Dual of the Le algebra and beta which is a PL
temperature classically is an element of the Le algebra okay so here all is
geometric uh Q is an element of the algebra the temperature the pl isbra and
we will see that the entropy will be a casmier function of thetic
C uh and I say is very successful geometric description and I disc with

00:51
which is a biograph and we have concluded no this is just an extension
of analysis so this tool is the extension of the for transform group
this is a work of last part of the last century they have extended for transform
on Le group and if you could could extend for transform on Le group or laas
transform on Le group you will be able to Define that which is also purely
analytic okay okay so in Soo he introduced this metric so the first time
I read this metric it was very difficult uh well if you have if you are a
geometer this answer this one is classical because it is linked to the
ks2 form so this part is linked to the KK to form but you have an additional
term which is linked to the cycle so so you introduce this tool the

00:52
moment map from the simple manifold to the coant orbit so the Dual of the
lebra and this moment map is a geometrization of the notor theorem okay
not theorem you have a symmetry you have an AV variant moment map the component
of the moment map as the are the invariant of the notor
theorem but in sometimes when you have a default of chology G indexed by the LI
bracket of XY is notal equal to the press on bracket of GX Dy okay and so
you have an additional term and you can make the link to the Theta to Theta
which is a a cycle and also you have the if you
take uh yes and the other relation is this one where the cosy
appear that proves that the coant operator is not equivalent the coant
operator will be equivalent you will not have this term but uh when you have an

00:53
Ain quadrant operator you have an additional term which is called the suo
cycle like that suyon uh so okay Q is indexed by Beta
when you have the action of the group you have the adant operator and the qant
J beta is coant operator JQ plus a cycle where uh no no you have no beta on is Q
is andex by Beta is parameterized by Beta beta is a PL temperature Q is a
itat so you have all the demonstration in my slide but I will not be the detail
now what is important is that the metric introduced you can proves that the

00:54
metric introduced by suo is the asan of the massio potential it proves that it
is the Fisher metric for Le group uh so you have two figures in the
book of Soo which are interesting this one we say you have the lebra you have
the group and you have the Dual of the lebra beta is in the Le algebra so when
the group act you have the co the adant operator and Q is parameterized by Beta
so we have q at Jo Beta And if you take the action of the group on q q is an
element of the Dual algebra you will have the coant operator on Q and when
you have not equality you have the coant orbit okay so this is a figure in the
book of su and you have this fundamental equation of
thermodynamic and the second figure is also very
interesting here you have the lebra so you have beta BL temperature an element

00:55
of the algebra when the group act you have the adant operator Q is in the Dual
of the Libra when the group act you have the coant operator plus the cycle and
here you have two function the mass potential on the entropy the
entropy with a mirle is invariant under the action of the group and Fe is not
you have a term which is a proportional to Beta but it doesn't matter because
the fish matric is the asan of Fe okay with respect to Beta so this term will
disappear and the fure metric will be invariant under the action of the
group so why we will see why entropy is invariant of those action of the group
we will see this is the property of the kazmier function so suo look for this

00:56
kind of group group The Gallo group if you are familiar with mechanic so you
have the two frame x xpre and x and so here you have
rotation you have boost in speed and you have uh space translation and you have
time translation the idea of Soo you can rewrite this transformation with an
additional Dimension as a matrices this is a theory of group representation and
this matrices is a Le group okay so this is Gallo group at this group as a
default of chology the coant operator is not equivalent for this
group uh and then in his book he tried to find what is the Gibs equilibrium for
the Gallo group it proves that there is no equilibrium action of the full gallon
group on the space of motion of an eated mechanical system is not related to any
G state so his idea is to take a one parameter subgroup of the gallion group

00:57
what is the idea you can parameterize this group by the Le algebra okay
exponential map of toe beta toe is a real value and beta is an element of the
Le algebra and you can parameterize the matrices by
two and then you can parameterize all equilibrium and you will take the the
most simple one the centrifusion it's very important to
understand I understood why the temperature is an element of the algebra
with this example so you have a centrifuge okay what are the equilibrium of the
centrifuse you have two equilibrium you have a first temperature which is a
classical PL temperature or thermal equilibrium but you have another
equilibrium which is the wall the wall is a reservoir of angular momentum and
each time you have a particle uh with a chalk

00:58
on on the wall the wall will transmit angular momentum to the particle and
with viscosity you will have a second equilibrium which is related to the
angular momentum okay so you have a thermal balance thermal equilibrium and
you have the angular momentum equilibrium and so the particle are
rotating in the centrifuge on they homogenize their temperature okay
so when I say to ro B this is a difficult problem and he said no I have
this problem solved in my in my book but Ro Balon how how Balon comput the Gibs
density for a centrifuge he say okay the energy should be constant so it will
give you the first parameter of flange the pl temperature I say okay we have
also the angular momentum should be constant and so I have a second

00:59
parameter of FL Ranch for the second equilibrium and I look for the maximum
entropy density given by thisum you you see you you have to make
some assumption and so on in Soo uh the G density is given by this expression
where here here is a moment map and here is a PL
temperature so you you recover the classical theramic if you take the pl
temperature as beta is a derivative of the entropy
by Q you find DS DQ by T which is bolman Clos entropy and the Gibs density is
equivalant under the action of the group okay I will skip that so I will
conclude with two things so the first thing that the
entropy I could uh so this notation is for Ain equivalent coant operator has

01:00
this property of the entropy that mean in fact the entropy is
a Casmir function of the to the pon bracket so Kazmir was a
PhD student of this B on eron Fest where we have developed this idea
inis okay and so you have us to introduce a good blon bracket so the
classical ton bracket is only this term but when you have a cycle you have to
add this term in person book so you can extend for a default of chology the
first term is given by the structure coefficient which is classical in Le
group and the second one is given by the default of chology on the moment map
okay that detail of the equation yes what is important explain

01:01
also the second principle of thermodynamic because DS will Beal to
theter and the theter is only the Fisher transer and as the Fisher transer is
positive definite you have the principle of thermodynamic second principle of
thermodynamic which is explained by that and you have a new equation which is
lrange equation of it this is a four year equation re written geometrically okay
but with the heat in F you can write the itat equation on the temperature on the
heat here you have the it equation in the model of of
Soo okay so with FR gelas we have also a
studies the stochastic model of this and you recover the famous Point car papers
where you have theer Point car equation which is given here which is exactly

01:02
this one because Q is a derivative of Fe and it is exactly the equation of this
is equation for the problem of [Music] variational okay I skip that so you will
find an example because it's very theorical but you can take a group
so very simple question what is the gion density in the point car unit dis okay
it's a very simple question what is the gion density in point car unit dis Goan
density is a density of Maximum entropy it is given by the model of suo in the
quank unit dis a group act it's su11 so you compute the the Le algebra
you compute the Dual of the Le algebra and the moment map is given by this
expression okay so the stic manifold is here the unit disk of qu car on the
coint orbit generated by the moment map are

01:03
this hyperbolic sheet in green okay this is the example of of the quadrant orbit
I invite you to read this paper of char Melle M interpreting
this moment map is from Zed is an element of the unit Point car dis and
the quadrant orbit is an element of the Dual of lebra because you can decompose
on the D the D of the lebra bases and if you look at the three components you
have this relation which are the hbic sheet of
this and uh and so you can also describe dissipation in this model because the
stic foliation is given by the hyperbolic sheet and in fact the the
transverse structure will give you a dissipative model for the point dis and

01:04
you can also compute the this is a Goan density for the P unit
dis given by the model of Sol to conclude uh so I make reference
to batist Cino for on morison for the metritic model but in the metritic model
they have made the assumption that the entropy is a kmir function of the pon
bracket it is a basic assumption in the soo model It Is by design that the
entropy is given by that okay I have given yes so to conclude you have just
to to remember these structures behind the thermodynamic but also behind the
information Theory and behind the information geometry you have these
structures with two foliation stic foliation which is are the level set of

01:05
enthropy and the transverse reman fation which are the level set of energy
okay the no just to to say that no I will the last slide last slide so I this is
interconnection between flow Theory relation information and foliation and
what is open is to characterize the transverse structures you can you have
different way to introduce the transverse structure of stic foliation
which are given in this references so liberman foliation and ier G
structure structures polar foliation defoliation and foliation based also on
moving frame the problem is not Clos you you
can work on this problem and to find a very

01:06
interesting problem so uh this topic is uh is a we are developing this topic in
two European action so caligola which is the cost act msca action on kalista
which is a cost action with the University of
bologna so we are studying a different inter relation on especially information
geometry and group damic and also uh the application in in machine learning if
you are interested also we have organized in SE last September Workshop
kalista workshop at e dein developing this this model and you have
different video and slide available on the website and also you have the talk
of quto uh which gives some illustration on application of

01:07
informal Network and if you are interested uh we organize in end of
October the S edition of GSI geometric science of information and the deadline
for submitted paper is the second of April thank you for your [Applause]
attention e