Topology and perception

mathematics and cognitive sciencePierre Baudot

X-SHS - October 27th 2015(issued from collaboration with Daniel Bennequin)

Two traditions:

What if our senses were true and mathematicaly perfect? What if our instantaneous sensations the faculty of the human brain to compensate for the brevty of life?

Pythagoras:

« Everything is number. »

« Hey what! everything is sensible. »

« What you are looking at when you rise your eyes is the

beautiful ». «Mathematics is a faculty of the human mind to

compensate for the brevity of life and the imperfection of

the senses » J.Fourier

A very old and good model already existAristotle in -350 BC…

What is a good model/theory for neuroscience/cognition ?

Mathematic Experience?

« It is, that the same attribute cannot at the same time belong and not belong to the same subject and in the same respect; ...This, then, is the most certain of all principles, since it answers to the definition given above. For it is impossible for any one to believe the same thing to be and not to be » .Aristotle.

⇒Experience is mathematic

Mathematic Experience?

NON-CONTRADICTION: A and nonA* => false (never happens)

Ising/neural models of associative memory

Hopfield Boltzmann networksComplex energy landscape

“It is certain that all bodies whatsoever, though they have no sense, yet they have perception … and whether a body be alterant or altered, evermore a perception precedeth operation; for else all bodies would be alike to one another” Francis Bacon 1967:

68-69 (in Monod, Wyman, Changeux, on the nature of allosteric transition, 1964

Mathematic vs. Physic vs. Cognition-Biology ?

Penrose :

cognition 5 physic 5 mathematic

math

biophycomplement of physic in mathematic = metaphysic

complement of cognition in physic = in-animated matter

cognition 6 physic 6 mathematicmathbio phy

Physic is about observable only (Heisenberg-Einstein)Math is a subset of physical and cognitive activity.

Descartes: “I think therefore I am”

Rimbaud: “It is false to say I think. I is someone else.”… “sorry for the wood that considers himself a violin”

phy

bio

math

The problem of cognition formalismis displaced to the problem of mathematical formalism

Adaptation-learningSymmetry and invariance

1854 Boole: probabilty,

non-contradiction, algebra,

Idempotence

Set theory: a cognitive theory?

Cantor: « A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought – which are called elements of the set. »

Whitehead and Russel: Principia Mathematica … intuitionist logic,

Fredge, analytic theory of mind

Hilbert Program and Gödel theorems, arithmetisation of logic, ZFC.« My theory of demonstration only simulate the internal activity of our understanding and record the proceedings of the rules that govern the functioning of our thoughts.» Hilbert, 1930.

Turing and Von Neumann: computabilty, Turing machine (Artificial Intelligence)“the computable numbers are those whose decimals are calculable by finite means ... For the present I shall only say that the justification lies in the fact that the human memory is necessarily limited” Turing 1937

Bourbaki: skeptikal on this question, even divided.

( Ω , * )

ObjectsSet

InteractionComposition law Function Ex: f(X,Y)=X+Y

Here, algebra of Random VariableObjects: Random Variables

Composition law: Join

Algebra

A theory

Group and Permutation: theory of AmbiguityGaloisGalois

Algebra (group theory)

Galois: Résolution of 2nd order equation: ax²+bx+c=0 ⇒Algebraic resolution: roots are given by rational function:⇒ if ∆<0 no real roots <=> irreducible in R. The Extension by the quantity i (i²=-1) makes the polynomial reducible in Complex numbers⇒When you’re faced to an apparently unsolvable situation extend your rationality and extend the real field to the imaginary field. intersection: unit circle x²+y²-1=0 and line Ax+By+C=0 => ax²+bx+c=0∆<0 A²+B²<C², ∆=0 A²+B²=C², ∆>0 A²+B²>C²

⇒ Permutation (Algebraic view of a Symmetry): « The Goup of an equation is the permutation Group of its roots ». Ex of a permutation (1,2,3) and (1,3,2)More exactly substitution (« Soit une équation donnée dont a, b, c, … sont les m racines. Il y aura toujours un groupe de permutations des lettres a,b,c,… qui jouira de la proposition suivante :1°. Que toute fonction des racines, invariable par les substitutions de ce groupe, soit rationnellement connue ;2°. Réciproquement, que toute fonction des racines, déterminable rationnellement, soit invariable par les substitutions.)

⇒ Algebraic indistinguishability-ambiguity (Q: why Galois defended Republic?)

⇒Factorisation: «une équation irréductible ne peut avoir aucune racine commune avec une équation rationnelle sans la diviser»: if an irreducible polynomial P has a common root with another polynomial F, then F can be written: F(x)=P(x).Q(x) (ex: P(x)=x²+1 irreducible in R, its roots are (i,-i), and F(x)=x3-x²+x-1, reducible in R, its roots are (1,i,-i), and we have F(x)=P(x).(x-1)

acbwith

ifa

iborif

a

b

4

02

02

2 −=∆

<∆∆±−

>∆∆±−

231

321

Algebra (group theory)⇒ Repeat Factorisation (reduce the group) until only one permutation exist, identity. ⇒ Cayley: any group is a subgroup of S ⇒Correspondence between the structure of numbers (fields) and structure of functions (polynomials)

Lattice of subfields

Polynomial: x3-2 over Q

Lattice of subgroups 3 2=ω3 1=θ

Whole field K

Base field Galois Group G

Trivial Group (identity)

Subfield Subgroup

« the group of the equation caracterise at a time, not what we know about the roots, but the objectivity of what we do not know about them. Inversely, this non-knowledge is no more a negative, an unsufficiency, but a law, a learning to which corresponds a fundamental dimension of the object » Deleuze, 2000 (in André, 2007).

Riemann (1854): Differential Geometry: Manifold (M measure free of prior in n dimension) = multiply extended magnitude +Metrics d(p,q): local distance differential « Inner and relative » geometry

Moebius (1827): Barycentric co-ordinates(relative Universal co-ordinates)

1600-1850: Projective geometry(Desargues, Pascal, Poncelet)

Gauss, Bolyai, Lobatchevski, Poincaré (1860-1900): 5th axiom undecidable, 3 geometries in 2D

Klein : Proj geom = mother geometry, geometry of a space = study of invariance under a group of transformations (transitive action).=> Homogenous Geometry (ex: group of euclidian isometries)

GEOMETRY

"The method ... for laying co-ordinate into the space-time continuum breaks down, and there seems to be no other way which would allow us to adapt systems of co-ordinates to the 4 dimensional universe so that we might expect from their application a particularly simple formulation of the laws of nature.So there's nothing for it but to regard all imaginable systems of co-ordinates, on principle, as equally suitable for the description of nature. The general laws of nature are to be expressed by equations which hold good for all systems of co-ordinates, that is are co-variant with respect to any substitutions whatever (generally covariant).It is clear that a physical theory which satisfies this postulate will also be suitable for the general postulate of relativity." (Einstein, 1916)

nFAspaceAffinexx

atpoxx

≈

∞≈

)1,...,,(

int)0,...,,(

21

21

τσστ

στ dxdxgds ∑=2

Projective invariant: Cross-ratio )).((

)).((

.

.),,,(

yxzt

zxyt

xytz

xztyzyxt

−−

−−==

),,,log(2

zyxtr

Cayley Klein Hilbert Metric

),...,,(),...,,(: 121121 ++ ≈ nn

nxxxxxxFP λλλ

GEOMETRY« Perceptive space is just an image of geometrical space, a deformed image by a kind of perspective, and we can only have a

represention of objects by bending them to the laws of this perspective… Moreover, when we say that we localise an object in a given

point of space, what does it mean? It silmply means that we represent the movement we have to do to reach the object. » Poincaré

(1895)

Koenderink & al (2002)Attempt of proof that perception is projective (see Koenderink): Gibson (1956): «the research of invariant is the fundamental fact of perception»

Eye & head movement

Straton (1896), Kohler (1961) : Inverting goggles=> Reflection invariance

Gibson (1933): curving goggles Diffeomorphism invariance

Galois’s Goggle: Position permutting goggles

Sur & al (2005): Visuo-auditive rewiring

Sur & Rubenstein (2005)

Klein’s cognitive Erlangen program

Adaptation is transformation invariance (compensate a change in evt)

Proposition: brain adaptation can achieve such

invariance (considering

developmental critical period of « adaptation »)

Eye & head movement invariance: translation and rotation (You don’t see your saccade)

Subjectiv space: continuous? Quantified?

« La première impression irraisonnée que nous donnent les phénomènes naturels et la matière est celle de continuité. Devant un morceau de métal ou un volume de liquide, l’idée s’impose à nous qu’ils sont divisibles à l’infini, qu’une portion si petite qu’elle soit aura toujours les mêmes propriétés. » (Hilbert, 1925)

Completion-binding-inference

Blind spot

« Mais partout où l'on a rendu suffisamment précises les méthodes de recherche dans la physique de la matière, on a trouvé des limites à la divisibilité, limites qui ne tiennent pas à l'insuffisance de notre étude, mais à la nature des chose…Or l'énergie elle-même, comme il est maintenant bien établi, ne peut être divisée à l’infini sans restriction: Planck a découvert les quanta d'énergie. Et le résultat est chaque fois qu'un milieu continu et homogène, indéfiniment divisible et réalisant ainsi l'infini de petitesse, ne se rencontre nulle part. La divisibilité à l'infini d'un milieu continu est une opération qui n'est possible que dans la pensée, c'est seulement une idée que contredisent les observations et les expériences de la physique et de la chimie. » (Hilbert, 1925)Et de la psychophysique…

Elementary percepts : sensory adaptation

Related Stevens law: the intensity of sensationis a power law of excitation:

=

0

lnI

Ikp

⇒ universal: applies to any sensory modality, also to complex systems⇒ Threshold and perception is quantified ⇒ Elementary adaptation ⇒ Introduce the notion of gain control, andcontextual modulation

Weber-Fechner law: the intensity of the sensationis proportional to the logarithm of excitation:

αkIp =

I

Ikp

∆=∆

“Quelque soit la raison de la durée des tremblements des tympans externes et internes, l'âme en extrait son logarithme“

(Mengoli, 1659).

Venus appears when it is 1/64 more shinny than surrounding sky

«We are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment. If

that be so, the rough data of experiment, which are our sensations, could be measured. We might, indeed, be tempted to believe that this is so, for in recent times there has been an attempt to measure them, and a law has even been formulated, known as Fechner's law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine the experiments by which the endeavour has been made to establish this law, we

shall be led to a diametrically opposite conclusion. It has, for instance, been observed that a weight A of 10 grammes and a weight B of 11 grammes produced identical sensations, that the weight B could no longer be

distinguished from a weight C of 12 grammes, but that the weight A was readily distinguished from the weight C. Thus the rough results of the experiments may be expressed by the following relations: A=B, B=C, A<C

, which may be regarded as the formula of the physical continuum. But here is an intolerable disagreement with the law of contradiction, and the necessity of banishing this disagreement has compelled us to invent the

mathematical continuum. We are therefore forced to conclude that this notion has been created entirely by the mind, but it is experiment that has provided the opportunity.” Poincaré, Science and Hypothesis, 1902

… and conclude that this principle of adaptation is the principle of Analysis-situs, and introducing “impression manifolds” further pursue in his following book

“ To have the mathematical continuum of n dimensions, it will suffice to take n like scales whose divisions correspond to different values of n independent magnitudes called coordinates. We thus shall have an image of the physical continuum of n dimensions, and this image will be as faithful as it can be after the determination not

to allow the contradiction of which I spoke above.”

Poincaré: From sensory adaptation to topology

Adaptation-learningSymmetry and invariance

Information theory

Measure - Set theory

1854: Riemann Integration

1901: Lebesgue and Borel, measure and additivityIf X and Y are disjoint then m(XUY)=m(X)+ m(Y)

σ-algebra: closed for complement and countable union

1933: Kolmogorov axiomatic of probability

Probability is a normalised measure-Volume

Solovay (1975): with a finite version of AC all sets are measurable. Diaconescu: in a topos AC excluded third

Information , set theory, and measures

Hu Kuo Ting theorem 1962: Information functions with the operation , (join) ; (mutual) and / (conditional) are in bijection with set additive functions.Information function are finite signed measures. Universality of information in Solovay axiomatic (with finite choice)

44444444444444 344444444444444 2143421

U

HRH

n

kjikji

nn

n

kji

n

jiji

ji

RHH

n

i

i

n

i

i SSSSHSSSHSSHSHSHSH

0

100

1,,

121

1

1,11

)...()1(...)()()()()( ∑∑∑≠≠

=−

−

≠=

−

==

∩∩∩−+−∩∩+∩−==

∑=

−=n

k

k

kISH

1

)1()(

∑∑∑≠≠

=−

−

≠===

∪∪∪−+−∪∪+∪−==n

kjikji

nn

n

kji

n

jiji

ji

n

i

i

n

i

i SSSSHSSSHSSHSHSHSI1,,

121

1

1,11

)...()1(...)()()()()( I

∑=

−=n

k

k

kHSI

1

)1()(

)...(...)()( 121 nnkjiji SSSSHSSSHSSH ∩∩∩≥≥∩∩≥∩ −

)...(...)()( 121 nnkjiji SSSSHSSSHSSH ∪∪∪≤≤∪∪≤∪ −

Optimal coding: The principle of efficient coding first proposed by Attneave in 1954, states that the goal of sensory perception is to extract the redundancies and to find the most compressed representation of the data-environement. In his princept paper, Attneave notably claimed that any kind of symmetry and invariance are information redundancies and that Gestalt principles of perception can be defined on information theoretic ground. Information topology, beside its own mathematical and physical motivations, is an attempt to formalize this intuition, and to provide an algebraic topology framework for adaptation and learning.

Shape construction (Gestalt)

(Field, Hayes & Hess, 1992)

Contour integration Association field

Gestalt: Laws of

perceptual binding

Compositionality (Arcimboldo)

The hole is more than the sum of its part

System-environment Adaptation

)/()();( 0

1

ESHRHSHSEI HSS

n

i

i −−=∑=

)/()();( ESHSHSEI −=

Consciousness theory“Et quoi! Tout est sensible” Pythagore E-motion: “what goes with motion”…

"When we think a given thought, then the meaning of this thought is expressed

in the shape of the corresponding neurophysiological process.“ Riemann

structural-complexity (logical depth, complex landscape) topologyMotivation

Atlan, Bennett…

Algebraic topology

What is algebraic topology?

Graphs (complex networks) are 1-chain “ topological“ complex

1-chain complex : algebraic “sum“ of 1-simplex (“edges“) ponderated by coefficients with value in a group, a field (…) alternative and generalization of adjacency and incidence matrix(e.g. 0,1 assimilated to the field with two elements F2)

In physic: Phase-space (or configuration space) of n-body (d=6n,3n)In mathematic: many complex phenomena only starts for d>2 (uniformisation theorem…)

Homology is a story of group…

(alternative to hypergraph)

Universal coefficient theorem: Co-homology accounts for the change of in

the coefficient‘s group

Since the work of René Thom, many works have underlined that algebraic topologyis a pertinent framework to handle and quantify complex systems.

TOPDRIM european Project (Topology driven methods for complex systems) all the developments in persistent homology and topological data analysis,

and the developments of spectral graph theory.

In some peculiar cases for example, the Euler characteristic or the Betti sequence capture quite well some notion of complexity.

I will not review here the pertinence of entropy and information in complex systems modelisation, prediction (…)

It is the core of statistical physic approach of complex systems and you can find it in text book such as Badii and Politi, see also Shalizi, Bialek, Gaspard (…)

What is “information topology”?Encounter of two domains, topology and information theory, that provide a Mathematical foundation and extension of Information theory (entropy is derived uniquely). Currently, in development, several different independent approaches, notably exhibiting some main structures of operad and motivic theory.

• Cathelineau, J. Sur l’homologie de sl2 a coefficients dans l’action adjointe, Math. Scand., 63, 51-86, 1988..

• Kontsevitch, M. The 1+1/2 logarithm. Unpublished note, Reproduced in Elbaz-Vincent & Gangl, 2002, 1995.

• Elbaz-Vincent, P., Gangl, H. On poly(ana)logs I., Compositio Mathematica, 130(2), 161-214. 2002.

•Bloch S.; Esnault, H. An additive version of higher Chow groups, Annales Scientifiques de l’École Normale Supérieure. Volume 36, Issue 3, May–June 2003, Pages 463–477 Bloch S.; Esnault, H. The Additive Dilogarithm, Documenta Mathematica Extra Volume : in Kazuya Kato’s Fiftieth Birthday., 131-155. 2003.

•Marcolli, M. & Thorngren, R. Thermodynamic Semirings, arXiv 10.4171/JNCG/159, Vol. abs/1108.2874, 2011. Marcolli, M. & Tedeschi, R. Entropy algebras and Birkhoff factorization, arXiv, Vol. abs/1108.2874, 2014].

• Baez, J.; Fritz, T. & Leinster, T. A Characterization of Entropy in Terms of Information Loss Entropy, 13, 1945-1957, 2011.

Here I will present another approach based on the geometrical and combinatorial structure of probability and random variable seen as partitions, developed with Daniel Bennequin. It will be underlined why topology provides some pertinent formalism to model and quantify complexity.

•Baudot P., Bennequin D. The homological nature of entropy. Entropy, 17, 1-66.

(and ref. therein, Fresse…)

Information cohomology

Information structures(Geometric and Algebraic expression of Random Variables & Probability)

Random Variables:

Partitions algebra-lattice

and complex

(see also

Fresse, 2003)

Probability

complex

Entropy:

H(X;P)

Information structures(Geometric and Algebraic expression of Random Variables & Probability)

Random Variables:

Partitions algebra-lattice

and complex

(see also Fresse, 2003)

Probability

complex

(finite probability fields)

Entropy:

H(X;P)

Information cohomology

(no convex or assymptotic assumption)

See, Hu, 1962, Yeung 2008, Matus

Hyperbolic volume (Lobachevsky) :

Projective metric Cayley-Klein-Hilbert:

−

−

−

−=∆

21

31

30

202)(

zz

zz

zz

zzLiVol

=

ax

ax

xa

xaLogxxd

'

''

')',(

Zaguier, 2007

=> Persepectives: Motivic integration-measure, Periods (better than computational classes), number theory…

Higher information Group: analog to polylog?

Beilinson et al, 1990

Aomoto polylog: n-forms,Lie cohomology,Hodge-Tate mixed motives

See Vincent-Elbaz and Gangl, 2001, 2015….

n-mutual information, links, synergy

n-mutual information, links, synergy

Information tree cohomology ~operad Related results: Baez & al, 2011, Marcolli & al, 2011

Complex system

What is data?

The origin of data topology

"The assumption that the sum of the three angles [of a triangle] is smaller than 180deg leads to a geometry which is quite different from our (Euclidean) geometry, but which is in itself completely consistent. I have satisfactorily constructed this geometry for myself so that I can solve every problem, except for the determination of one constant, which cannot be ascertained a priori. The larger one chooses this constant, the closer one approximates Euclidean geometry. . . . If non-Euclidean geometry were the true geometry and if this constant were comparable to distances which we can measure on earth or in the heavens, then it could be determined a posteriori. Hence I have sometimes in jest expressed the wish that Euclidean geometry is not true. For then we would have an absolute a priori unit of measurement." Gauss

_ Riemann : Homology and electromagnetism

_ Gauss : Links and electromagnetism

Kircher,

“the world is bound with secret knots” (1631, Natura Magnetism)

_ "Is there any sense - and I mean any physical sense, not metaphysical sense - in which one can speak of absolute data? Is one justified in saying that the coordinate x = p cm where p = 3.1415. . is the familiar transcendental number that determines the ratio of the circumference of a circle to its diameter? As a mathematical tool the concept of a real number represented by a nonterminating decimal fraction is exceptionally important and fruitful. As the measure of a physical quantity it is nonsense. If p is taken to the 20th or the 25th place of decimals, two numbers are obtained which are indistinguishable from each other and the true value of p by any measurement. According to the heuristic principle used by Einstein in the theory of relativity, and by Heisenberg in the quantum theory, concepts which correspond to no conceivable observation should be eliminated from physics. This is possible without difficulty in the present case also. It is only necessary to replace statements like x = p cm by: the probability of distribution of values of x has a sharp maximum at x = p cm; and (if it is desired to be more accurate) to add: of such and such a breadth.'' Born

_ Hilbert “on infinity“ (1925): ''the infinity more than any other notion, needs to be elucidated'‘… “a necessity for the honour of

Human spirit“. I think that Grothendieck would have said it indeed became a necessity for Human practical survival:

Energy, time, space, speed, money, life are finite (in practice and in measure).

_ Kolmogorov (1933): Finite probability field is the generalised theory of probability. _ Solovay (1975): with a finite version of AC all sets are measurable. Diaconescu: in a topos AC excluded third_ Cartesian Product is not associative. For example R3: R*(R*R)<>(R*R)*R

_ Topos: “This is the theme of the topos which is the "bed" where come to marry geometry and algebra, topology, and

arithmetic, mathematical logic and category theory, the world of the continuum and the one of "discontinuous" or

"discrete” structures. It is the largest construction I have designed to handle subtly, with the same language rich geometric

resonances, a common “essence” to some of the most distant situations.” A. Grothendieck, Récoltes et Semailles,

Data : points in ?

(current res. @ INSERM) Data analysis: Uncovering the statistical structure in data

System-environment Adaptation-Learning

Environment

S

Complex Information Flow

The minimal information structure such that theorem1 holds, and hence such that we have a cohomology, has a fundamental interpretation in terms of complex systems self-organisation and cognition. This minimal condition is that the information structure is composed of 2 random variables with a probability space of 4 atomic events. Interpreting those two variables as classically in biology or neuroscience, as the variables associated to a (neural) system and to the environment, this condition simply state that for information to exist as a cohomological invariant it is necessary that the information structure contains at least two entities, called the system and the environment.

Machine Learning (Descartes)

ADN sequence Chromosome 13 Human Genome

...ATTGCTATATATATAGC...

Physic Scales

Space-time

Hippocampus

Axons (ax, green)

Dendrites (de, yellow), Boutons

(bo, purple),

Spines (sp, red),

Glia (gl, gray);

unclassified (white) is mostly

extracellular space.

Chklovskii, Schikorski &

Stevens, Neuron, 2002

Human V1, Preuss & Coleman, Cereb cort, 2002

Electronic

microscopy &

2D

crystalography.

10 A resolution

Steven, Belnap,

2005

Spatial structures over scales

Protein amino-

structure (Poly(A)-binding protein

bound to mRNA)

Protein amino-structure

(Beta-Ketoacyl-Synthase I)

Protein amino-structure (Potassium channel

Doyle, McKinnon & al)

Biochemical reaction (Krebs cycle) Mitchal & al, 2013

Biochemical reaction (mammal), Mitchal & al, 2013

Grill-Spector & Malach, 2004Visula area atlas and macoscopic flow

Hierarchisation and specialisation

Felleman & Van Essen (1991) .

Dynamique temporelle et échelles adaptatives

Engramme?

Spatial and temporal structures and scales

What and where is memory (engram)?

where is adaptation-learning ?

where is decision ?

To understand a complex structure: Follow its development and trajectories (History)

= its structure in space-time

STIMULUS RESPONSE

Molecule (K+ channel) LTP-LTD (BCM), STDP…

Compartment (glut synapse)

I-V curve

Cell (pyr neuron)

X

Network (V1 neuron)

Receptive Field

STA

Individual

Cortical areas

FUNCTION

Seen, not seen

(etc.) Structure

Function(Algebraic Geometry!)

Stability

instability

reproducibility-

signal

Variability-noise

Agregation-

grouping-

Synchro

differentiation-

segmentation-

Threshold-

Transition

Common

Principles

Neuron

Network

STA

Individual

Cortical areas

Molecule (Na+K+ channel)Simmons, Van Steveninck (2005)

Mainen, Sejnowski (1995), Bryant & Segundo (1976)

Van Steveninck (1997), Machens (2004), our results

Compartment (synapse)

Hasson & al (2004,2010)

Reliability Discrete (finite #) States and transition

?Atomic

Hodgkin Huxley (1952)

MacKinnon(2003)

Bi , Poo (1998)

STDP

Fluctuation-Dissipation ( return to equlibrium from inner fluctuation relaxation from external stimulation )

Linear and nonlinearResponse =>impulsion

Failures

Spike

In statistical physic, the renormalisation of a system comes to a coarse-graining of the partition function with resolution-scaling factor λ (or lattice spacing) giving an energy configuration energy ,

with a partition function summing across the Boltzmann weight . This transformation can be iterated as long as the lattice spacing (something like the minimum distance between partitions) remains much smaller than the correlation functions . Considering a spin S(X) at space x and spin S(Y) at space Y , the correlation ξ describes the exponential decay of the two point correlation function <S(X,S(Y)> in the disordered phase

The nth iteration defines effective on the lattice spacing

The recursion relation:

is a renormalisation group transformation, (link between odd and even function and covariant and

contravariant expressions).

Hamiltonian flows. Scaling operators: Let us consider an infinitesimal dilatation which leads from the scale λ to the scale

The variation of the Hamiltonian , takes the form of a differential equation which involves a mapping of the

space of hamiltonians into itself and a real function defined on the space of hamiltonians:

, which is a RG transformation in differential form. Moreover, we look only for markovian flows as a function

of the “time" log λ , that is such that does not depend on λ . A fixed point hamiltonian is then a

solution of the fixed point equation .

.

)(2 SH λ

∑=

−=N

i

TSHeZ

1

/)(λ

λ

ξ

1)()(log−≈

− ∞→− yxyx

ySxS

)(2

SH n λ λn2

[ ])()( 122SHSH nn λλλ −ℜ=

)/1( λλλ d+

λℜ

[ ]λλλ

λλ H

d

dHℜ=

∗H

[ ] 0=ℜ ∗Hλ

λℜ

partitionrenormalisation

Consider eachpartition as a node thencompute the degree of the network distribution with partitions at diffeent scales

Partition length

Total Nb of partition

Song, Havlin, Makse, 2006

Auto-similarityRevealed byrenormalisation

FRACTALS

NON-FRACTALS

WWW, protein interactions, metabolic, genetic, taxonomy, tree of life, protein homology network.

Internet,Social networks

Renormalisation in Complex networks

Renormalisation proceduregoes backward in time

<=> module-cluster genesis

Song, Havlin, Makse, Nature Physics, 2006Evolution temporelle

Ancestralnode

Renormalisation

1present

Renormalisation flow is a reverse heat flow (Polchansky)

Information and Space-time structure?

Irreversibilty- dissipation and relativistic space-time structure_ The light cone is the physical subspace where information transmission occur and entropic cost_ ∆H<>0 Time like relation?_ In minimal and negative => Ik,=0 k<n, (Entangled – Link) independence until order n and ∆H=0 (Coboundary)⇒ No Communication 2-Channel exist => Space like? Entangelment => Space? (EPR solution)⇒ Information topology => General relativity : Differential (non metric) space-time distinction:Space is the coboundary between future and past? (Continuity-Gluing condition between past and future submanifolds)

"Tomorrow, we will have learned to understand and express all of physics in the language of information." Wheeler

Information and Space-time structure

Quantum Case: reinterpretation of Cartan, Dirac and Penrose argument, from spinors to vectors, with probability:

Classical: Ω=[0,1] parameterizes probas for a variable with two values

The analog of probability laws are positive definite hermitian forms. For instance, if a basis is chosen:

More generally, Pauli matrices:

questions?

And thanks D.Bennequin, J.M. Goaillard, F.Chavane, M.Levy, O.Marre, C.Monier (…)

Team Homeostasis UNIS1072, Team Géométrie et dynamique,

ISC-PIF and Max Planck Institut for Mathematic in the sciences

« We are such stuff as dream are made on »

So i whish you very nice dreams …

Thank You,