SUJETS DE THESES

(ou de Master, en version Òsurvol problŽmatiqueÓ)

 

Giuseppe Longo

 

Centre Cavaills, RŽpublique des Savoirs,

CNRS, Collge de France et Ecole Normale SupŽrieure, Paris,

and Department of Integrative Physiology and Pathobiology,

Tufts University School of Medicine, Boston

http://www.di.ens.fr/users/longo

firstname.lastname (at) ens.fr

 

Chacun de ces thmes propose diffŽrentes thses possibles, ˆ caractre plus ou moins mathŽmatisŽ. 

 

1 – LÕalŽatoire et le temps : algorythmique vs. physique.

In classical physical systems (and by this we mean also relativistic ones) randomness may be defined as Ôdeterministic unpredictabilityÕ. That is, since PoincarŽÕs results and his invention of the geometry of dynamical systems, deterministic systems include various forms of chaotic ones, from weak (mixing) systems to highly sensitive ones to border conditions. Randomness can then be viewed as a property of trajectories within these systems, namely as unpredictability in finite time. Moreover, ergodicity (a la Birkhoff) provides a relevant way to define randomness asymptotically, that is for infinite trajectories, still in deterministic systems but independently of finite time predicatbility.

Also recursion theory gave us a proper form of asymptotic randomness, for infinite sequences, in terms of Martin-Lšf randomness. This has been extensively developped by Chaitin, Calude, Schnorr and many others.

A third form of randomness should be mentioned: the randomness intrinsic to quantum theories. This randomness is intrinsic to quantum measure and indetermination, two principial issues in quantum mechanics, as, according to the standard interpretation, it cannot be viewed as a form of (hidden or incomplete) determination. Technically, it differs from classical randomness in view of Bell inequalities and their role in its probability measure.

It may be shown that these three forms of randomness differ in finite space and time. Yet, by hinting to some recent results obtained in the team of the author and by T. Paul, we will see that they merge, asymptotically. This poses several open questions as for the correlations in finite time of classical, quantum and algorithmic radomness, an issue extensively studied by many, where these asymptotic analyses may shed some further light.

A major question is then to be asked, namely whether and how finite time randomness and time irreversibility are related, in the various physical contexts.

 

 

2 - AlŽatoire biologique entre mathŽmatique, biologie et philosophie.

A common characteristic in the various forms of physical randomness is the predetermination of the spaces of possibilities: random results or trajectories are given among already known possible ones (the six sides of a dice, the spin-up/spin-down of a quanton...). In fact, in quantum physics, even in cases where new particles may be created, sufficiently ÓlargeÓ spaces are provided up- stream (the Fock spaces of which Hilbert spaces): FockÕs spaces capture all the possible states, infinitely many in general. The classical methods transfer successfully in molecular analysis in Biology, where only physical processes are observed, even though there are meant to happen within cells.

In System Biology, however, phase or reference spaces (that is, the spaces of possible evolutions) are far from being predetermined. Typically, the proper biological observables of Darwinian Evolution, namely phenotypes and species, are not pre-given or there is no way to give them in advance within a space of all possible evolutions, in a sound theory. And, of course, there is no way to pre-give the possible molecular interactions (internal and systemic) as well as the feedbacks, from the forthcoming ecosystems onto molecular cascades. An analysis of Species Evolution, in terms of a diffusion equation (thus of underlying random paths) is given in a paper below. The attention to the problem of randomness in System Biology is stimulated and may be partly clarified by these analogies and differences w. r. to Physics.

Finally, the issue may be raised concerning the kind of randomness we may need of in biology, where complex interactions between different levels of organization, in phylogenesis in particular, seems to give even stronger forms of unpredictability than the ones analyzed by physical or algorithmic theories.

 

 

3 – GŽnŽrique vs. spŽcifique, biologie vs. physico-mathŽmatique.

In physics, within a given phase space, the set of Ò conceivableÓ trajectories is generic, while the effective trajectory, defined by the geodesic principle, is specific (critical, stable or unstable, meaning minimal for the Lagrangian action, or, for example, in the particular case of optics, minimal for the optical path). In other words, effective physical phenomenality is specific and is enframed within a pertinent phase space (a great part of the physicistÕs job is actually to characterize this space and its metrics). It is doubtlessly this which confers to physical theory a great mathematical force as well as a possibility, by means of abstraction, to characterize the physical objects using very general properties, despite the singularity of each specific experience of which the conditions are not always exactly reproducible: the ÒtrajectoryÓ in the broadest sense (that is whatever is the intended phase space) of any object will be specific and its analysis is related to the phase space (abstract and general).

In contrast, it appears that for biology, the cells of an organism, the organisms of a species, the species of an environment are concerned by Ògeneric trajectoriesÓ : the ÒpossibleÓ ones that are and remain compatible with the ecisystem. It would be the falling back upon the specificity of this generic which would cause it to lose its biological character each time a physical reduction is attempted.  In short these dualities beween physics and biology may be summarized as follows :

 

 

         PHYSICS                              BIOLOGY

 

variation

(Gaussian)

variability

(individuation process)

Specific trajectories (geodetics)

and generic objects

 

  generic trajectories (possibles/compatibles with ecosystem) and specific objects

 (Schršdinger)

energy is an operator (Hf),

time is a parameter f(t,x)

    Energy is a parameter (allometry),

time is an operator

(measured by

entropy and anti-entropy production)

  pointwise criticality

extended criticality

reversible time

(or irreversible for degradation-simplified thermodynamics)

double irreversibility of time

(thermodynamics and complexity constitution )

    random is non deterministic (QM) or

deterministic non predictability (CM)

within a space phase

randomess is intrinsic indetermination of the phase space changes  (phylogenesis and ontogenesis)

 

 

Figure 1. Theorical differentiation between theories of the inert and of the living state of matter, described through conceptual dualities.

 

 

 

These conceptual (and mathematical ?) dualities raise several questions, from the issue of their mathematization, in general and in specific cases (Evolution, in particular), to the epistemological issue, namely their role in the foundation of these different forms of knowledge and/or their relation to the intended (existing) mathematization.

 

 

References, see the papers in:

3 -   Theoretical  Biology

4 -   Interfaces Computability, Physics and Biology

in http://www.di.ens.fr/users/longo/download.html 

 

and/or :

G. Longo, M. MontŽvil, Perspectives on Organisms: Biological Time, Symmetries and Singularities, Springer, 2014.      
       - Foreword by D. Noble and Introduction (chapter 1).pdf 

 

F. Bailly et G. Longo, MathŽmatiques et sciences de la nature. La singularitŽ physique du vivant.   Hermann, Paris, 2006.