SUJETS DE THESES
(ou de Master, en version
Òsurvol problŽmatiqueÓ)
Centre Cavaills,
RŽpublique des Savoirs,
CNRS, Collge 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 thmes propose diffŽrentes thses possibles, ˆ caractre
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 MartinLšf 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 spinup/spindown 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 pregiven 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 pregive 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. physicomathŽmatique.
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 antientropy production) 

pointwise criticality 
extended criticality 

reversible time (or
irreversible for degradationsimplified 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.