SUJETS DE THESES
(ou de Master, en version Òsurvol problŽmatiqueÓ)
CNRS,
DŽpt. Informatique – ENS,
et
CREA, Polytechnique, Paris
http://www.di.ens.fr/users/longo
name (at) di.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 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 :
F. Bailly et G. Longo, MathŽmatiques
et sciences de la nature. La singularitŽ physique du vivant.
Hermann, Paris, 2006.