In 1913, Niels Bohr wrote his groundbreaking paper "On the Constitution of Atoms and Molecules", were he already mentioned quantum jumps between energy levels. Later on, he was also the leader of the Copenhagen interpretation of measurement. A century later, thanks to major progresses in fast electronics and low temperature physics, the delicate manipulation of simple quantum systems, and the observation of quantum jumps and quantum trajectories, have become a reality. These observations teach us important things about measurement, with deep theoretical implications, but also with practical stakes for the conception of the still elusive quantum computers. After a brief overview, we shall focus on a few recent experiments dedicated to simple quantum systems and on their theoretical interpretation which involves some remarkable probabilistic results and structures.

I shall  introduce Bohmian Mechanics and present some basic notions for the analysis of the theory. Among them the notion of typicality, which is basic  for establishing Born’s statistical law in a Bohmian universe. The talk ends with a view on relativistic quantum physics seen from a Bohmian perspective.

The physics of many-body systems strongly depends on their dimensionality. With the realization of quantum wells for example, it has been possible to produce two-dimensional gases of electrons, which exhibit properties that dramatically differ from the standard three-dimensional case, some of them still lacking a full understanding.
 
During the last decade, a novel environment has been developed for the study of low-dimensional phenomena. It consists of cold atomic gases that are confined in tailor-made electromagnetic traps. The talk will discuss some experimental aspects of this research, including dynamical features like the emergence of coherence in the gas when it is rapidly cooled across the superfluid transition.

L'effet Hall fractionnaire est un des phénomènes les plus surprenants de la physique de la matière condensée. Il se manifeste via les propriétés de transport de gaz d'électrons bi-dimensionnels soumis à des champs magnétiques intenses. Cette physique pourrait également être reproduite dans des expériences avec des atomes froids, par exemple dans un gaz de bosons en rotation rapide.

La fonction d'onde de Laughlin, proposée comme une approximation pour le fondamental de tels systèmes, est à la base de notre compréhension actuelle de ce phénomène, mais certaines de ses propriétés fondamentales sont encore mal comprises d'un point de vue mathématique. Cette fonction d'onde décrit un fluide quantique hautement corrélé et il est en particulier crucial d'élucider l'aspect robuste de ses corrélations.

Dans cet exposé on étudiera un modèle pour la réponse de la fonction de Laughlin aux variations d'un potentiel extérieur. Cela nous conduira à une famille de problèmes variationnels d'un type nouveau. Nos résultats principaux sont des estimations d'énergie rigoureuses indiquant une forte rigidité dans la réponse de l'état  de Laughlin à la variation du potentiel extérieur.

Travail commun avec Jakob Yngvason.

Mixed twistor D-modules are D-modules with mixed twistor structure. The notion of twistor structure is a generalization of that of Hodge structure, introduced by C. Simpson. As in the Hodge case, various operations for D-modules are enhanced to those for mixed twistor D-modules. It implies that some important mathematical objects are naturally equipped with mixed twistor structure. For example, the D-modules associated to families of Laurent polynomials, called the GKZ hypergeometric systems, are naturally equipped with mixed twistor structure. It looks natural to pursuit their roles in the Hodge theoretic aspect of mirror symmetry. In this talk, I am planning to discuss the degeneration of twistor structure associated to the degeneration of Landau-Ginzburg models, and to explain how the isomorphism of Givental induces an isomorphism of mixed TEP-structures in the local mirror symmetry for Fano toric manifolds.

I shall give a quick introduction to the model of repeated quantum interactions and their continuous-time limit. I shall also connect them to quantum trajectories. This approach to open quantum systems makes quantum noises appearing very naturally and it shall be the occasion to present these tools, which are not so commonly shared by the physicist community.

In a celebrated 1961 paper, Landauer formulated a fundamental lower bound on the energy dissipated by computation processes. Since then, there have been many attempts to formalize, generalize and contradict Landauer's analysis. The situation became even worse with the advent of quantum computing. In a recent enlightening article, Reeb and Wolf set the game into the framework of quantum statistical mechanics,  and finally gave a precise mathematical formulation of Landauer's bound. I will discuss parts of this analysis and present some extensions of it that were obtained in a joint work with V. Jaksic.

Concepts, familiar from pure states in quantum mechanics, such as "superposition" and "transition probability", are shown to be also meaningful for generic states in infinite systems, described by funnels of type I_{infty} algebras. In the physically important case of states of Connes-von Neumann type III1, these concepts also have a physically significant operational interpretation in terms of primitive observables. (Joint work with Erling Störmer)

Many cellular and developmental processes are tightly controlled by regulatory logic that is similar across different species. Is this similarity the result of common ancestry or is it due to convergent evolution?
We address this question using in silico modeling of genetic networks controlling organ development in flowers. First, we determine the number of networks implementing a given logic and point to an open mathematical problem. Second, we take a computational approach based on Markov Chain Monte Carlo and sample uniformity that highly constrained space of networks. The production of a large number of samples in that ensemble reveals how structural features of genetic networks are shaped by the imposed regulatory logic. Thus, network structural features are expected to be shared across different species through convergent evolution.