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.

I will give a proof of an integrality of p-adic multiple zeta values. I would also like to explain how it can be applied to give an upper bound of the dimension of finite multiple zeta values.

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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)

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.

The substrate for heredity, DNA, is chemically rather inert. However, it bears one of the elements of information that specify the form of the organism. How can a form be specified, starting from DNA ? Recent observations indicate that the dynamics of transcription — the process that decodes the hereditary information — can imprint forms of a certain topological class onto DNA. This topology allows both to optimize transcription and to facilitate the concerted change of the transcriptional status in response to environmental modifications. To the best of our knowledge, this morphogenetic event is first on the path from DNA to organism. It inspires new rules to optimize networks of transcriptional interactions 'à la manière de' synthetic biology. The latter new interdisciplinary domain will be briefly presented.

We investigate two aspects of large random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees. Starting with a given tree (called the seed), a random sequence of trees is built by adding vertices one by one, connecting them to one of the existing vertices chosen randomly with probability proportional to its degree. Bubeck, Mossel and Racz conjectured that the law of the trees obtained after adding a large number of vertices still carries information about the seed from which the process started. We confirm this conjecture using an observable based on the number of ways of embedding a given (small) tree in a large tree obtained by preferential attachment. Next we study scaling limits of such trees. Since the degrees of vertices of a large preferential attachment tree are much higher than its diameter, a simple scaling limit would lead to a non locally compact space that fails to capture the structure of the object. Yet, for a planar version of the model, a much more convenient limit may be defined via its loop tree. The limit is a new object called the Brownian tree, obtained from the CRT by a series of quotients.

In this talk, I will present a rigorous probabilistic construction of Liouville Field Theory on the Riemann sphere with positive cosmological constant, as considered by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings". Then, I will explain some of the fundamental properties of the theory like conformal covariance under PSL$_2(C)$-action, Seiberg bounds, KPZ scaling laws, the KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula. If time permits, I will also explain the construction in the disk. This is based on joint works (some on arxiv and others in progress) with F. David, Y. Huang, A. Kupiainen, R. Rhodes.

Séminaire Laurent Schwartz — EDP et applications