A twin building is a simplicial complex associated to a group G with a twin BN-pair. A complex hyperbolic Kac-Moody (KM) group G is associated with a hyperbolic KM Lie algebra g = g(A), where A is a hyperbolic type Cartan matrix. The invariant symmetric bilinear form (. , .) on the standard Cartan subalgebra h in g has signature (n-1,1) on the split real form of h, providing a Lorentzian geometry. The Cartan-Chevalley involution on g gives a « compact" real form k of g, a real Lie algebra whose complexification is g, whose Cartan subalgebra t also has Lorentzian form (. , .), and there is also a corresponding compact real group K. We are able to embed the twin building for G inside the union of all « lightcones" {x in k | (x,x) <= 0} which is in the union of all K conjugates of t.  This provides a geometrical realization of the twin building of G closely related to the structure of all Cartan subalgebras in k, and sheds light on the geometry of the infinite dimensional groups G and K. This is especially interesting in the case of rank 3 hyperbolic algebras whose Weyl groups are hyperbolic triangle groups, so that the building is a union of copies of the tesselated Poincaré disk with certain boundary lines identified. This is joint work with Lisa Carbone (Rutgers University) and Walter Freyn (TU-Darmstadt).

I will give an overview of a project aimed at understanding the motivic nature of l-adic sheaves. I will survey motivating questions about independence of l and past results of Lafforgue, Drinfeld, and others. I will then discuss how to incorporate correspondences into the theory, recent results, and open questions.

MicroRNAs are recently discovered regulators of gene expression, coordinating many biological processes. They are especially important for developmental processes, including cell differentiation, while their deregulation causes diseases like cancer.
In particular, microRNAs are key regulators of muscle differentiation, a complex process necessary for the formation of muscle fibers. I’ll describe the regulation of skeletal muscle differentiation by the miR-98 – E2F5 molecular pathway.

After overviewing the representation of biological molecules, I introduce the representation of interactions between modules in the proteins with the SO(3) rotation of the peptide unit introduced by Penner et al.. Using protein data in PDB, I will show our representation with the SO(3) rotation is useful in characterizing the structure and its change due to the mutation or the native dynamical change with time. I also discuss some possible potential extensions of our approach to the chromatin structure.

Deninger and Werner developed an analogue for p-adic curves of the classical correspondence of Narasimhan and Seshadri between stable bundles of degree zero and unitary representations of the fundamental group of a complex smooth proper curve. Using parallel transport, they associated functorially to every vector bundle on a p-adic curve whose reduction is strongly semi-stable of degree 0 a p-adic representation of the fundamental group of the curve. They asked several questions: whether their functor is fully faithful and what is its essential image; whether the cohomology of the local systems produced by this functor admits a Hodge-Tate decomposition; and whether their construction is compatible with the p-adic Simpson correspondence developed by Faltings. We will answer these questions in this talk.

This talk starts with the Bianchi orbifolds, which arise from the action of the Bianchi groups (SL2 matrix groups over imaginary quadratic integers) on hyperbolic 3-space (their associated symmetric space). Studying these orbifolds, one can obtain :
– dimensions of spaces of Bianchi modular forms;
– Algebraic K-theory of rings of imaginary quadratic integers;
– Group (co)homology and equivariant K-homology of the Bianchi groups;
– Chen-Ruan orbifold cohomology of the Bianchi orbifolds. The latter cohomology ring is conjectured to be isomorphic to the cohomology ring of a crepant resolution for the orbifold, which makes it interesting for string theory.
What makes Bianchi modular forms interesting, is that there are deep number-theoretical reasons for expecting that the Taniyama-Shimura correspondence can extend to the cuspidal Bianchi modular forms, attaching Abelian varieties to them. For the dimension calculations, use is made of the speaker’s constructive answer to a question of Jean-Pierre Serre on the Borel-Serre compactification of the Bianchi orbifolds, which had been open for 40 years. This has so far permitted heavy machine calculations joint with M. Haluk Sengun, locating several of the very rare instances of Bianchi modular forms in a large array of dicriminants and weights, such that these forms are not lifts of classical modular forms.
Complementary to this, studies of the torsion part of the cohomology of the Bianchi groups have given rise to a new technique (called Torsion Subcomplex Reduction; some procedures of the technique had beforehand been used as ad hoc tricks by Soulé, Mislin and Henn) for computing the Farrell-Tate cohomology of discrete groups acting on suitable cell complexes. This technique has not only already yielded general formulae for the cohomology of the tetrahedral Coxeter groups as well as, above the virtual cohomological dimension, of the Bianchi groups (and at odd torsion, more generally of SL2 groups over arbitrary number fields, in joint work with Matthias Wendt), it also very recently has allowed Wendt to refine the Quillen conjecture.
This talk will further discuss the adaptation of this technique to Bredon homology computations, yielding the equivariant K-homology of the Bianchi groups. The Baum-Connes conjecture, which has been proved for large classes of groups, constructs an (iso)morphism from the equivariant K-homology of a group to the K-theory of its reduced C*-algebra. For the Bianchi groups, this yields the isomorphism type of the mentioned operator K-theory, which would be very hard to compute directly.

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.