Institut de recherche mathématique avancée

Résumé : Consider closed holomorphic curves in symplectic $G$-manifolds, with respect to a $G$-invariant almost complex structure. We should not expect the moduli space of such curves to be a manifold (after all, transversality and symmetry are famously incompatible). However, we can hope for a clean intersection condition: the moduli space decomposes into countably many disjoint strata which are smooth manifolds, whose dimensions are explicitly computable.

I present this decomposition for simple curves, and indicate how to extend this to multiple covers. These are the first steps towards a well-behaved theory of equivariant holomorphic curves. This has applications to the 3-body problem and real Gromov-Witten theory.

Résumé :  Any closed, flat Riemannian manifold is finitely covered by the torus, by Bieberbach's classical theorem. Similar classifications have been pursued for closed, Riemannian conformally flat manifolds, as well as for closed, flat Lorentzian manifolds. I will present the classification of closed, Lorentzian conformally flat manifolds with unipotent holonomy. This is joint work with Rachel Lee.

Résumé : It is well known that cluster structures and Poisson structures in the algebra of regular functions on a quasi-affine variety are closely related. In this talk, I will discuss this connection for Poisson structures on a simple simply connected complex Lie group G defined by a pair of classical R-matrices. The key element of the construction is a rational Poisson map from the group with a bracket defined by a pair of suitably chosen standard R-matrices to the same group with an arbitrary pair of R-matrices. In the case of G=SL_n one can build explicitly the corresponding cluster structure and prove its regularity and completeness.

Résumé : In this talk, we discuss the construction cluster structures on the symplectic groupoid of upper-triangular structures and related structures on the Teichmuller spaces. This is a joint project with L. Chekhov.

Résumé : After providing a short overview of the interdisciplinary research we are carrying on in the field of Structural Music Information Research (SMIR), we focus on two very active axes: the application of mathematical morphology to music analysis and the algebraic combinatorics of a special family of rhythmic structures. Although mathematical morphology was originally developed for image processing, it has been successfully applied to pattern discovery in the case of symbolic music representations (i.e. representations such as MIDI that are not based on the audio signal). We will present some recent approaches aiming to extend the classical morphological operators (erosion, dilation and opening) to the case of pattern discovery up to variations. These constructions can also be applied to the study of musical meters and rhythms, together with some other approaches that are more algebraically oriented. We will focus in particular to the family of so-called "perfectly balanced rhythms", i.e. rhythms that are obtained as sums or differences of regular rhythms seen as regular polygons within a cyclic group of a given order. Surprisingly, the classification of these algebraic structures remains an open problem in mathematics.

Moreno Andreatta holds diplomas in mathematics from the University of Pavia, piano performance from the Novara Conservatory and computational musicology from the EHESS in Paris. Senior researcher in maths & music at CNRS/IRMA he is also associate researcher at IRCAM. He is at the present the coordinator of the SMIR (Structural Music Information Research) and LaMaMu (LaboMathéMusique) projects at IRMA.

Paul Lascabettes is a postdoctoral researcher at IRMA at the University of Strasbourg, working on the discovery of musical patterns and the generation of musical rhythms. He holds a PhD in mathematics applied to music, obtained at IRCAM and Sorbonne University under the supervision of Isabelle Bloch and Elaine Chew.