Institut de recherche mathématique avancée

Résumé : The term „Diatonicity“ stands for a conceptual network around the properties of the diatonic scale and their role for the understanding of modality and tonality. The term is also used in the study of pitch class profiles for the magnitude of the fifth Fourier coefficient. The lecture provides an overview of selected algebraic and computational approaches to this domain of investigation and also presents considerations on their integration. This includes the study of chords, scales and rhythms as first and second-order maximally even sets, the transformational modeling of well-formed and pairwise wellformed modes, and the inspection of the Fourier-qualities of these musical objects. The suitability of the Fourier transformation also provides a challenging link to a recent quantum-theoretical approach to the modeling of tonal attraction

Résumé : Julien Dardennes (Toulouse) Title: The coarse distance from dynamically convex to convex Abstract: Chaidez and Edtmair have recently found the first examples of dynamically convex domains in that are not symplectomorphic to convex domains, answering a long-standing open question. In this talk, we present new examples of such domains without referring to Chaidez-Edtmair’s criterion. We also show that these domains are arbitrarily far from the set of symplectically convex domains in R^4 with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity (joint work with J. Gutt, V. Ramos and J. Zhang). Arnaud Maret (Paris) Title: Complex projective spaces via surface groups representations Abstract: My plan is to explain how complex projective spaces can be identified with components of totally elliptic representations of the fundamental group of a punctured sphere into PSL(2,R). I will explain how this identification realizes the pure mapping class group of the punctured sphere as a subgroup of the group of Hamiltonian diffeomorphisms of the complex projective space. Luya Wang (Stanford) Title: Deformation inequivalent symplectic structures and Donaldson's four-six question Abstract: Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan.

Résumé : En 1981, E. Ghys a montré que tout flot d'Anosov sur un fibré en cercles M au dessus d'une surface S est orbitalement équivalent à un revêtement fini le long des fibres du flot géodésique de S. Dans cet exposé, je montrerais que si le degré du revêtement est impair, tous ces relevés sont orbitalement équivalents les uns aux autres, et si le degré est pair, il y a deux classes de conjugaisons orbitales de tels relevés. Il s'agir un travail en commun avec Sergio Fenley.

Résumé : We can define a Tamarkin category for an open set in a cotangent bundle using the microlocal geometry of sheaves. Some computational evidence indicates that the so-called Chiu-Tamarkin invariant is deeply related to the Hochschild cohomology of the Tamarkin category and symplectic (co)homology. In this talk, we will explain the relation precisely: All of them are isomorphic in a suitable filtered sense. The talk is based on a joint work with Christopher Kuo and Vivek Shende.

Résumé : Résumé : je parlerai de ma nouvelle prépublication arXiv:2311.18594, où j'ai démontré un résultat qui généralise - en utilisant un nouveau point de vue sur les opérades à boucles - le théorème de Loday-Quillen-Tsygan sur l'homologie de l'algèbre de Lie gl(A) et le théorème de stabilité de Fuchs sur l'homologie de l'algèbre de champs de vecteurs sur R^n à coefficients tensoriels.

Résumé : Dans le cas classique, sur la cohomologie d'une variete projective lisse munie d'une classe de cohomologie ample on a une forme bilineaire naturelle donnée par le cup produit avec la classe ample suivi de l'integration. On montre que sur les morceau primitifs de la cohomologie cette forme bilineaire est definie (de signe precis). Dans cet exposé nous montrerons une generalisation de ce theoreme au cas relatif, i.e., d'un morphisme X-->Y.

Le gradué de la filtration perverse est une structure de Hodge (d'apres l'exposé précédent) et nous montrerons qu'une certaine forme bilineaire naturelle associée a un fibre ample en bas et a un fibre relativement ample donne des polarisations. En explicitant ce que cela signifie sur chaque morceau primitif on obtient des analogues des relations de Hodge--Riemann classiques.