Institut de recherche mathématique avancée

Résumé : TBA

Résumé : We present two UV-complete formulations of Euclidean JT gravity on finite-size geometries of fixed topology. One is based on discretization, reducing the problem, on the disk topology, to counting self-overlapping polygons. The other is a direct approach in the continuum, which reveals a deep and perhaps surprising relationship between JT quantum gravity and Liouville quantum gravity techniques. I explain the path integral, probabilistic, formulation of the problem, and also write down the associated JT CFT. This is a conformal field theory that plays the same role for JT gravity as Liouville CFT does for Liouville gravity. The three versions of JT, in negative, zero, or positive curvature, can be discussed in parallel, the three models having the same UV structure. A fundamental conjecture is that the the continuum description is equivalent to the scaling limit of the discrete model, paralleling the expected equivalence between triangulation-based and continuum formulations of Liouville quantum gravity. Another conjecture is that negative curvature JT on finite size geometries converges to the Schwarzian field theory in an appropriate large size limit. The Schwarzian theory is the leading term in a long wavelength, hydrodynamic-like approximation to the UV complete theory, in which a notion of smooth boundary emerges. Similarly, the usual canonical quantization à la Wheeler-DeWitt of JT gravity should be seen as a long-distance effective description of the full quantum gravity model. If established, the emergence of the Schwarzian theory at long distance provides a concrete mechanism for the emergence of time.

Résumé : Many complex projective surfaces admit elliptic fibrations over a curve. When such a fibration admits a section, the classical Weierstrass model provides a well-understood birational description inside a projective bundle over the base curve. However, in the absence of a section, finding analogous models becomes more subtle. In this talk, I will present an analogue of the Weierstrass model for elliptic surfaces equipped with a line bundle of relative degree two. Under suitable assumptions, this construction gives a determinantal model inside a weighted projective bundle. In particular, I will focus on the case where the base curve is the projective line and explore some relevant applications of our construction in this setting. The results that I am presenting are part of joint work with S. Rollenske and A. Zanardini.

Résumé : Dans cet exposé, on va discuter de ce que sont les processus ponctuels, en particulier les processus de renouvellement, et on verra le cas particulier des processus de Poisson. L'étude de ces processus nous permettra de construire une méthode statistique afin de déterminer la présence ou non d'une connexion entre deux neurones, à partir de l'observation de leur spike (ou potentiel d'action).

Résumé : TBD