Seminars, for informal dissemination of research results, exploratory work by research teams, outreach activities, etc., constitute the simplest form of meetings at a Mathematics research centre.
CAMGSD has recorded the calendar of its seminars for a long time, this page serving both as a means of public announcement of forthcoming activities but also as a historic record.
We are going to present recent results and ongoing research concerning the Ising chain (i.e., the Ising model in dimension 1) with homogeneous (large) spin-spin interaction, but subjected to a random external field, the latter being sampled from an i.i.d sequence. The framework is that of disordered systems. We will first present the pure model (homogeneous external field) which is exactly solvable. Then we will turn to the disordered model, showing that for this model the disorder is strongly relevant, through estimates on the free energy and a description of the typical configurations. There are two distinct cases with qualitatively different behaviours: the cases of centered or uncentered disorder. We will also present a continous analogue of the model, obtained by a weak disorder limit, which has the advantage of allowing some explicit computations. Our approach confirms and deepens claims made in the physics literature by D. Fisher and collaborators, based on a study of the Glauber dynamics of the model using the renormalisation group method.
A classical result by McDuff shows that the space of symplectic ball embeddings into many simple symplectic four-manifolds is connected. In this talk, on the other hand, we show that the space of symplectic bi-disk embeddings often has infinitely many connected components, even for simple target spaces like the complex projective plane, or the symplectic ball. This extends earlier results by Gutt-Usher and Dimitroglou-Rizell. The proof uses almost toric fibrations and exotic Lagrangian tori. Furthermore, we will discuss natural quantitative questions arising in this context. This talk is based on joint work in progress with Grigory Mikhalkin and Felix Schlenk.
In this talk, I will present some recent results for general non-local branching particle process or general non-local superprocess, in both cases, with and without immigration. Under the assumption that the mean semigroup has a Perron-Frobenious type behaviour, for the immigrated mass, as well as the existence of second moments, we consider necessary and sufficient conditions that ensure limiting distributional stability. More precisely, our first main contribution pertains to proving the asymptotic Kolmogorov survival probability and Yaglom limit for critical non-local branching particle systems and superprocesses under a second moment assumption on the offspring distribution. Our results improve on existing literature by removing the requirement of bounded offspring in the particle setting and to include non-local branching mechanisms. Our second main contribution pertains to the stability of both critical and sub-critical non-local branching particle systems and superprocesses with immigration. At criticality, we show that the scaled process converges to a Gamma distribution under a necessary and sufficient integral test. At subcriticality we show stability of the process, also subject to an integral test. In these cases, our results complement classical results for (continuous-time) Galton-Watson processes with immigration and continuous-state branching processes with immigration.
