| 14h - 15h |
Denis Périce (Constructor University Bremen) |
Mean-field limit of the Bose-Hubbard model in high dimension
Abstract: The Bose-Hubbard Hamiltonian effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott-insulator transition. In the physics literature, the mean field theory for this model is known to provide qualitatively accurate results in three or more dimensions. In this talk, I will present results that establishes the validity of the mean-field approximation for bosonic quantum systems in high dimensions. Unlike the standard many-body mean-field limit, the high-dimensional mean-field theory exhibits a phase transition and remains compatible with strongly interacting particles. |
| 15h15 - 16h15 | Matthias Täufer (Valenciennes) |
Recent developments in shape optimization on quantum graphs: infinite graphs, heat content, torsional rigidity
Abstract: In this talk, I discuss several recent developments in geometric analysis of metric graphs. Metric graphs are networks of intervals, joined at their endpoints to form a network-like space and self-adjoint differential operators on them are usually called quantum graphs. Shape optimization,that is the question which metric graphs will maximize or minimize particular quantities such as the n-th eigenvalue, has a long history and in the last decades, many analogues of classical results such as the Faber-Krahn inequality have been proved. I will give a brief overview of this history and then pass to more recent results. In particular, I will introduce infinite metric graphs (of finite diameter) where the spectral theory becomes much richer and also discuss questions of shape optimization for quantities such as the torsional rigidity and the heat content which have recently attracted interest. Based on joint work with Partizio Bifulco, Marco Düfel, James Kennedy, Delio Mugnolo, Sedef Özcan and Marvin Plümer. |
| 14h - 15h | Rayan Fahs (Rennes) |
Local decay and asymptotic profile for the damped wave equation in the asymptotically Euclidean setting
Abstract: We consider the question of energy decay for the (possibly damped) wave equation in the asymptotically Euclidean setting. In even dimensions, we go beyond the optimal decay by providing the large time asymptotic profile, given by a solution of the free wave equation. In odd dimensions, we improve the best known estimates, and in particular, we go beyond the decay rate which is optimal in even dimensions. The analysis mainly relies on a comparison of the corresponding resolvent with the resolvent of the free problem for low frequencies. Moreover, all the results hold for the damped wave equation with a short-range absorption index. This is a joint work with Julien Royer. |
| 15h15 - 16h15 |
Jérémy Faupin (Univ. de Lorraine) |
Propagation estimates for the boson star equation
Abstract: In this talk we will study the long-time behavior of the boson star equation, a semi-relativistic Klein-Gordon equation with a Hartree-type non-linearity describing the dynamics of a large number of gravitating bosons. After recalling results giving the existence and uniqueness of solutions under general conditions on the interaction potential, we will discuss the maximal velocity of these solutions and, in some regime, their scattering and minimal velocity. This is joint work with Sébastien Breteaux and Viviana Grasselli. |