Pour tout renseignement complémentaire, veuillez contacter les
organisateurs, Hakim
Boumaza, Mathieu
Lewin
ou Stéphane
Nonnenmacher.
| 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. |