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 describe new boundary conditions for $AdS_2$ in Jackiw-Teitelboim gravity. The asymptotic symmetry group is enhanced to $\operatorname{Diff}(S^1) \times C^{\infty}(S^1)$, whose breaking to $SL(2, R) \times U(1)$ controls the near-$AdS_2$ dynamics. The action reduces to a boundary term which is a generalization of the Schwarzian theory. It can be interpreted as the coadjoint action of the warped Virasoro group. We show that this theory is holographically dual to the complex SYK model. We compute the Euclidean path integral and derive its relation to the random matrix ensemble of Saad, Shenker and Stanford. We study the flat space version of this action, and show that the corresponding path integral also gives an ensemble average, but of a much simpler nature. We explore some applications to near-extremal black holes.

In the talk we discuss two Dirichlet problems ("formally" in duality)\begin{align}\label{(CP)}\tag{CP}u\in W_0^{1,q}(\Omega): \; &\operatorname{div}(M(x)\nabla {u})+a(x)\,{u}=-\operatorname{div}({u}\,E(x)) +f(x); \\ \label{(DP)}\tag{DP}\psi\in W_0^{1,q}(\Omega): & - \operatorname{div}(M(x)\nabla \psi)+a(x)\,\psi= E(x)\cdot\nabla \psi +g(x)\end{align}where $\Omega$ is a bounded open set in $\mathbb{R}^N$, $M(x)$ ia bounded elliptic matrix, $f$, $g$ are functions belonging to $L^m(\Omega)$, $m\geq1$, $E\in(L^N(\Omega))^N$, $0\lt \alpha_0\leq a(x)\in L^1(\Omega)$.

In the first part we briefly recall some recent results:

existence and summability properties of weak solutions ($q=2$), if $m\geq\frac{2N}{ N+1}$;

Calderon-Zygmund theory ($q=m^*$, infinite energy solutions), if $1\lt m \lt \frac{2N}{ N+1}$;

uniqueness;

the case $|E|\leq\frac{A}{|x|}$, where $E\not\in(L^N(\Omega))^N$;

the case $E\in(L^2(\Omega))^N$.

Then we show:

a new (simpler) existence proof, thanks to the presence of the zero order term, for \eqref{(CP)};

a straight duality proof for \eqref{(DP)};

continuous dependence of the solutions with respect to the weak convergence of the coefficients;

regularizing effect of dominated coefficients ($|f|\leq Q\,a(x)$ or $|g|\leq Q\,a(x)$, $Q\gt 0$);

"weak" maximum principle: if $f(x)\geq0$ [$g(x)\geq0$] and, of course, not =0 a.e., then $u(x)\geq0$ [$\psi(x)\geq0$] and the set where $u$ [$\psi$] is zero has zero measure (at most).

Work in progress: obstacle problem; nonlinear principal part. Open problem: "strong" maximum principle.

Eliashberg, Kim and Polterovich constructed nontrivial partial orders on contactomorphism groups of certain contact manifolds. After recalling their results, the subject of this talk will be the remnants of these partial orders on the orbits of the coadjoint action on their Lie algebras.

Recent experiments on large chains of Rydberg atoms [1] have demonstrated the possibility of realising one-dimensional, kinetically constrained quantum systems. It was found that such systems exhibit surprising signatures of non-ergodic dynamics, such as robust periodic revivals in global quenches from certain initial states. This weak form of ergodicity breaking has been interpreted as a manifestation of "quantum many-body scars" [2], i.e., the many-body analogue of unstable classical periodic orbits of a single particle in a chaotic stadium billiard. Scarred many-body eigenstates have been shown to exhibit a range of unusual properties which violate the Eigenstate Thermalisation Hypothesis, such as equidistant energy separation, anomalous expectation values of local observables and subthermal entanglement entropy. I will demonstrate that these properties can be understood using a tractable model based on a single particle hopping on the Hilbert space graph, which formally captures the idea that scarred eigenstates form a representation of a large $\operatorname{SU}(2)$ spin that is embedded in a thermalising many-body system. I will show that this picture allows to construct a more general family of scarred models where the fundamental degree of freedom is a quantum clock [3]. These results suggest that scarred many-body bands give rise to a new universality class of constrained quantum dynamics, which opens up opportunities for creating and manipulating novel states with long-lived coherence in systems that are now amenable to experimental study.

Driven by physical and technological applications, during the last five years the study of nonlinear evolution on branched structures (graphs, networks) has undergone a fast development. We review on the main achievements and on the open problems. This is a joint project with several people, among which Simone Dovetta, Enrico Serra, Lorenzo Tentarelli, and Paolo Tilli.