Let (M,g) be a closed negatively curved manifold. We introduce a new invariant, the Marked Poincaré determinant (MPD) which associates to each free homotopy class of closed curves in M a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. This invariant can be seen as a weighted version (by a function called the unstable Jacobian) of a well-known invariant of (M,g): the marked length spectrum. We prove a local MPD rigidity result in dimension 3: if g is sufficiently close to a hyperbolic metric g0 and both metrics have the same MPD, then they are homothetic (i.e. isometric up to rescalling).The proof relies on a geometric fact of independent interest, namely, we show the Lichnerowicz Laplacian of g0 is injective on the space of trace-free divergence-free symmetric 2-tensors, which, to our knowledge, is the first result of its kind in negative curvature.

This is joint work with Karen Butt, Alena Erchenko, Thibault Lefeuvre and Amie Wilkinson.

The finite volume method is a discretization method for solving partial differential equations (PDE) where the degrees of freedom approximate the average of the PDE solution over control volumes. In this talk, we will apply this method to the Euler equations, a system of non-linear hyperbolic PDEs governing the dynamics of a compressible, adiabatic and inviscid fluid. Particular attention will be paid to the robustness and stability of the approximation and to ensure, at the discrete level, some fundamental physical principles (e.g., conservation, positivity of some quantities, second law of thermodynamics).

Matter, and more specifically molecules, are constantly moving... However, a mathematical model can explain this movement, which is both ordered and chaotic : the Langevin equation. So let Ω ⊂ R^d be a bounded smooth domain and b : Ω→ R^d be a smooth vector field. We focus on the associated overdamped Langevin equation : \partial_t Xt = b(Xt ) + h^1/2Bt in the low temperature regime h→ 0 and in the case where b admits the decomposition b = −∇ f− ℓ with ∇ f· ℓ= 0 on \partial Ω. To study this equation, we analyse the spectrum of the infinitesimal generator of the dynamics: Lh = −∆ + ∇ f · ∇ + ℓ · ∇ with Neumann boundary conditions. In this case, moving particles will remain trapped inside the domain and more precisely the process remains trapped, for some time, in a certain region of the domain before going to another area. These regions are called metastables and correspond to neighborhoods of minima of f. Finally, thanks to spectral theory and more specifically small eigenvalues of Lh , we can describe the return to equilibrium of this metastable dynamic.

State-Space Models (SSMs) are deterministic or stochastic dynamical systems defined by two processes. The state process, which is not observed directly, models the transformation of the states over time. On another hand, the observation process produces the observables on which model fitting and prediction are based. Ecology frequently uses stochastic SSMs to represent the imperfectly observed dynamics of population sizes or animal movement. However, several simulation-based evaluations of model performance suggest broad identifiability issues in ecological SSMs. Formal SSM identifiability is typically investigated using exhaustive summaries, which are simplified representations of the model. The theory on exhaustive summaries is largely based on continuous-time deterministic modelling and those for ecological SSMs have developed by analogy. While the discreteness of time does not constitute a challenge, finding a good exhaustive summary for a stochastic SSM is more difficult. The strategy adopted so far has been to create exhaustive summaries based on a transfer function of the expectations of the stochastic process. However, this evaluation of identifiability does not allow to take into account the possible dependency between the variance parameters and the process parameters. We show that the output spectral density plays a key role in stochastic SSM identifiability assessment. This allows us to define a new suitable exhaustive summary. Using several ecological examples, we show that usual ecological models are often theoretically identifiable, suggesting that most SSM estimation problems are due to practical rather than theoretical identifiability issues.