6 problems

Man'e's conjecture. For a generic Lagrangian $L$ on a closed manifold $M$, there exist a dense open set $U_0$ of $H^1(M,\mathbb{R})$ such that for every $\omega\in U_0$,…

Let $W$ be a $C^1$ Lagrangian torus invariant under the geodesic flow of a Riemannian metric on $\mathbb{T}^n$, containing $n$ orbits whose canonical projections have well-defined,…

Let $H:T^*\mathbb{T}^n\to\mathbb{R}$ be a Tonelli Hamiltonian, and consider Lagrangian invariant tori at supercritical energy levels. A Hedlund Lagrangian torus is one for which th…

Made's weak conjecture. For a generic Lagrangian $L$, there is an open dense subset $U$ of $H^1(M,\mathbb R)$ such that, for every $c\in U$, there is a unique $(L,c)$-minimizing me…

Made's strong conjecture. For a generic Lagrangian, there is a unique minimizing measure, and this measure is supported on a periodic orbit or a fixed point of the Euler–Lagrange f…

Let $M$ be a manifold and let $L$ be a Tonelli Lagrangian on $TM$. For a cohomology class $c\in H^1(M,\mathbb{R})$, write $\mathcal{M}(L,c)$ for the Mather set associated with $L$…