13 problems

Unstable periodic geodesics conjecture. The particular geometry of unstable periodic geodesics should play a key role in spectral-projector estimates, especially in determining opt…

Let $(M,g)$ be a compact Riemannian manifold with boundary of dimension $n\geq 3$. Its scattering relation records, for geodesics leaving the boundary, when and where they next mee…

Let $n$ be a positive integer, and consider the moduli space of $n$ BPS monopoles equipped with its $L^2$ metric. Manton's conjecture. The slow motion of $n$ BPS monopoles is well…

Let $X$ be a compact Riemannian manifold, let $H:=\sqrt{-\Delta}$, and let $Q(f)$ be a pseudodifferential operator of order zero with principal symbol $f_P$ on $ST^*X$. Suppose the…

Let $M$ be a locally symmetric space of non-compact type with finite Riemannian volume. For $p\in M$, let $S_p(M)$ denote the unit tangent sphere at $p$, and let $E^+(p,\infty)$ be…

Let $M$ be a $d$-dimensional Riemannian manifold with or without boundary, where $d\geq 2$, and let $W_t(P)$ denote the wave front at time $t$ from a point $P\in M$. The wave front…

Let $2\leq k<n$, and consider a coordinate system on $S^n$ obtained by extending a coordinate system from $S^k$. Consider the integrable system originating from the geodesic flow o…

Let $G$ be a metabelian Carnot group, let $A\triangleleft G$ be an abelian subgroup with $[G,G]\subseteq A$, and let $V_1$ denote the first layer of $\operatorname{Lie}(G)$. Assume…

Let ${\,\overline{\!A\,}\,}$ be a multi-parameter with $A_k\in[P_k,Q_k]$. Let $F_{\,\overline{\!A\,}\,}$ be the associated map, let $\Omega_{\,\overline{\!A\,}\,}$ be its conjectur…

Polar self-focal point conjecture. The universal bounds can only be achieved at polar self-focal points.

Let $(M,g)$ be a pinched negatively curved manifold. Let $(\mu_n)_n$ be a sequence of invariant probability measures converging vaguely to $\mu$, and let $F\in C_b(T^1M)$ be a unif…

Veech's conjecture. The upper-bound constant equals the entropy:

Let $M$ be an $n$-dimensional Riemannian manifold, and let $\gamma$ be a closed stable geodesic. Write $\lambda$ for the eigenvalue parameter of an eigenfunction. Toth–Zelditch's s…