33 problems

Let $M$ be a compact manifold of dimension $2$, and let the $L^2$-metric be the right-invariant metric on the group of volume-preserving diffeomorphisms induced by the $L^2$-length…

Sato's conjecture. Any soliton equation, or completely integrable system, is obtained in this way. Moreover, classification of soliton equations should be reduced to classification…

Let $\mathcal{H}$ be the space of Kähler metrics, and let $K_M$, $K_G$, and $K_C$ denote the sectional curvatures of the $L^2$ metric, the gradient metric, and the Calabi metric, r…

Let $\operatorname{Diff}(S^1)$ denote the group of diffeomorphisms of the circle $S^1$, equipped with its group structure. A Poisson-Lie structure is a Poisson structure for which…

Physicists' conjecture. Fix a connection $\nabla_A\in\mathcal{C}$. Then for every $\nabla_{A'}\in\mathcal{C}$, there exists a unique $g\in\mathcal{G}$ such that…

Let $M$ be an infinite-dimensional isoparametric submanifold in a Hilbert space, or one of its parallel manifolds, and let the setting and hypotheses of Theorem 1 apply, including…

Existence conjecture. There exist infinite-dimensional Kähler manifolds satisfying this condition and different from the projective spaces.

Let $K$ be an infinite-dimensional convex body with $V_1<\infty$, and define … The sequence $m_k$ is decreasing. Extremal decay conjecture. Either … or … The Brownian-motion body…

Let $M$ be a complex curve equipped with two families of cuts satisfying the conditions in the preceding conjecture. For each integer $n\geq0$, let the associated set of $C^n$-poin…

Let $\mathcal{O}$ be a $\operatorname{Diff}_0(M)$-orbit in the explicit shape space $\mathcal{X}$, let $\mathcal{F}_{\mathcal{O}}\coloneqq \Pi^{-1}\mathcal{O}$ be the entire fiber…

Conjecture on ultra-log-concave intrinsic-volume sequences. The sequence $(a_k)_{k\geq 0}$ is the intrinsic-volume sequence of some infinite-dimensional GB-set if and only if

Gao–Vitale conjecture. Either

Let $\Delta$ be the $\ell^2$-simplex equipped with the Fisher–Rao metric, and let the gradient flow lines of $F_{(c_n)}$ be given by the integral curves in Proposition. In finite-d…

Geodesic uniqueness conjecture. Given two spacelike Cauchy hypersurfaces $S_0,S_1\in\mathfrak{C}(M,g)$, there is a unique geodesic path $c:[0,1]\to\mathfrak{C}(M,g)$ such that…

Infinite-dimensional extension conjecture. The results in Sections 4 and 5 should also hold when the ambient manifold is infinite-dimensional.

Let $K$ be a field, let $n$ be a positive integer, and let $G$ act on the infinite affine space ${\mathbb A}^n_{K,\infty}$. Write $\textnormal{KQ}({\mathbb A}^n_{K,\infty}/G)$ for…

Let $X$ be a Banach space, and let $(M_n)_{\mathbb{N}'}$ and $(P_n)_{\mathbb{N}'}$ be sequences of subspaces of $X$, with $(P_n)_{\mathbb{N}'}$ finite-dimensional. For $\lambda<1$,…

Hilbert-cube four-point conjecture. The set

Let $\mathcal{G}$ be a proper étale Lie groupoid modelled on an infinite-dimensional space; such a groupoid is called an orbifold groupoid. An orbifold in local charts is a corresp…

Let $G^1$ and $G^2$ be smooth, weak Riemannian metrics on an infinte-dimensional Banach manifold $\mathcal H$. Assume that $G^2$ is uniformly stronger than $G^1$, both metrics admi…

Local 1-affineness conjecture. The prestack $\LocSys_\cG(\oD)$ is 1-affine.

Geometric proof conjecture. The same proof technique should also apply in the smooth, infinite-dimensional category of optimal transport.

Consider the space of parametrized curves with structure group $\mathrm{Diff}(S^1)$, and the connection obtained by projecting the vector field moving a small circle along a large…

Let a $\Psi{\rm DO}^*_0$-bundle be a bundle whose structure group is the group of bounded invertible classical pseudodifferential operators of nonpositive order with bounded invers…

Conjecture 1.3. There is a submanifold $\mathfrak{J}_{c}\subset\mathfrak{J}_{\xi}$ passing through $J$ and a neighborhood $\mathfrak{U}$ of $J$ in $\mathfrak{J}_{\xi}$ such that…