13 problems

Main Conjecture. The subset of points $x$ in $J(n,m,k)$ such that the map $\Psi_k^0$ is non-transversal to the stratum containing the point $\Psi_k^0(x)\in Mat^0(n,m,k)$ has codime…

Metric-line classification conjecture. The metric lines in $\mathcal{J}^k(\mathbb{R},\mathbb{R}^n)$ are precisely the line, homoclinic, and direct-type geodesics.

Let $X$ and $Y$ be smooth manifolds, let $J^k(X,Y)$ be the $k$-jet space of smooth maps from $X$ to $Y$, and let $cSigma^{r_1,\ldots,r_k}(f)$ denote the successive rank-drop strata…

Let $J^k(\mathbb{R},\mathbb{R})$ be the space of $k$-jets of real functions of one real variable, equipped with its sub-Riemannian distance. A geodesic…

Let $X$ be a quasi-compact complex manifold, let $\mathcal{F}$ be a codimension-$1$ foliation on $X$, let $\mathfrak{V}(\mathcal{F})$ denote the set of hypersurfaces that are smoot…

Finite generation conjecture. The ring of invariant polynomials

Let $\pi:\mathbb{H}\rightarrow S$ be the uniformizing maps of a Shimura variety $S$ corresponding to a semisimple group $G$. Let $k\geq 2$ and let…

Fiber-structure conjecture. For $n\gg0$, the fiber of $p_n$ over the image of $\gamma$ is homeomorphic to a quotient of

Wild jet-moduli conjecture. The moduli spaces $G\text{-}\mathrm{Cov}(D)$ and $J_n^GV$ exist and have the finiteness properties required to define the motivic measure.

Wild moduli-space conjecture. The sets $G\text{-}\mathrm{Cov}(D)$, $J_n^{G,E}V$, and $J_n^GV$ for $0\le n<\infty$ are realized as $k$-schemes admitting stratifications with at most…

Let $J^k(E,n)$ be the $k$-jet space of $n$-dimensional submanifolds of a manifold $E$, equipped with its Cartan distribution $\mathcal C^k$. A solution of a PDE…

Let $\mathcal{X}$ be a Deligne–Mumford stack of finite type over $D$, and let $\mathcal{J}_{\infty}\mathcal{X}$ be its moduli space of twisted arcs. For $n\in\mathbb{Z}_{\ge0}$, a…

Let $\mathcal P^k(n)$ denote the $k$-level Monster tower over $\mathbb C^n$. For a point $p$, write $\operatorname{Germ}(p)$ for the germs of curves representing $p$. The jet ident…