10 problems

Let $f$ be a nonlinearity in the smooth asymptotic class of the dimensional-bound theorem, let $T_f$ denote its blow-up level, and define … For a corresponding stable solution…

Necessary-dimension conjecture. A necessary condition for the existence of such solutions is

Let $X\in\text{Alex}^n(-1)$. For $0\le k\le n-1$, define the $k$-singularity function by setting $\theta_k(x)=2\pi$ for $x\in\mathcal{S}^{k-1}(X)$ and…

Let $Y$ be a 3-manifold, let $p=(g,B)\in\mathcal P$ be a parameter pair with smooth metric $g$, and let $(\mathcal Z,\ell,\Phi)$ be an element of the moduli space of $\mathbb Z_2$-…

Let $(X,\mathsf{d},o)$ be the pointed Gromov–Hausdorff limit of a sequence of closed Riemannian manifolds $\{(M_\alpha^n,\mathsf{d}_{g_\alpha},o_\alpha)\}_\alpha$ satisfying the un…

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, let $(X,p)\in\operatorname{Alex}^n(-1)$, and set … Here $\mathcal{S}^k_{\epsilon_i}$ denotes the $\epsilon_i$-singular strat…

Let $n\in\mathds{N}$, let $k$ be a nonnegative integer, and let $(X,p)\in\operatorname{Alex}^n(-1)$, where $\mathcal{S}^k_\epsilon$ denotes the $\epsilon$-singular stratum. Hausdor…

Let $X^n$ be a noncollapsed limit space with Ricci curvature bounded below, and let $S$ denote its singular set. A point is regular if at least one tangent cone is Euclidean; the t…

Large-singular-set conjecture. If a minimizer has a large singular set, for example of Hausdorff dimension one, then a Lavrentiev gap must occur between the infimum over absolutely…

Let $M$ be a three-dimensional manifold with metric $g$, and let $\Sigma$ be the zero locus of a $\mathbb{Z}/2$-harmonic spinor, so that $\Sigma$ is a closed set of Hausdorff dimen…