13 problems

Let $n$ be the dimension of a spacelike hypersurface and let $\eta>0$ measure the loss from harmonic decay in the Carlotto–Schoen construction, which currently provides control in…

Alexandrov gluing conjecture. The gluing produces an Alexandrov space $Y$ if and only if the gluing is by isometry and, for every $p\in Y$, $T_p(Y)$ is a metric cone over…

Let $X_0$ and $X_1$ be noncollapsed $RCD(K,n)$ spaces with nonempty boundaries $\partial X_0$ and $\partial X_1$, where the boundaries are understood in the sense of Gigli and Pasq…

Closure conjecture. The set $\mathcal A$ is closed under gluing: if $A,B\in\mathcal A$ admit a gluing, then

Largeness conjecture. Every gluing

Let $(\hat\gamma,\hat k)$ denote boosted subextremal Kerr black hole initial data with parameters $\hat{\mathfrak{m}}$ (mass) and $\hat{\mathfrak{a}}$ (specific angular momentum).…

Let $X\in\operatorname{Alex}_{\amalg}^n(\kappa)$ satisfy the paper's condition, let $Y$ be a connected length metric space, and let $f\colon X\to Y$ be a 1-Lipschitz onto map. Glui…

RCD gluing conjecture. The glued metric measure space

Let $(S,M,A)$ be a graded marked surface with a full arc system $A$. Say that it has enough marked intervals when … and, for each $k$, the inclusion $\iota_k:(D^2_k,M_k)\to(S,M)$ i…

Let $(S,M)$ be a graded marked surface with an arc system $A$ such that the inclusions of disks $(D_i,\Lambda_{m_i})\hookrightarrow (S,A)$ in an arc decomposition … determine monom…

Let $\pi_a:M\to B_a$ be two Lagrangian fibrations defining real polarizations $P_1$ and $P_2$, and let $U_{P_2,P_1}:H_{P_1}^{1/2}\to H_{P_2}^{1/2}$ be the formal integral operator…

Let $G$ be a complex reductive group and let $X$ be a smooth complex projective curve. Let $\mathcal Shv_{\mathcal N}(Bun_G(X))$ be the Betti category of nilpotent sheaves. Automor…

Let $M_1$ and $M_2$ be manifolds whose states $psi_{M_1}$ and $psi_{M_2}$ are glued along a common boundary submanifold $\Sigma$ to form a state $\psi_M$. Let $\Omega_1$ and…