26 problems

Let $Y$ be the object under consideration, let $p$ be a point, and let $TC_pY$ denote its tangent cone at $p$. Write $0_p$ for the zero element in this tangent cone, and let…

Let $Y$ be an exact special Lagrangian submanifold in $\mathbf C^{n+1}$ with tangent cone $C\times\mathbf R$ at $0$. Assume that $C$ is stable, meaning that the only linear and qua…

Let $(X^4,\mathsf{d})$ be a noncollapsed Ricci limit space, and consider the set of points whose tangent-cone cross-sections are not homeomorphic to $S^3$. Discrete singularity con…

Ricci-flat tangent cone conjecture. If one tangent cone at infinity is Ricci flat, then $M$ is Ricci flat. Consequently, the tangent cone at infinity should be unique; analogous qu…

Consider the horospherical tangent cones at infinity of symmetric spaces, with restricted root systems whose factors determine the relevant types. Regularity conjecture. The horosp…

Splitting conjecture. Every tangent cone at infinity $M_\infty$ splits isometrically as

Let $H = \langle a_1, a_2, \ldots, a_n \rangle$ be a numerical semigroup, and let $I_H^*$ be the defining ideal of the tangent cone of the semigroup ring $K[H]$. Denote by … the nu…

Let $(X_j,g_j,p_j)$ be a sequence of Einstein metrics converging with non-collapsing volume to a singular Einstein metric $(X_\infty,g_\infty,p_\infty)$, and let…

Let $R={\mathbb C}[{\bf z}^{(v)}]$, let $I_{v,w}$ be the Kazhdan–Lusztig ideal defining the local chart of $X_w$ at $E^{(v)}_\bullet$, and let $\overline I_{v,w}$ be its ideal of l…

Let $X\subset\mathbb{C}^n$ be a pure $d$-dimensional entire complex analytic subset. Suppose that there exists a sequence $T=\{t_j\}_{j\in\mathbb{N}}$ of real positive numbers such…

Unique-expansion conjecture. The elements of $\operatorname{Ap}(\Gamma_m,a)$ have unique expressions. Hence the tangent cone of $k[[\Gamma_m]]$ is Cohen–Macaulay. This would extend…

Let $\mathbf{C}^3$ carry a Calabi–Yau metric whose tangent cone at infinity is $\mathbf{C} \times A_2$. Identify metrics that differ by biholomorphism and scaling. Székelyhidi's pa…

Tangent-cone uniqueness conjecture. There is a unique tangent cone to $T$ at $p$.

Let $A$ be an admissible Hermitian–Yang–Mills connection on $(\mathcal{E},B)$, and let $\hat{\mathcal{E}}$ be any chosen optimal extension of $\mathcal{E}$ at $0$. An analytic tang…

Let $B$ be a ball in $\mathbb C^n$ and let $\mathcal E$ be a reflexive sheaf over $B$ with an isolated singularity at $0$. For an admissible Hermitian–Yang–Mills connection $A$ on…

Disk-decomposition uniqueness conjecture. Under these hypotheses, $\Sigma$ has a unique tangent cone at infinity.

Partial-transposition conjecture. If

Intrinsicness conjecture. The filtration

A minimizing hypercone is a codimension-one cone that is area-minimizing, equivalently mass-minimizing. The paper excludes trivial examples in low dimensions. Strict area-minimalit…

Let $H=\langle a_1,\dots,a_r\rangle$ be a numerical semigroup, and let $\widetilde H$ be its interval completion, the numerical semigroup generated by all integers in the interval…

Let $H$ be a numerical semigroup, let $I_H^*$ denote the defining ideal of its tangent cone, let $\mu(H)$ be the number of minimal generators of $H$, and let $\width(H)$ be the dif…

Homeomorphism-type conjecture.

Nonuniqueness-locus conjecture.

Let $M^n$ be a Ricci-flat manifold, and let $\alpha$ be the exponent appearing in Corollary. Quadratic curvature blow-up conjecture. $\alpha$ in Corollary can be chosen to be $1$. A…

Let $X_u^v=X_u\cap X^v$ be a Richardson variety in $GL_n/B$, let $wB\in X_u^v$, and let the tangent cone at $wB$ be defined by the associated graded local ring of $X_u^v$ at that p…