425 problems

Let $\mathbb{R}\langle \xi\rangle$ be the real closed field of algebraic Puiseux series, let $\mathcal{P}=\{P_1,\ldots,P_s\}\subset \mathbb{R}[X_1,\ldots,X_n]$, and set … For a bou…

Setting. For a Navier-Stokes velocity field with vorticity $\omega$, direction $\xi = \omega/|\omega|$, and peak set $P(t)$ (the high-vorticity region), define the scale-invariant…

Let $X$ be a $d$-dimensional algebraic variety with at worst Gorenstein canonical singularities. The bounded denominator conjecture asserts that the denominator of the rational num…

Let $X$ be a variety, let $i$ be a nonnegative integer, and write $\underline{\Omega}^{i}_{X}$ for the $i$-th graded piece of the Du Bois complex of $X$. For a sheaf or complex, le…

Fix $n\in\mathbb{N}$, $\varepsilon>0$, and let $I\subseteq [0,1]$ be a finite set. For a klt singularity $x\in (X,\Delta)$, let $\operatorname{mld}^{\mathrm{K}}(x,X,\Delta)$ denote…

Optimal index conjecture. The index $m$ of $X$ is the largest possible index among all terminal Calabi-Yau varieties of dimension $n$, and also among all canonical Calabi-Yau varie…

Let $A\in\mathbb Z^{n+2}$ be a finite set of full dimension containing $0\in\mathbb Z^{n+2}$ and satisfying Assumption. For generic polynomials $f_1,\ldots,f_{n+1}\in\mathbb C^{A}$…

Bifurcation-locus conjecture. The bifurcation locus of the periodgon of this family is precisely $\Sigma$.

Shokurov's codimension bound conjecture. The following inequality holds:

Let $(X,B)$ be a log variety, and let $a(x)=a(x;X,B)$ denote the minimal log discrepancy at a closed point $x\in X$. Lower semicontinuity conjecture. For every closed point…

Let $\tau$ be a collection of singularity types, let $X_\tau$ be the classifying space for $\tau$-cobordisms, let $\mathcal K_\tau$ be the Kazarian space, let $\mathcal K^+_\tau$ b…

Let $a\geq0$ and $d$ be a nonnegative integer. The sets $\mathcal S_d\text{(global)}$, $\mathcal S_d^0\text{(global)}$, $\mathcal S_{a,d}\text{(local)}$,…

Let $R$ be an analytic Cohen–Macaulay local ${\mathcal C}$-algebra of countable Cohen–Macaulay type. The Schreyer conjecture. $R$ has at most a one-dimensional singular locus, name…

Let $A$ be the coefficient ring in the reduction-modulo-$p$ setup, let $\frak{a}\subset A[x_1,\ldots,x_n]$ be an ideal, let $c$ be its log canonical threshold, and let $c_p$ denote…

Let $X$ be a variety of dimension $n$ and let $Y\subset X$ be a subvariety. The log canonical threshold of the pair $(X,Y)$ is denoted by $\operatorname{lc}(X,Y)$. Shokurov's conje…

Let $k$ be a field of characteristic zero, let $X$ be a variety of dimension $d$, and let $J\subseteq X$ be the Jacobian subscheme. Denote by $J^{(d-1)}$ its $(d-1)$-th neighborhoo…

Finite-order translation conjecture. The characteristic variety is a union of translated subtori, with each translation given by a point of finite order.

Let $(X,Y)$ be a pair, where $Y$ is a formal rational or real linear combination of closed subvarieties of $X$. For a log resolution $f:X'\xrightarrow{\ \ } X$, write … and define…

Let $F(x,y)$ be a polynomial whose zero set is a plane curve consisting of the product of $d$ real lines. Consider the critical points of $F$ having the same non-zero critical valu…

Let $B$ be a closed geodesic ball in a Riemannian three-manifold, and let $M$ be a minimal surface with boundary $\partial M \subset \partial B$. Suppose that $M$ is properly embedde…

cDV smoothing conjecture. $V$ has a smoothing by a small deformation. In particular,

Quasi-adjunction characteristic-variety conjecture. The Zariski closure of $\exp(\mathcal F)\subset\operatorname{Char}H_1(U,\mathbb Z)$ is a component of the characteristic variety…

Translated-subgroup support conjecture. The support of $\pi_n(\partial B_{\epsilon}-D)$ is a union of translated subgroups of $\operatorname{Char}\pi_1(\partial B_{\epsilon}-D)$ by…

Let $A$ be a one-dimensional local domain essentially of finite type over a field $K$ of characteristic zero, and let $\Omega_{A/K}$ denote its module of Kähler differentials. Berg…

Let $A$ be a local one-dimensional domain with field of fractions $F$. Geller's conjecture. The ring $A$ is regular if and only if the map … is injective. The conjecture gives a cr…