66 problems

Let $G$ be the ambient reductive group, let $P$ be a parabolic subgroup of $G$ containing $B$, with Lie algebra $\mathfrak{p}$ and opposite parabolic $\overline{P}$. Let $M$ be a…

Let $G$ be a gauge group with gauge coupling $\tau$, and let $G^\vee$ be a group whose Lie algebra satisfies $\operatorname{Lie}(G^\vee)=\mathfrak{g}=\operatorname{Lie}(G)$. Let…

Dubédat's boundary duality conjecture. The right boundary of $SLE^\sigma(\kappa,\kappa-4)$ is an $SLE(\kappa',(\kappa'-4)/2)$. The conjecture is a precise proposed realization of S…

Let $d>2$, let $E_{d+1(d+1)}(\mathbb{Z})$ be the U-duality group, and let $\mathcal{E}_{\mathrm{string};3/2}$, $\mathcal{E}_{\mathrm{particle};d/2-1}$, and…

Let type II string theory be compactified on $T^d$, equivalently M-theory on $T^{d+1}$, and let $E_{d+1(d+1)}(\mathbb{Z})$ be the U-duality group. Denote by…

Entropy conjecture. The exact location of the multicritical point is given by

Multicritical-point conjecture. The exact location of the multicritical point is determined by $x_0=x_0^{*}$ with $K=K_p$, equivalently

Let $\mathfrak{F}^{(a,b)}$ be the infinite binary biperiodic Fibonacci word with parameters $a,b\geqslant 1$. There is an explicit, effectively computable map $\theta$ on infinite…

Cartier--Nishi duality conjecture. The Cartier--Nishi theorem holds for $\Upsilon$.

Double-duality and exact-sequence conjecture. For every triangular $\mathbf{t}$-module $\Upsilon$,

Let $\tau$ be a preorder of size $n$, let $P_\tau$ be its lattice-point poset, and let $\overline M_{P_\tau}$ be the transmuted $M$-triangle. Transmuted M-triangle duality conjectu…

Let $\tau$ be a preorder and let $\tau^\ast$ denote its dual preorder. Preorder h-vector duality conjecture. One has … for every preorder $\tau$. The statement substantially genera…

Finite q-duality conjecture. For every word $w=u_{1}\cdots u_{k}\in\mathfrak{h}^{0}$,

Let $n,r,s$ be non-negative integers, and let ${\mathcal C}^{\psi}_{{\mathcal N}=2}(n,r|s)$ denote the corresponding N=2 coset vertex algebra. Procházka–Rapčák duality conjecture.…

Precluster–support duality conjecture. For conservative mass-action networks that do not have two species with exactly the same rates and have at least one positive steady state, t…

Let $X$ be a perfectoid boundary carrier equipped with a condensed sheaf/module $\mathcal{F}$ of sources and operators, and let $X^{\flat}$ be its tilt. Assume the holographic gene…

Let $X$ be a smooth, proper, geometrically connected surface over a finite field of characteristic $p$. Write … for its Brauer group, let $\operatorname{Br}(X)_{\nd}$ be the quotie…

Ito's conjecture. There is an isomorphism of vertex superalgebras

Let $Y$ be a smooth partially proper rigid analytic variety over $K$ of dimension $d$, let $\partial Y$ denote its boundary, and let…

Let $K$ be a finite extension of ${\bf Q}_p$, let $Y$ be a smooth partially proper rigid analytic variety over $K$ of dimension $d$, and let…

Let $q\geq 2$ with $3\nmid q$, and define the generalized vertex operator algebras … … Duality means that each simple vertex (super)algebra can be equipped with a module structure…

Sphere-spectrum duality claim. The result established for coherent, eventually coconnective commutative dg algebras over $\C$ also holds over $\mathbb{S}$: the restricted duality f…

Let $Q$ be the quotient of the Witt vector ring stack $W$ by its formal subgroup $\hat W$, let $T_F(Q)$ denote the associated Tate module, and let ${}^{s}W$ be the extension of $W$…

Let $\pi\colon X\rightarrow\operatorname{Spec} k$ be a smooth complete surface over a perfect field $k$ of characteristic $p\neq0$, and let $R\pi_*\mu_{p^n}$ be the derived flat-co…

Let $\pi\colon X\rightarrow\operatorname{Spec} k$ be a smooth complete surface over a perfect field $k$ of characteristic $p\neq0$. Let $H_{\mathrm{fl}}^i(X,\mu_p)$ denote flat coh…