157 problems

Let $(X,H)$ be a smooth polarized surface with $H_1(X,\mathbb{Z})=0$ and $h^{2,0}(X)>0$. Let $r$ be prime and let $c_1\in H^2(X,\mathbb{Z})$ be algebraic. Let…

Let $\mathcal M_k^0$ be the reduced moduli space of $\operatorname{SU}(2)$ monopoles of magnetic charge $k$ on $\mathbb R^3$, and let $\widetilde{\mathcal M}_k^0$ be its universal…

Let $G$ be a gauge group and let ${}^LG$ denote its Langlands dual group. The associated Yang–Mills theories are compared at the quantum level. Montonen–Olive conjecture. The Yang–…

Mirror-duality conjecture. The diagram

Let $K$ and $T$ be centrally symmetric convex bodies in $\mathbb{R}^n$, and let $K^\circ$ and $T^\circ$ denote their polar bodies. Write $N(K,T)$ for the minimal number of translat…

In an ${\mathcal N}=1$ supersymmetric quiver gauge theory, suppose that different gauge theories have the same infrared moduli space, and that this moduli space is toric. Toric dua…

Mizuno's duality conjecture. The dual quadruple $(A^*,B^*,C^*,D^*)$ should also be modular. The supplied status evidence reports that Wang discovered counterexamples to this conjec…

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…

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

Duality conjecture.

Type IIB supergravity has an $SL(2,\mathbb R)$ duality group, with the axio-dilaton transforming under this symmetry. Type IIB S-duality conjecture. The subgroup $SL(2,\mathbb Z)$…

The Koornwinder–Macdonald polynomials have five parameters; the self-duality condition used earlier restricts these to a four-parameter subfamily, while the case $n=1$ already exhi…

Let $X$ be a variety in the generality considered in the source, with etale motivic cohomology with compact supports and etale motivic Borel–Moore homology related by the natural p…

Let $\gamma$ be a path sampled from the relevant SLE process, and define its trunk by the set of points where the driving function and force-point process agree. Let $\beta$ be the…

Let $K$ and $T$ be symmetric convex bodies in $\mathbb{R}^n$, and let $\gamma_p(K,T)$ denote Talagrand's functional associated with the metric induced by $T$. The preceding theorem…

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…

Correspondence conjecture. There is a one-to-one correspondence between finite quantum groups and three-dimensional topological field theories satisfying the squeezing property, wi…

Consider the five basic superstring theories and the web of $S$, $T$, and $U$ dualities relating certain pairs of them. M-theory unification conjecture. The five superstring theori…

U-duality conjecture. The discrete subgroup $\mathcal{E}_{d(d)}(\mathbb{Z})$ lifts to a symmetry of the full string theory.

The theory is formulated on the phase space $\mathbb{R}^{2n}$, with Euclidean metric $G$ and a regulated generating functional whose duality invariance has been established perturb…

Let $\mathcal C$ and $\mathcal C'$ be codes in $\mathbb F(\!(z)\!)^n$, and let $\bar{\Lambda}(\mathcal C)$ denote the generalized adjacency matrix of $\mathcal C$. Dual invariance…

Let $\mathcal C$ and $\mathcal C'$ be convolutional codes with generalized adjacency matrices $\bar{\Lambda}(\mathcal C)$ and $\bar{\Lambda}(\mathcal C')$. MacWilliams duality conj…

For a group code $\mathbb{{\mathcal{C}}}$ with symbol groups $G_k$ and restrictions $\mathbb{{\mathcal{C}}}_{|\{k\}}$ at each time $k$, equipped with the appropriate topologies, Du…