35 problems

Let $G$ be a rank-one compact Lie group with maximal torus $T$, let $\mathfrak{M}$ be the moduli stack of $G$-bundles, and let $\mathfrak{M}_T$ be the corresponding moduli stack of…

Let $R$ be a ring spectrum and let $E$ be a commutative localized quotient of $R$. For a finite cell $R$-module $M$, let $\hat{\mathrm{L}}^R_E M$ denote the completed $E$-localizat…

Nonzero-filter localizability conjecture. The filter $mid 0$ is localizable over $Q$.

Restricted-product conjecture. The ring of quasi-endomorphisms satisfies

The DT–PT virtual structure sheaf localization conjecture. There exists a canonical halving $\mathbb{E}_{1/2}$, given explicitly in the source's Definition, such that

Thread-set conjecture. Let ${\mathbb{A}}$ and ${\mathbb{B}}$ be tuples of subsets of $\operatorname{Spc}({\mathcal{T}}^c)$. If they have the same thread sets, then there is a canon…

Quenched localization conjecture. There exists an environment-measurable process $(\Gamma_t)_{t\geq 0}$ with $\Gamma_t\subseteq\omega$ and

Scalar-fluctuation upper-bound conjecture. If $\varepsilon>0$ is small, $|\lambda|<\varepsilon$, $M\geq C_\varepsilon$, and $1<k<N$, then

Let $cvarphi^{\rm loc}$ be the local function on the patch $comega$, let $cgamma_{\partial_n}$ denote the normal-derivative trace on $cpartial\omega$, and let $cell$ be the localiz…

Pseudo-localization conjecture. The respective cokernels $\mathcal{C}_{\bullet}$ and $\mathcal{C}'_{\bullet}$ of these two restriction morphisms are acyclic.

Let $d\in\{1,2,3\}$. For each $n\ge 1$, let $\varphi_{n,L}$ be the corresponding eigenfunction, let $U_{n,L}\in[-L/2,L/2]^d$ be a point where $|\varphi_{n,L}|$ reaches its global m…

Let $0 \le i \le n+1$. For $i \le n$, let $\mathcal D_i$ be the full subcategory of dualizable objects $X$ in $\Sp_{k,n}^{\dual}$ that are retracts of $Y \wedge Z$ for some dualiza…

Let $L$ be a right exact localization on the category of groups, and let $G$ be a nilpotent group. A homomorphism $f:H\to G$ is a Nikolov–Segal map if … Here $[G,G]$ is the commuta…

Let $L$ be a localization on the category of groups, let $G$ be a group, and let $G\to LG$ be the localization map. The cokernel of this map is the quotient of $LG$ by the normal c…

Let $L$ be a localization on the category of groups, and let $G$ be a finite $p$-group, meaning a finite group whose order is a power of the prime $p$. Write $G\to LG$ for the loca…

Let $L$ be a localization on the category of groups, meaning a functor equipped with a natural transformation from the identity satisfying the idempotence conditions for a localiza…

Localization Dichotomy Conjecture. The following statements are equivalent: $P$ admits a generalized Wannier basis that is exponentially localized; $P$ admits a generalized Wannier…

Let $d>1$, let $u_{\text{\tiny\rm gs}}$ be the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP), and let $V_\varepsilon$ be a disorder potential satisfying assumptio…

Let $d>1$, and let $u_{\text{\tiny\rm gs}}$ denote the ground state of the Gross–Pitaevskii eigenvalue problem (GPEVP). Higher-dimensional ground-state bound. The ground state sati…

Let $R$ be a ring spectrum, let $v\in \pi_*R$ be a homogeneous element, and consider the localization map … Here $R[v^{-1}]$ denotes the localization obtained by inverting $v$, and…

Let the Hamiltonian be convex and let its associated Lagrangian have superlinear growth. A generalized directed polymer is the polymer process associated with the corresponding ran…

Let $A$ be a random banded matrix with independent entries and bandwidth $b$, and let $l$ denote its eigenvector localisation length. Localization-length conjecture. The eigenvecto…

Let $\Theta_{g,k}$ denote the FJRW invariants for the quintic indexed by genus $g$ and $k$, and let the relations … . Effective determination conjecture. The equations … , effectiv…

Let $\epsilon_{H,S,T,V}$ be the Rubin–Stark element and let $\epsilon_{H,S,T,V}^{\mathrm{an}}$ be the analytic Rubin–Stark element defined in the inverse-limit localized module. Le…

Let $\Sigma$ be a marked surface, let $\Delta$ be a triangulation, and let $\mathcal{A}_\Sigma$ be the associated algebra. The natural homomorphism … comes from the inclusion of…