136 problems

Let $H$ be the Anderson model in dimension $d=2$. Two-dimensional Anderson localization conjecture. The spectrum of $H$ is pure point with dynamically localized eigenfunctions. Thi…

Let $p$) be a prime, and let $L_h$ denote chromatic localization at height $h$. The $K(n)$-local sphere is expected to exhibit a precise transchromatic decomposition under the loca…

Let an $N\times N$ random band matrix (RBM) have bandwidth $W$. The eigenvectors may be completely delocalized for large bandwidth and exponentially localized for small bandwidth,…

Let $A$ be a Gaussian random band matrix with bandwidth $W$, normalized so that its eigenvalues $E$ are concentrated in $[-2,2]$ and its asymptotic density of states is expected to…

Consider a stronger-than-linear edge-reinforced random walk whose transition probabilities are proportional to a positive function $f$ of the number of edge crossings, under the hy…

In the strong disorder phase of a directed polymer in a random environment on ${{\mathbb Z}}^d$, trajectories are considered under the polymer measure in an environment that may co…

A generalized Hamilton–Jacobi polymer is a directed polymer model with a random potential whose path weights are governed by a generalized Hamilton–Jacobi structure. In this settin…

Let $W$ denote the band width and let $\ell$ denote the localization length of a two-dimensional random band matrix; let $L$ be the linear size of the system and $W_c$ its critical…

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…

Bremsstrahlung-function conjecture. The Bremsstrahlung function is given by

Let $\mathsf{T}$ be a compactly generated triangulated category. A localization of $\mathsf{T}$ is smashing if its kernel is a strictly localizing subcategory and its corresponding…

The Chalker–Coddington model is a random unitary operator on $l^{2}({\mathbb Z}^{2})$ with real tunneling amplitudes $r$ and $t$ satisfying $r^{2}+t^{2}=1$. The parameter $|t|$ is…

Input and output localization conjecture. The class $m\cdot\overline{\chi}^P(X,{\mathcal F})$ depends only on $(\hat X',{}{\mathcal F}{|\hat X'})$, and it is the class of a project…

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…

Extension and fixed-point formula conjecture. (1) The class $\mu(C)$ extends to a class in

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…

All-energy localization conjecture. As for $d=1$, localization holds at all energies and for arbitrary disorder in dimension $d=2$.

Consider the random Schrödinger evolution … where $x\in\mathbb{R}^3$, $V(x)$ is a random potential with short-scale correlation, and $\lambda$ is a coupling constant. Diffusive del…

Fix a radius $R_0>0$. The localization theorem provides balls $B_1,\dots,B_n$ in $\mathbb{R}^3$, each of radius $R_0$, and a normalized vector $\Psi$ in the physical Hilbert space…

All-orders trivalent-vertex conjecture. The two generating functionals agree to all orders. The claim is motivated by agreement of the expansions checked in the paper, including se…

Fractional-framing conjecture. The localization generating functional and the topological vertex agree to all orders when the topological vertex is evaluated at the corresponding f…

Tree-decay conjecture. The quantity $\mathscr{T}_{t,\mathbf{x}}^{(n)}$ captures the correct spatial decay of the expectation of…

Higher-dimensional localization length conjecture. The localization length is of order $N$:

Higher-dimensional PRBM regime conjecture. This model exhibits three distinct regimes according as $\alpha<0$, $0<\alpha<d$, or $\alpha>d$.

PRBM localization length conjecture. The localization length satisfies