243 problems

Dubrovin's universality conjecture. Near a generic gradient-catastrophe point, the dispersive solution should be modeled by this particular solution $\mathcal{D}(X;T)$.

A GOE matrix is a real symmetric random matrix with Gaussian entries up to the orthogonal symmetry constraint; more generally, Hermitian random matrices belong to the unitary symme…

A Dirichlet series is a series of the form $\sum_{n\geq1}a_n n^{-s}$, viewed as a function of the complex variable $s$. Linnik–Ibragimov's conjecture. Any reasonable Dirichlet seri…

Consider non-Hermitian random matrices with independent and identically distributed entries. Their microscopic eigenvalue statistics in the bulk and at the soft edge exhibit the sa…

A random matrix ensemble is classified by its symmetry class: real symmetric, complex Hermitian, or quaternion. Local eigenvalue statistics, including eigenvalue gaps and other sta…

Let $\alpha$ be an algebraic irrational parameter, and let the Hurwitz zeta function be $\zeta(s,\alpha)$. Gonek's universality conjecture. Universality also holds for the Hurwitz…

For non-Hermitian random matrices, consider local bulk statistics of complex eigenvalues away from spectral edges, symmetry axes, and symmetry points. A universality class is a dis…

Consider the high-temperature Ising model and critical Bernoulli percolation as two planar lattice models. High-temperature Ising–percolation universality conjecture. The universal…

One motivation for studying martingale problems for singular SPDEs is that they provide a tool for deriving equations as scaling limits. Weak KPZ universality conjecture. A wide ra…

Renormalization is a microscope describing the dynamics of a system on a smaller scale, and classical systems such as unimodal maps and critical circle maps exhibit rigidity and pa…

Edelman's universality conjecture. The same limiting distribution should hold for other entry distributions, such as Bernoulli entries; in particular, for fixed $\varepsilon>0$,

Universality conjecture. The displayed formula, originally obtained for products of Gaussian matrices, continues to hold for the wider class of matrices with independent entries fu…

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles, and let $G_{\gamma_1,\gamma_2}$ and $H_{\gamma_1,\gamma_2}$ denote the associated maps between them. Let…

Let $g$, $h$, and $h^{-1}$ be the conjugacies associated with a critical invariant circle, and let $\kappa(\cdot)$ denote their regularity. A numerical characteristic is universal…

Let $R$ denote a critical invariant circle, and let $\kappa(R)$ denote its regularity. A numerical characteristic is universal when it takes the same value on an open set of functi…

Consider the renormalization operator acting on the class of twist maps under discussion, and call a property universal when it holds on an open set of functions. A fixed point is…

Let $\mu$ be the unique normalized invariant measure of a cubic critical homeomorphism of the circle, and let $HD(\mu)$ denote the Hausdorff dimension of its measure-theoretical su…

Universality conjecture. When $\nu>2$, the parameter scalings $\delta_n$ tend to a universal limit depending only on $\nu$.

Let $F_{n,2}^{G,L}(t)$ denote the largest-eigenvalue distribution function for the Gaussian unitary ensemble or Laguerre unitary ensemble under the scaling in the preceding expansi…

Let $A_{n,k}$ be an $n\times k$ random matrix with independent and identically distributed entries, and let … Here $Y_{n,k}$ is a sample covariance-type matrix, and its coefficient…

Consider discrete orthogonal polynomial ensembles whose equilibrium measure has a band edge, and let the associated boundary-curve fluctuations be studied in the scaling limit near…

Universality conjecture. Such scaling limits should be universal for a wide class of Hermitian random matrices. This predicts that the same limiting random point fields, including…

Mehta's universality conjecture. The local statistics of the eigenvalues of $A$ are, in the limit $n\to\infty$, determined only by the overall symmetries of the ensemble and are in…

Let $p^{(n)}_{l,\beta}(\lambda_1,\dots,\lambda_l)$ be the $l$-point marginal density obtained by integrating the joint eigenvalue density of an orthogonal ensemble with potential…

Let $v_0$, $(x_0,t_0)$, $a_0=a(v_0)$, $a_0'=a'(v_0)$, $c_0=c(v_0)$, and $\kappa=-(a”'(v_0)t_0+b”'(v_0))$ be the quantities associated with a generic solution, where $kappa eq0$ and…