Mathematics Problem Archive

For an ordinal $\alpha$, determine the length (maximal order type) of the well-quasi-order $L_\alpha$ of countable linear orders of Hausdorff rank bel...

Determine the reverse-mathematical strengths of Laver's labeled-linear-order theorem $\mathsf{LAV}$ and the Nash–Williams bqo transfinite-sequence the...

Is weak-$L_{\omega_1,\omega}$-$\mathsf{CA}$ equivalent to weak-$\Sigma^1_1$-$\mathsf{AC}_0$?...

Is the open mapping theorem for separable Banach spaces provable in $\mathsf{RCA}_0$, or at least in $\mathsf{WKL}_0$?...

Determine the exact reverse-mathematical strength of the Krein–Šmulian theorem for separable Banach spaces....

Is Szemerédi's theorem provable in $\mathsf{ACA}_0$? More generally, determine its reverse-mathematical strength....

Determine the reverse-mathematical strength of Kříž's labeled-tree generalization of Kruskal's theorem....

The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?...

Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?...

Is the theory of the field of Laurent series over $\mathbb{Z}_p$ decidable? of the field of polynomials over $\mathbb{C}$?...

Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?...

What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) for second-order arithmetic, ZFC, or stron...

Give a mathematically precise meaning to an exact formula comparable to Onsager's formula for the two-dimensional Ising model, and prove or disprove t...

For the Edwards-Anderson Ising spin glass on $\mathbb{Z}^d$ with i.i.d. mean-zero finite-variance nearest-neighbor couplings, prove or disprove the ex...

For the two-dimensional square-lattice model of independent electrons at density $1/4$, does magnetic flux $\pi/2$ per plaquette minimize the ground-s...

Prove that Bose-Einstein condensation occurs in a continuum model of a weakly interacting Bose gas, or determine whether the long-held assertion fails...

Prove that the Anderson model has purely absolutely continuous spectrum in dimension $\nu\geq 3$, for suitable disorder width $b-a$, in some energy ra...

Prove that the spectrum of the Anderson model in dimension $\nu=2$ is dense pure point....

For the Anderson model in dimension $\nu\geq3$ and disorder strengths $|b-a|$ admitting absolutely continuous spectrum, prove that $\sum_{n\in\mathbb ...

For the $N$-electron Coulomb ground-state energy $E(N,Z)$, determine the asymptotics of the ionization energy $\delta E(Z)=E(Z,Z-1)-E(Z,Z)$ as $Z\to\i...

Give a mathematically rigorous formulation and justification of the nuclear shell model....

Prove that, as the number of nuclei tends to infinity, the ground state of some neutral system of nuclei and electrons approaches a periodic limit, es...

What I have in mind here is not a specific problem, but a project, a very large scale project. The six (nonlinear) Painlev\'e equations form the core ...

The fact that RMT, and the Tracy-Widom distributions, arise in so many problems in so many different areas leads one to the following question: how ca...

In 2002, P. Zhou analyzed the behavior of solutions of the Cauchy problem for perturbations of the defocusing NLS equation i u_t + u_{xx}- 2|u|^2 u - ...

Give a rigorous explicit solution of the fixed-frequency scalar-wave diffraction problem for two finite slits in the plane, including asymptotics of t...

Construct a set of at least four mutually unbiased bases in dimension six, or prove that there are no seven mutually unbiased bases in $\mathcal{H}_6$...

Determine whether there exist bound entangled bipartite quantum states with negative partial transpose....

Given $n\in\mathbb{N}$, can a prime $p>n$ be found in deterministic polynomial time?...

Given coprime $a,n\in\mathbb{N}$, can a prime $p\equiv a\pmod n$ be found in deterministic polynomial time?...

Is complete integer factorization $C_5$ in deterministic polynomial time $P$?...

Is complete integer factorization $C_5$ in randomized polynomial time $R$?...

Does there exist a set $S\subset\mathbb{N}$ of positive lower asymptotic density for which complete factorization of every input $n\in S$ is in determ...

Given $n$, can one find $r,s\in\mathbb{N}$ with $n=r^2s$ and $s$ squarefree in deterministic polynomial time?...

Is complete integer factorization randomized polynomial-time reducible to computation of the squarefree part?...

Can $\omega(n)$, the number of distinct prime factors of $n$, be computed in deterministic polynomial time?...

Can one decide in deterministic polynomial time whether a coprime integer $a$ is a square modulo a composite $n$?...

Is complete integer factorization randomized polynomial-time reducible to deciding quadratic residuosity modulo a composite?...

Given a prime $p$, can a quadratic nonresidue modulo $p$ be found in deterministic polynomial time?...

Given a sign vector $\varepsilon\in\{-1,1\}^k$, can the least prime $p$ satisfying $(p_i/p)=\varepsilon_i$ for every $i\le k$ be found in deterministi...

Given a prime $p$ and a quadratic residue $a$, can a square root $x^2\equiv a\pmod p$ be found in deterministic polynomial time?...

Given a prime $p$ and $f\in(\mathbb{Z}/p\mathbb{Z})[x]$ known to have a root, can a root be found in deterministic polynomial time?...

Given a prime $p$ and $f\in(\mathbb{Z}/p\mathbb{Z})[x]$, can the complete irreducible factorization of $f$ be found in deterministic polynomial time?...

Given a prime $p$ and $b$, can one decide in deterministic polynomial time whether $b$ generates $(\mathbb{Z}/p\mathbb{Z})^*$?...

Is recognition of primitive roots modulo a prime in randomized polynomial time $R$?...

Given a prime $p$, can a generator of $(\mathbb{Z}/p\mathbb{Z})^*$ be found in deterministic polynomial time?...

Given a prime $p$ and $a$ coprime to $p$, can $\operatorname{ord}_p(a)$ be computed in deterministic polynomial time?...

Given a prime $p$ and elements $g,b$ with $b$ in the subgroup generated by $g$, can an exponent $x$ satisfying $g^x\equiv b\pmod p$ be found in determ...

Given $g,b,n$ such that $g^x\equiv b\pmod n$ has a solution, can such an exponent $x$ be found in deterministic polynomial time?...

Given $k,m,n$ with odd $n$ and $\gcd(km,n)=1$, can integers $x,y$ satisfying $x^2-ky^2\equiv m\pmod n$ be found in deterministic polynomial time?...