Mathematics Problem Archive

Over $\mathsf{RCA}_0$, does the Furstenberg–Zimmer structure theorem imply $\Pi^1_1\text{-}\mathsf{CA}_0$?...

Over $\mathsf{RCA}_0$, is $\mathsf{EXT}(\omega^*)$ — the assertion that every well-founded partial order has a well-ordered linearization — equivalent...

Over $\mathsf{RCA}_0$, is Fraïssé's conjecture for countable linear orders equivalent to $\mathsf{ATR}_0$?...

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 there a subsystem weaker than $\mathsf{ATR}_0$ that proves that the three-element antichain is a better-quasi-order?...

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....

Over $\mathsf{RCA}^*_0$, is Ramsey's theorem for triples equivalent to $\mathsf{ACA}_0$, as it is over $\mathsf{RCA}_0$?...

Do spin correlations in the classical Heisenberg model, and other $O(N)$ models with $N>2$, decay exponentially at every nonzero temperature, or do th...

(A) Prove long-range order for the quantum Heisenberg ferromagnet in dimension $D>2$ at temperature $T>0$. (B) Prove long-range order for spin $1/2$ i...

Establish, in some energy range, the existence of extended eigenstates or continuous spectrum for linear operators with extensive disorder, such as a ...

Determine the large-distance behavior of the one-particle reduced density matrix for a one-dimensional Fermi gas with spin and attractive interaction....

Give a fundamental and experimentally accessible definition of the entropy creation rate for general classical systems in stationary nonequilibrium st...

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...

Give a mathematically precise meaning to determining the energy levels of the helium atom exactly, in the sense that hydrogen energy levels are exact,...

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 ...

Let $V$ be a function on $\mathbb R^\nu$, $\nu\geq2$, satisfying $\int |x|^{-\nu+1}|V(x)|^2\,d^\nu x<\infty$. Prove that $-\Delta+V$ has absolutely co...

For the $N$-electron Coulomb Hamiltonian with nuclear charge $Z$, let $N_0(Z)$ be the least $N$ after which adding electrons no longer lowers the grou...

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 justification of the techniques used to determine molecular configurations from first principles....

Prove that the integrated density of states $k(E)$ is continuous as a function of the energy $E$....

For spatial dimension $\nu=1$ and $1/2<\gamma<3/2$, determine the optimal constants $L_{\gamma,1}$ in the Lieb-Thirring inequality as predicted by the...

In many situations one is concerned with the asymptotic behavior of Riemann-Hilbert problems with exponentially varying data of the form $e^{in\phi(z)...

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 ...

Random matrix theory was introduced into theoretical physics by Wigner in the 1950s in his study of neutron scattering resonances, but as a subject, R...

Much is now known, theoretically and also numerically, about the spectrum of self-adjoint operators. This is in great contrast to the situation regard...

The long-time behavior of the solution of the Cauchy problem for a great many integrable systems on the line is now well-understood, using, for exampl...

A number of so-called ``transportation'' problems have now been analyzed in terms of RMT. These include: the ``vicious'' walker problem of M. Fisher, ...

Let $J$ be a random tridiagonal matrix drawn from the tridiagonal Gaussian orthogonal ensemble and evolve it by the finite nonperiodic Toda flow, so i...

The bijective mapping properties of the scattering transform for a variety of integrable systems on the line between appropriate weighted Sobolev spac...

The asymptotic behavior of a variety of combinatorial problems has now been analyzed in great detail. Here we have in mind Ulam's problem for the leng...

In the late 1990s Fokas introduced a new, more flexible approach to inverse scattering theory, and in recent years a number of researchers (Fokas, Its...

There has been considerable progress (Kuijlaars,...) in understanding basic statistics such as the correlation functions for the 2-matrix random matri...

Standard algorithms to compute the eigenvalues of a random matrix $H$ are completely integrable Hamiltonian systems (see ). The question raised in was...

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

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 one decide in deterministic polynomial time whether an integer $n$ is squarefree?...

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