Mathematics Problem Archive

The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solutio...

Lebesgue's universal covering problem on the minimum-area convex shape in the plane that can cover any shape of diameter one...

Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?...

Does every higher-dimensional tiling by translations of convex polytope tiles have an affine transformation taking it to a Voronoi diagram?...

What constant $1.1 < a \leq 10.76$ governs the lower bound of a closed knot $K$'s minimum ropelength $L(K) \geq a\operatorname{Cr}(K)^{3/4}$?...

Is the upper bound of a closed knot's minimum ropelength linear to its crossing number?...

The Thomson problem – what is the minimum energy configuration of $n$ mutually-repelling particles on a unit sphere?...

Determine the automorphism group of the partial order of Turing degrees....

Classify reasonable increasing functions on the Turing degrees; Martin's conjecture predicts that they are essentially iterates of the Turing jump....

Over $\mathsf{RCA}_0$, is $\mathsf{IPT}^2_2$ equivalent to $\mathsf{RT}^2_2$?...

Over $\mathsf{RCA}_0$, is Hindman's theorem equivalent to $\mathsf{ACA}^+_0$, equivalent to $\mathsf{ACA}_0$, or strictly between them?...

Determine the reverse-mathematical strength of the dual Ramsey theorem $\mathsf{DRT}^k$....

Determine the reverse-mathematical strength of the Carlson–Simpson infinite-variable-word lemma $\mathsf{CS}$....

Over $\mathsf{RCA}_0$, does the Lebesgue differentiation theorem imply $\mathsf{WWKL}_0$?...

Over $\mathsf{RCA}_0$, is the Auslander–Ellis theorem equivalent to $\mathsf{ACA}_0$?...

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 $\mathsf{ATR}_0$?...

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

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

The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\aleph_1$-saturated models of a countable theory....

Shelah's categoricity conjecture for $L_{\omega_1,\omega}$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals...

Shelah's eventual categoricity conjecture: For every cardinal $\lambda$ there exists a cardinal $\mu(\lambda)$ such that if an AEC K with LS(K)${} \le...

Does every simple first-order theory have stable forking?...

The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under ...

Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality $\aleph_{\omega_1}$ d...

If the class of atomic models of a complete first order theory is categorical in the $\aleph_n$, is it categorical in every cardinal?...

Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?...

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...

Find an asymptotic expression for the splitting of the separatrix of a quasiperiodically forced pendulum in the regime where the perturbation series i...

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

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...

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)...

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...

Random point processes corresponding to $\beta$-ensembles, or, equivalently, log gases at inverse temperature $\beta$, are defined for arbitrary $\bet...

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...

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...

Consider the Korteweg--de Vries (KdV) equation u_t + u u_x + u_{xxx} =0 with initial data u(x, t=0) = u_0(x),\qquad x\in \mathbb{R}. In the 1970's, Mc...