Mathematics Problem Archive

Are infinite sequences of flips possible in dimensions greater than 3?...

The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint ...

The Erdős–Oler conjecture: when $n$ is a triangular number, packing $n-1$ circles in an equilateral triangle requires a triangle of the same size as p...

The disk covering problem about finding the smallest real number $r(n)$ such that $n$ disks of radius $r(n)$ can be arranged in such a way as to cover...

The Kobon triangle problem on triangles in line arrangements...

The Kusner conjecture: at most $2d$ points can be equidistant in $L^1$ spaces...

The McMullen problem on projectively transforming sets of points into convex position...

Opaque forest problem on finding opaque sets for various planar shapes...

For each arrangement of points in which the rectilinear crossing number is minimized, is the number of halving lines maximized?...

The Atiyah conjecture on configurations on the invertibility of a certain $n$-by-$n$ matrix depending on $n$ points in $\mathbb{R}^{3}$...

Connelly’s blooming conjecture: Does every net of a convex polyhedron have a blooming?...

What is the lowest number of faces possible for a holyhedron?...

Can every spherical non-convex polyhedron that tiles space by translation have its faces grouped into patches with the same combinatorial structure as...

Is there a general expression for the minimum ropelength of an arbitrary closed knot?...

Is there a general expression for how much the ends of a long rope of radius 1 get closer when a tight open knot is tied into it?...

Is there a non-convex polyhedron without self-intersections with more than seven faces, all of which share an edge with each other?...

Babai's problem: which groups are Babai invariant groups?...

Is the complement of the finite spectrum of every first-order sentence also a finite spectrum? Equivalently, is $\mathrm{NE}=\mathrm{coNE}$?...

Does there exist a reasonable logic strictly stronger than first-order logic that has both compactness and Craig's interpolation property?...

Prove a superpolynomial lower bound on the size of Frege proofs; in particular, do some tautologies require exponentially large Frege proofs?...

For any finite sets $X$ and $Y$ of published mathematical theorems expressible in second-order arithmetic, is either $\mathsf{RCA}_0+X$ interpretable ...

Are the following statements equivalent to $\Pi^1_1\text{-}\mathsf{CA}_0$? (i) If countable torsion abelian groups $G,H$ satisfy $G\oplus G\cong H\opl...

Determine the reverse-mathematical strength of Alexandroff's one-point compactification theorem for countably based MF spaces....

Determine the reverse-mathematical strength of the assertion that a proper MF space is metrizable if and only if it is regular....

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

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 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 - ...