Mathematics Problem Archive

Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory...

Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic curve to pass through a collection of very general point...

Nagata–Biran conjecture that if $X$ is a smooth algebraic surface and $L$ is an ample line bundle on $X$ of degree $d$, then for sufficiently large $r...

Nakai conjecture: if a complex algebraic variety has a ring of differential operators generated by its contained derivations, then it must be smooth....

Parshin's conjecture: the higher algebraic K-groups of any smooth projective variety defined over a finite field must vanish up to torsion....

Section conjecture on splittings of group homomorphisms from fundamental groups of complete smooth curves over finitely-generated fields $k$ to the Ga...

Virasoro conjecture: a certain generating function encoding the Gromov–Witten invariants of a smooth projective variety is fixed by an action of half ...

Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points...

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

Prove resolution of singularities for algebraic varieties over fields of positive characteristic in arbitrary dimension....

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...

Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets...

Square packing in a square: what is the asymptotic growth rate of wasted space?...

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?...

Tripod packing: how many tripods can have their apexes packed into a given cube?...

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?...

Dissection into orthoschemes – is it possible for simplices of every dimension?...

The values of the Hermite constants for dimensions other than 1–8 and 24...

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

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?...

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

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

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

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?...

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?...

Does every convex polyhedron have Rupert's property?...

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

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

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

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

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

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 $\mathsf{CS}$....

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 Lebesgue differentiation theorem imply $\mathsf{WWKL}_0$?...

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