Mathematics Problem Archive

(The global isometric embedding problem, Yau [189] 1993; Gromov [82]). Can everyC∞ 2-dimensional Riemannian manifold be isometrically embedded in R4?...

For which setsA⊂ Sn is there an immersionf: M→ Rn+1 such that Gf(M)⊂A?...

LetM,M′⊂ R3 be smooth orientable closed surfaces. Suppose there exists a diffeomorphism f: M→ M′ which preserved the Gauss curvature and the Gauss map....

LetP, P′⊂ R3 be polyhedral surfaces. Suppose that the faces of P and P′ are parallel and have the same area. Does it follow then that P and P′ are con...

Is every convex polytope unfoldable?...

Does there exist a reasonably simple algorithm for detecting the edges of a convex polyhedron intrinsically?...

Does there exist a convex polyhedron with a pseudo edge graph which is not unfoldable....

Of all convex surfaces with a fixed intrinsic diameter, is the one with the greatest area a doubled disk?...

Let S ⊂ R3 be a closed surface of constant width and fixed area. How small can the volume of S be?...

LetS⊂ R3 be a closed surface of diameter d. Suppose that there exists a constant h < dso that whenever a pair of planes separated by a distance of h i...

Let $\Gamma$ be a closed curve of fixed length $L$ in $\mathbb{R}^3$. Determine the maximum possible volume of the convex hull of $\Gamma$....

Let $\Gamma$ be a closed curve of fixed length $L$ in $\mathbb{R}^3$, and let $A$ be the area of its convex hull. Prove that $A$ is maximized when $\G...

Is every compact connected minimal surface bounded by a pair of convex planar curves topologically an annulus?...

Are there any complete negatively curved surfaces embedded in the unit ball?...

Does there exist any complete negatively curved surfaces with negative Euler characteristic contained in between a pair of parallel planes in R3....

Let M be a complete noncompact convex surface in R3, with principal curvatures k1, k2, then show that inf M|k1−k2| = 0....

Bass conjecture: for every finitely generated $\mathbb Z$-algebra $A$ and every $n\geq0$, is the G-theory group $K'_n(A)$ finitely generated? Equivale...

Fröberg conjecture on the Hilbert functions of a set of forms....

Fujita conjecture regarding the line bundle $K_{M} \otimes L^{\otimes m}$ constructed from a positive holomorphic line bundle $L$ on a compact complex...

General elephant problem: do general elephants have at most Du Val singularities?...

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....

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

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?...

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 $\Pi^1_1\text{-}\mathsf{CA}_0$?...