Mathematics Problem Archive

1. Does ${\rm Sing}_b (T)$ have zero $(m-1)$-dimensional Hausdorff measure? 2. If yes, does ${\rm Sing}_b (T)$ have (Hausdorff) dimension at most $m-2...

Which manifolds can be modeled on an orbit of a real form in a space of flags?...

What is the homotopy classification of totally real immersions of real $3$-manifolds in the complex full flag manifold $F_{12}$?...

For ${2n+1}>3$ and $0<k<{2n+1}$, are projective subspaces obtained by projection of the $k$-dimensional equatorial spheres minimal submanifolds of the...

Assume that a left-invariant Riemannian metric is given on a compact connected Lie group $G$ such that the index of any geodesic segment is even. Must...

Let $M$ be a connected closed manifold with finite fundamental group of odd order. Assume that the universal cover of $M$ admits a metric of positive ...

Let $M$ be a connected closed manifold admitting a metric of positive scalar curvature. Does this imply that $M$ is $p$-atoral for all odd $p$?...

Find more numerical invariants of metric spaces that are nondecreasing under coarse embeddings....

Find applications of the harmonic map approximation of coarse embeddings....

For solvable Lie groups $G$, show that solvsolitons are the only local maxima of the Ricci pinching functional $g\mapsto F(g)=\frac{Scal(g)^2}{|Ric(g)...

Explain the existence and the role of the symplectic nature of the groupoid/algebroid and its relevance for the geometry of the moduli spaces of geome...

; Do there exist embedded minimal surfaces with finite genus and Morse index $4$?; More focussed: in the $1$-parameter deformation of Costa's surface,...

- Construct congruence lattices which are norm$_1$ and norm$_1*$- isospectral in all dimensions (see ). - Are there families of $p$-isospectral lens s...

Prove that geodesic spheres provide the least-perimeter way to enclose prescribed volume in $CP^2$....

Prove that the pictured standard triple soap bubble is the least-perimeter way to enclose and separate three given volumes in $\mathbb{R}^3$....

1. If the normal holonomy group of an irreducible and full homogeneous submanifold $M^n$ of the sphere with $n \geq 2$ does not act transitively, then...

; Do CMC spheres about a point $x$ in such a space form a foliation of $X-\{x\}$?; Could this be a way of proving embeddedness of CMC spheres in gener...

; Calabi-Yau problem. For an embedded minimal surface in $\mathbb{R}^3$, does complete imply proper?; Hoffman-Meeks conjecture. For a complete embedde...

Is every Laplacian algebra of polynomials maximal?...

Prove the lower bound \[ d\le \mbox{Index}(\Phi_{\mathcal A})+\mbox{Null}(\Phi_{\mathcal A}), \] where $\mbox{Null}(\Phi_{\mathcal A})$ is the {\it nu...

Prove that there exists infinitely many distinct minimal branched 2-dimensional immersions in $N^n$....

Does the normalized Ricci flow converge in Gromov-Hausdorff sense to a generalized K\"ahler-Einstein space?...

Does Frankel's theorem hold for symmetric Finsler metrics?...

Is this true without the bumpy assumption?...

Describe the combinatorics of B2(K6); B3(K4) and B3(K5)....

Describe all the possible different types of strata for 10 points....

Compute the number of different types of strata for n points with arbitrary n. 4 OLEG KARPENKOV v1 v2 v3v4 v5 v6 K3;3 q1 q2 q3 p1 p2 p3 p4 p5 p6 q3 q2 ...

Which subgraphs of Kn define the same stratifications?...

Find all strata of codimension more than 1 that are not defined as an intersection of the closure of several codimension 1 strata....

Which Cayley algebra systems define the same strata?...

Are negatively curved annuli bounded by a pair of fixed convex planar curves rigid?...

Let Γ be a smooth closed curve immersed in R3. Suppose that Γ has a continuous binormal vector field B which is one-to-one. Does it follow then that th...

Does every curve bounding a surface of positive curvature in 3-space have (at least) four points where the torsion vanishes?...

Does connectedness of the shadows imply that f(M) is convex?...

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?...

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...

What is the shortest curve in R3 with a given width or inradius?...

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...

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...

Bass–Quillen conjecture: if $A$ is a regular Noetherian ring, is every finitely generated projective module over $A[t_1,\ldots,t_n]$ extended from a p...

Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex....

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?...

In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?...