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 $F_{12}$?...

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....

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...

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

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 $N^n$....

Is this true without the bumpy assumption?...

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?...

Write (if exist) Cayley algebra systems defining the strata for the following graph: Currently this example is a strong candidate for a counterexample ...

Develop theory of geometric conditions for strata in multidimensional case....

Does there exist a closed C2 surface in Euclidean space R3 which is flexible?...

Are all smooth tight surfaces in R3 rigid?...

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

Given a metric of positive curvature on the disk what is the condition on a space curve to form the boundary of an isometric embedding of the disk?...

Are there some nonconvex surfaces which remain rigid after finitely many points of them have been deleted. For instance, are punctured analytic tight s...

(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...