Mathematics Problem Archive

Is there any algorithm for recognition whether two graph-links are equivalent or not? Our conjecture is “no”. The idea behind that is that graph-links...

Can one construct a projection from the set of graph-links to the set of realizable graph-links?...

Construct a parity on graph-links by using Bouchet's criterion about the realizability of a graph. Try to find a parity which is responsible for the c...

Construct generalizations of the Frobenius extension and the Rasmussen for “rigid” graph-links with orientable atoms....

Is it true that two equivalent realizable graph-links are equivalent in the class of realizable graph-links? If it is not true, then construct an exam...

Construct a group for graph-links which is analogous to the group from ....

Construct “graph-braids”....

Problems on Free Knot Cobordism: The methods used for proving the fact that the invariant $L$ gives an obstruction to the sliceness are not immediatel...

Is string topology sensitive to smooth structures on 4-manifolds?...

Present explicit obstructions to the Lyashko–Looijenga covering in terms of braid groups. Which braids cannot be lifted to the space ${\mathbb{C}}^{\m...

Let a non-simple singularity $f$ be given and the Dynkin diagram of it be defined by an easily distinguished system of paths connecting 0 with critica...

Are there more refined restrictions to the collision of critical values? Is it true that for any two vanishing cycles, whose intersection number is eq...

Give more general lower bounds of the dimension of $\mu=const$ strata in terms of intersection forms of vanishing cycles....

What are the obstructions to the realization of these chains of formal changes by paths in the parameter space ${\mathbb{R}}^{k}$?...

Is there any convenient topological characteristic of the function $f_{-\varepsilon}$ which allows to predict these indices?...

What is the minimal number of open sets $U_{i}$ covering ${\mathbb{R}}^{6}$ such that for any $U_{i}$ there is a continuous map $\varphi_{i}:U_{i}\to{...

The same questions concerning the approximate solutions. That is, for any $i$ and any $(a,b)\in U_{i}$, the value $\varphi_{i}(a,b)$ should be not nec...

Is the complement of the essential ramification set in ${\mathbb{R}}^{d}$ a $K(\pi,1)$-space?...

Are these geometric obstructions sufficient to solve the above problem?...

Is it true that any hypersurface from the space $P(d;N)$ can be connected with a trivial one by a generic path in this space in such a way that it exp...

A version of the previous problem, in which the complexity measure is not purely topological: namely, it is the lowest number of critical points of Mo...

Give an upper bound for the function $T\mapsto F$....

Do non-singular real plane projective curves of an odd degree consisting of a single connected component form a connected set (i.e., are they rigid is...

Is it true that any real Morsification of $f$ can be connected with one of complexity $\rho(f)$ by a generic path in the base of a versal deformation,...

What can be said about the number $\rho(f)$?...

Is it true that any component of the complement of the discriminant variety of a versal deformation contains a Morsification, whose all $\mu(f)$ criti...

Give a similar universal estimate of the radius of convergence for multidimensional Newton’s method of [Shub and Smale1993]....

Let $M$ be a $3$-manifold whose fundamental group acts on a simplicial tree without global fixed points. In the covering space associated to an edge s...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with only double curves of intersection?...

Does every closed hyperbolic $3$-manifold contain a closed $\pi_1$-injective surface with the 1-line property: in the universal cover, every pair of p...

Do cusped finite-volume hyperbolic $3$-manifolds have closed quasi-Fuchsian surfaces that cubulate except for the cusps? Equivalently in the stated ge...

Are finite-volume hyperbolic $3$-manifolds virtually semi-fibered?...

Let $M$ be a $3$-manifold, let $\phi:\pi_1M\to\mathbb{Z}$ be dual to $(\Sigma,\partial\Sigma)\subset(M,\partial M)$, and let $N$ be obtained by cuttin...

Characterize hyperbolic $3$-manifolds with infinitely generated fundamental group. In particular, is there a $3$-manifold that is locally hyperbolic, ...

Do there exist fibered hyperbolic $3$-manifolds that are homology $S^2\times S^1$ and have arbitrarily large injectivity radius? Are there hyperbolic ...

Characterize the Thurston norm polytopes of finite-volume hyperbolic $3$-manifolds....

Does every hyperbolic $3$-manifold have a finite-sheeted cover homeomorphic to a CAT(0) cube complex?...

Fix $\mu$ below the three-dimensional Margulis constant. For hyperbolic $3$-manifolds of volume less than $V$, drill all closed geodesics of length le...

If $M_1$ and $M_2$ are cusped hyperbolic $3$-manifolds, does there exist a cover $M'_1\to M_1$ and a nonzero-degree map $M'_1\to M_2$ taking cusps to ...

Build a combinatorial model for hyperbolic $3$-manifolds with explicit bilipschitz constants....

Let $M$ be a hyperbolic $3$-manifold and let $n$ be any positive integer. Does $M$ admit an ideal triangulation with $m\geq n$ positively oriented tet...

Let $M$ be a finite-volume hyperbolic $3$-manifold. If $M$ contains infinitely many immersed totally geodesic surfaces, must $M$ be arithmetic?...

Do Haken hyperbolic $3$-manifolds have strongly irreducible Heegaard splittings?...

For a hyperbolic $3$-manifold with torus boundary, does its profinite completion determine whether it is a knot complement?...

Is there a classification of the strongly irreducible Heegaard splittings of a given $3$-manifold?...

Do higher-dimensional smooth manifolds always admit multisections? What is the correct generalization of uniqueness up to stabilization for multisecti...

A band joining the components of a two-component link $L\subset S^3$ is called complicated if either its core cannot be isotoped to meet a splitting s...

Study the function Ψ: Mg→ [−log(µg),∞) as a (rational) Morse function. Classify its rational critical points. Deduce properties of the rational cohomo...

Determine the finiteness properties of Ig. For which k is Hk(Ig) finitely gen- erated? For which k is there a K(Ig, 1) with finite k-skeleton (one say...

Letγ1,···,γ 2g be the standard basis of Z2g. Study the function L = ∑ Lγi:Yg→ [0,∞) as a Morse function. Find critical sets and deduce properties of t...