Mathematics Problem Archive

For a hyperbolic knot $K$, the limit of normalized colored Jones polynomials equals the hyperbolic volume of the knot complement....

Apply Perelman's proof of the Poincaré conjecture to materials fabrication across scales....

Is every aspherical closed manifold whose fundamental group has no non-trivial perfect normal subgroups a $K(\pi, 1)$ space?...

Can unknots be recognized in polynomial time?...

Are Berge knots the only knots in S³ admitting lens space surgeries?...

Can unknots be recognized in polynomial time?...

Is every connected subcomplex of a 2-dimensional aspherical CW complex also aspherical?...

Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?...

Wild Virtuals: Create the category of “wild virtual knots” and establish its axiomatics. In particular, one needs a theorem that states when a wild eq...

(Mod g is Kahler). Forg≥ 3, the group Modg is a Kahler group, i.e. it is isomorphic to the fundamental group of a compact Kahler manifold. It was show...

Ig is finitely presented for g≥ 4. One thing we do know is that, in contrast to Mod g, neither Ig norKg has a classifying space which is homotopy equi...

Compute L(Modg) explicitly for small g≥ 2. In principle L(Modg) can be computed for any given g. The point is that one can first bound the degree of L...

Every subgroup of finite index in ModS contains a congruence subgroup. V. Voevodsky had indicated (in a personal communication) a beautiful applicatio...

s it true that any normal subgroup is commensurable with such a subgroup? Recall that two subgroups Γ 1, Γ2 of a group G are commensurable if the inte...

Does ModS has the Kazhdan property (T)? A positive answer would imply the positive answer to the previous question, but this problems seems to be much...

Find special features of the monodromy of algebraic surfaces. There is some good motivation for this coming from at least three directions • The probl...

Is the mapping class group a-T-menable? 14. Geometric properties of the mapping class group 245...

(Billiards in general polygons). Does every billiard table have at least one regular periodic trajectory? If the answer is affirmative, does this trajec...

Forg≥ 2, does Γg contain a purely pseudo-Anosov surface subgroup? The paper is organized as follows. In §2, we discuss the existence of surface subgro...

LetH be a convex cocompact subgroup of Γg. Is Γg H-separable? Just focusing on surface subgroups, we can ask:...

Does there exist a closed hyperbolic 4-manifold X that is the total space of a smooth fiber bundle Σg→X→ Σh? We will call such an X a surface bundle o...

Let M be a closed hyperbolic 4-manifold. Then all the Seiberg-Witten invariants of M vanish. The relevance of this is given in the following propositi...

Forg,h≥ 2, does there exist a short exact sequence: 1→π1(Σg)→ Γ→π1(Σh)→ 1 for which Γ is a word hyperbolic group? Arguing as in the proof of Theorem 4...

Do mapping class groups or Out(Fn) satisfy the Baum-Connes conjecture? Does Out(Fn) satisfy the Novikov conjecture? An approach to proving these conje...

Determine whether the Torelli group Ig (g≥ 3) is finitely presentable or not (note that Ig (g≥ 3) is known to be finitely generated by Johnson [42])....

The generating hypothesis, which asserts that the stable homotopy functor is faithful on the category of finite spectra. That is, if f is a map of fin...

The Johnson question. This says that if X is a space, and x is in BP_n (X), then x is not v_n torsion. My guess is that one should consider this quest...