Mathematics Problem Archive

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...

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...

What is the smallest n = n(g) such that MCG (Sg) admits a properly dis- continuous action on Rn? on a contractible n-manifold? (The answers are expect...

(Fast word problem). Is there a sub-quadratic time algorithm to solve the word problem in Modg? One might guess that n logn is possible here, as there...

(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...

f Γ is an irreducible arithmetic group of rank ≥ 2, then every homomorphism Γ→ ModS has finite image. For many arithmetic groups Γ the conjecture can ...

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...

[Ehrenpreis Conjecture] Given two closed Riemann surfaces, there are finite un- branched covers with homeomorphic total spaces which are arbitrarily c...

Study, systematically and with the help of computers, the finite quotients of Mg which do not factor through Sp (2g, Z). We remark that Problem 3.1 wo...

Is there a faithful finite dimensional matrix representation of Mg,b,n for any value of the triplet (g,b,n ) other than (1, 0, 0), (1, 1, 0), (1, 0, 1...

Find a candidate for a faithful finite-dimensional matrix representation of Mg orMg,1,0....

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 linear? A locally compact group Γ is said to satisfy the Haagerup approximation property or is a-T- menable if there exists...

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...

(Orbit closures for moduli spaces). Determine the closures of the orbits for the GL+(2, R)-action on H(α) andQ(β). Are these closures always complex-a...

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...

Let M = H3/Γ be a finite volume hyperbolic 3-manifold, does Γ contain a surface subgroup. This was answered in [ 8] for non-compact but finite volume ...

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...

Forn> 3, does Aut(Fn) have property (T)? The corresponding question for mapping class groups is also open. If Aut( Fn) were to have Property (T), then...

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...

Find some geometric meaning for elliptic cohomology. I believe this problem may be solvable--we keep learning new things about it. One thing I will sa...

On the same theme, find some way of doing index theory related to elliptic cohomology. This is not really algebraic topology, but would have a major i...

The chromatic splitting conjecture, which is considerably more complicated to state. Basically nothing is known about this, and so this one may be mor...

Classify all finite loop spaces. This is the long term project of Bill Dwyer and Clarence Wilkerson. The theory, I believe, is that the Lie groups are...

Say something general about the stable or unstable homotopy groups of spheres. For example, Ravenel has suggested that the size of the nth homotopy gr...

Does the possible Kervaire-invariant-one element $\theta_6\in\pi_{126}^{S}$ exist; equivalently, is $h_6^2$ a permanent cycle in the mod-2 Adams spect...

Once again, I am not sure whether this problem deserves to be called major, but it is annoying that the the R. Cohen - Goerss result proving that h_0 ...

Show that pi_* L_K(n) S^0 is finitely generated over the p-adics in each degree. This would follow from the chromatic splitting conjecture, I think. (...

Show that the Picard group is finitely generated over the p-adics. I don't think this is known even for the algebraic Picard group, which is obtained ...

We now know that Morava E-theory admits an action of the stabillizer group S. This is the famous Hopkins-Miller result, which one day I hope will see ...

Understand the relationship between the K(n)-local category and some sort of (algebraic) derived category of E_*-S-modules. Jens Franke has claimed th...

One of the corollaries of the Hopkins-Miller theorem, together with the Devinatz-Hopkins fixed point business, is that the famous class zeta in contin...

Introduce stable homotopy theory into the world of C^*-algebras, like Voevodsky has done in algebraic geometry. More specifically, find a model struct...

Investigate Voevodsky's stable homotopy category of schemes from a homotopy theorist's point of view. This is obviously a huge, unstructured problem, ...

Stefan Stolz showed that a simply connected Spin manifold of dimension at least 5 admits a metric of positive scalar curvature if and only if its imag...

Try to carry out Stolz's plan for metrics of positive Ricci curvature. Here we expect the obstruction to lie in elliptic cohomology rather than K-theo...

Improve on Benson-Carlson-Rickard. Recall their theorem: if G is a finite p-group and k is an algebraically closed field, then thick subcategories in ...

Classify the localizing subcategories of the stable k[G]-module category. These should be in 1-1 correspondence with arbitrary subsets of Proj H^*(G,k...

Extend the results of Benson-Carlson-Rickard to connected, cocommutative Hopf algebras over a field, like A(n). Hovey-Palmieri have achieved some part...

In our memoir, we give a conjecture for the thick subcategories in a Noetherian stable homotopy category C--they should be in 1-1 correpondence with s...