Mathematics Problem Archive

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

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...

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...

Is it true that, given any pseudo-Anosov φ∈ Modg, there exists n =n(φ) such that the normal closure of φn is free? Gromov discovered the analogous phe...

(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...

(Density of pseudo-Anosovs). LetP denote the set of pseudo-Anosov ele- ments of Modg. Then d(P) = 1. J. Maher [ Mah] has recently proven that a random...

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...

Construct any representations of Mg, finite or infinite, which do not factor through Sp(2g, Z). In a very different direction, every mathematician woul...

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....

Determine the metric completion of the Gromov boundary of C(S) and relate this metric completion to the geometry of C(S). There is yet another way to ...

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...

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...

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...

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 ...

As a rule, I am not happy about the arbitrary nature of some of the constructions in the K(n)-local category. Consider the spectral sequence, for exam...

Find the shadow of the thick subcategory theorem in the K(n)-local category. There is only one thick subcategory of small spectra in the K(n)-local ca...

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...

Bousfield has give a description of the E(1)-local category in terms of algebraic data related to K-theory. Franke claims to have generalized all this...

Almost everyone who has ever thought about elliptic cohomology ends up thinking it has something to do with 2-categories. If you think about vector bu...

Dennis McLaughlin and Jean-Luc Brylinski also thought along these lines. They wanted to use gerbes, or 2-gerbes maybe, instead. I could never understa...

Yet another idea is to go back to a decription of cobordism I once heard. I think this description is in print somewhere, but I don't know where or wh...

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...

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...

Characterize the stable homotopy category up to equivalence. This has been done for categories that are homotopy categories of model categories by Sch...

In one of Bob Thomason's last papers, he determined the thick subcategories of finite objects in the derived category of a scheme. For the derived cat...

John Palmieri has determined the E_2 term of the Adams spectral sequence up to nilpotence--at least he has found a computable ring which is f-isomorph...

My general feeling about stable homotopy categories is that they are like commutative rings. Follow this up; define Spec C for example, for a stable h...

The equivariant stable homotopy category is not treated very well in our memoir. That is, we assume that the generators have to be dualizable. This is...

From an axiomatic point of view, I don't understand Grojnowski's equivariant elliptic cohomology. This theory takes values in an abelian category that...

Suppose G is a self-equivalence of the stable homotopy category. Must G be some iterate of the suspension functor? If G commutes with the suspension, ...

Figure out how to do equivariant stable homotopy theory without restriction on the group. Here you are going to have to change the current setup a lot...

As a simpler model of the equivariant stable homotopy category, construct a derived category of Mackey functors over a Green functor, and analyze its ...

A scheme is a generalization of a ring, in the same way that a manfold is a generalization of R^n. So maybe there is some kind of model structure on s...

My general theory is that the category of model categories is not itself a model category, but a 2-model category. Weak equivalences of model categori...

The 2-category of simplicial model categories is supposed to be (according to me) 2-Quillen equivalent to the 2-category of model categories. Even wit...

In the appendix to my book on model categories, I said maybe what we are doing in associating to a model category its homotopy category is the wrong t...

Determine the v_1 -exponents for the spheres. Recall that Cohen, Moore, and Neisendorfer showed that the p-torsion in the homotopy of S^2n+1 is all ki...

Suppose X is a simply connected finite complex. Do the Steenrod reduced powers P^t act trivially on the mod p cohomology of the loop space of X when p...

Build MU from the moduli stack of formal groups. This has got to be doable somehow, though it is an old problem (I first heard it in Ravenel's green b...

Classify all possible Bousfield classes of E-infinity ring spectra. I know very little about this problem. Note that the Spanier-Whitehead dual of the...