Mathematics Problem Archive

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...

Currently we know how to do equivariant stable homotopy theory only when the structure group G is compact Lie. But I bet we can do it when the group G...

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...

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...

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...

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...

Is every $n$-dimensional homogeneous absolute neighborhood retract a topological manifold?...

For every fibration $F\to E\to B$ of simply connected spaces whose fiber $F$ is rationally elliptic with nonzero Euler characteristic, does the ration...

For every algebraic variety $X$ defined over $\mathbb{Q}$, does the closure of $X(\mathbb{Q})$ inside the real locus $X(\mathbb{R})$ have only finitel...

Does every wild knot have infinitely many quadrisecants, that is, lines meeting the knot in at least four distinct points?...

Let $M$ be a closed manifold. Is every closed exact Lagrangian submanifold of the cotangent bundle $T^*M$ Hamiltonian isotopic to the zero section?...