Mathematics Problem Archive

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...

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

Show that there is only a set of localizing subcategories. It is known that there is only a set of Bousfield classes (Ohkawa; Strickland simplified hi...

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, ...

What is the endomorphism ring of the identity functor on the stable homotopy category? The ring Z splits off this ring, including by multiples of the ...

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...

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

The safest sort of problem to work on with model categories is building one of interest in applications. The essential idea is: whenever someone uses ...

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...

Every stable homotopy category I know of comes from a model category. Well, that used to be true, but it is no longer. Given a flat Hopf algebroid, St...

Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the category of monoids in C is again a model categ...

The second step: show that the category of algebras over a cofibrant operad admits a model structure, where the fibrations and weak equivalences are t...

Find conditions under which algebras over a noncofibrant operad admit a model structure that generalize the monoid axiom of Schwede-Shipley. This woul...

Let A be a cofibrant operad as above. Use the above results to construct spectral sequences that converge to the homotopy groups of the space of A-alg...

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...

Is every monoidal model category Quillen equivalent to a simplicial monoidal model category? This would remove the loose end in my book on model categ...

Charles Rezk has a homotopy theory of homotopy theories. This is just a category, though it is large. The objects are generalizations of categories wh...

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...

Find a model category you can prove is not cofibrantly generated. This is just an annoyance, not a very significant problem, but it has been bugging m...

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...

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...

This one is due to Mike Hopkins. Generalize the whole Thom spectrum business as follows. Take an A-infinity ring spectrum E. Look at the space of A-in...

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?...