Mathematics Problem Archive

Determine the abelianization of the group Hg,1. Is it trivial? Also determine the second homology group H2(Hg,1; Z). Is the rank of it equal to 1 give...

Generalize the infinitesimal presentation of the Torelli Lie algebra given by Hain [29] to the case of the group of homology cobordism classes of homo...

Prove that the above characteristic classes induce surjective homomorphism H3(BDiffδ +Σg; Z)−→R2 for any g. The cohomology classes in (22) are stable w...

Study whether the homology groups of BDiffδ +Σg stabilize with respect to g or not. The same problem for the group SympδΣg. 374 S. Morita Acknowledgmen...

Which properties of the braid groups can be extended to the mapping class groups?...

The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory should go, analogous to the Weil conjectures in alge...

Elucidate the connection between the Morava stabilizer groups and the K(n)-local category. The first such problem, which is certainly not very hard an...

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...

Presumably one should be able to form a category of E-S module spectra; spectra with an action of the ring spectrum E and a compatible action of the g...

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...

Neil Strickland points out that several different moduli spaces are used in differential geometry and physics. For example, there is the moduli space ...

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 ...

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...

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...

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...