AMR Open Problem Lists

The Kelvin problem on minimum-surface-area partitions of space into equal-volume cells, and the optimality of the Weaire–Phelan structure as a solution to the Kelvin problem

A superpermutation on $n$ symbols is a string containing every permutation of the $n$ symbols as a contiguous substring. Determine the minimum possible length of a superpermutation…

Let M be a matroid on ground set S, and suppose that S can be partitioned into k bases. Is it true that there is a cyclic ordering of the elements of S such that any $|S|/k$ consec…

Do there exist fibered hyperbolic $3$-manifolds that are homology $S^2\times S^1$ and have arbitrarily large injectivity radius? Are there hyperbolic homology spheres of arbitraril…

Let $f$ be real analytic near the origin, with $f(0,0)=0$, and let $n>1$. Suppose the origin is an isolated equilibrium of … Prove that the index of this vector field at the origin…

We consider the range of the random function … ($F$ chosen at random in the natural way) defined in $\mathbb{D}$, where $\sum|a_n|^2=\infty$. Is the image of $w=F(z)$ with probabil…

Is there a uniform bound on the Godbillon–Vey invariants of the taut foliations of a hyperbolic manifold in terms of its volume?

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

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 finitely many connected components?

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-infinity self equivalences of E.…

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 suspension spectrum of a spac…

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 book). Note that we have some 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 is sufficiently large? This q…

Determine the v1 -exponents for the spheres. Recall that Cohen, Moore, and Neisendorfer showed that the p-torsion in the homotopy of S^2n+1 is all killed by p^n, but not p^n-1, for…

The Johnson question. This says that if X is a space, and x is in BPn (X), then x is not vn torsion. My guess is that one should consider this question as a test case for whether 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 me for a while. The obvious can…

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 thing. Maybe we should be assoc…

Charles Rezk has a homotopy theory of homotopy theories. This is just a category, though it is large. The objects are generalizations of categories where composition is not associa…

Is every monoidal model category Quillen equivalent to a simplicial monoidal model category? This would remove the loose end in my book on model categories, where I am unable to sh…

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 without having all the definition…

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 categories are Quillen equivalences, a…

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-algebra structures on a given obj…

Find conditions under which algebras over a noncofibrant operad admit a model structure that generalize the monoid axiom of Schwede-Shipley. This would include the case where every…

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

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 category (with underlying fibration…