List of mathematical questions: On this page, you can find a list of questions I don't know the answer to. These are questions that sound fun, but are not part of my research, per se. It's far from exhaustive (the exhaustive list would be infinite), and for each of the questions, I do not know if it is an open problem or if someone knows the answer. If you know the answer or would just like to chat about one or some of them, please let me know !
About ring spectra :
Let A, B be two commutative ring spectra, and assume they are Morita equivalent, that is, their oo-categories of modules are equivalent (without a multiplicative structure). Does it follow that they are equivalent as associative ring spectra ? What if they are assumed to be connective ? Note that if we allow ourselves to work fully K(n)-locally, there are counterexamples.
Let R be an ordinary ring. There are two Z-algebra structures on the smash product Z smash R - coming from each factor. For which R are these two structures (abstractly) equivalent ? For R = Z,Q, they obviously are, while for R = F_p, they are not.
Let G, H be (finite ?) groups. Suppose the corresponding group rings Z[G] and Z[H] are isomorphic as rings. Does it follow that the corresponding spherical group rings S[G] and S[H] are equivalent as ring spectra ?
About spaces :
Let X be a connected space (homotopy type), and LX its free loop space. Can LX be finitely dominated without X being contractible ?
Let X be a space (homotopy type), and let D_X be its dualizing spectrum following Klein (so D_X is an X-parametrized spectrum, with D_X(x) = lim_X ( S[Omega(X,x)]) ). It is known that if X is compact and D_X is pointwise dualizable, then it is in fact pointwise invertible. Can D_X be pointwise dualizable and not invertible for non-compact X ?
About traces :
Let C be a small symmetric monoidal stable oo-category, and f : X -> X a nilpotent endomorphism of some object of C. Must the Hattori-Stallings trace of f be nilpotent in THH(C) ? If not, must the symmetric monoidal trace of f be nilpotent in the endomorphisms of the unit ? What if C is Perf(R) for some commutative ring spectrum R ? Or R is only E_n for some n bigger than 2 ?
Let R be a ring spectrum. There is a morphism of spectra R -> THH(R), and a corresponding morphism of groups on pi_0. Is the latter surjective on the image of pi_0 KEnd(Perf(R)) -> pi_0 THH(R), i.e. is the Hattori-Stallings trace of any endomorphism of some perfect R-module equal to the trace of some endomorphism of R ? If R is commutative, this endomorphism must be exactly the symmetric monoidal trace of the endomorphism. If R is connective, R -> THH(R) is simply surjective on pi_0. This question was solved in the negative by Logan Hyslop (see the file "Fun with traces")
Can one compute (unstable) THH of pi-finite spaces in the same way that one can compute unstable THH of finite sets ?
About the structure of categories :
Let S be the oo-category of spaces. If C is a compactly generated presentable oo-category, the canonical morphism Fun^L(C,S) otimes C -> Fun^L(C,C) is an equivalence (otimes is the Lurie tensor product). For an arbitrary presentable C, what can one say about this functor ? Is it fully faithful ? What is its essential image ? What if one works in a stable setting and replaces S with Sp ? The answer is no, stably. I will try to write down a detailed counterexample at some point.
Related to the previous question: let C,D,E be stable presentable oo-categories, and f : D -> E a fully faithful left adjoint. Is f otimes C : D otimes C -> E otimes C also fully faithful ? Variants of this are true: if we ask instead that f have a fully faithful right adjoint; or if we ask for f's right adjoint to preserve colimits. Variants of this are also false: without the word stable, Set and the inclusion Sp^con -> Sp provide a counterexample. The answer is no, stably. It is related to the previous point.
Two questions of the same kind: a- can there be a semi-additive presentable oo-category C whose additivization is already stable, without C being stable ? b- can there be a presentable oo-category whose semi-additivization is already additive, without C being additive ? Can these things be 0 without C being 0 ?
One can construct an action of BPic(S) semidirect Z/2 on the oo-category Cat_st of stable oo-categories. Is BPic(S) semidirecit Z/2 the whole automorphism space of Cat_st ?