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 \(\infty\)-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 \(\mathbb{Z}\)-algebra structures on the smash product \(\mathbb{Z} \otimes R\) - coming from each factor. For which \(R\) are these two structures (abstractly) equivalent ? For \(R = \mathbb{Z},\mathbb{Q}\), they obviously are, while for \(R = \mathbb{F}_p\), they are not.
Let \(G, H\) be (finite ?) groups. Suppose the corresponding group rings \(\mathbb{Z}[G]\) and \(\mathbb{Z}[H]\) are isomorphic as rings. Does it follow that the corresponding spherical group rings \(\mathbb{S}[G]\) and \(\mathbb{S}[H]\) 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 \(D_X(x) = \mathrm{lim}_X ( \mathbb{S}[\Omega(X,x)]) \) ). 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 \(\infty\)-category, and \(f : X \to X\) a nilpotent endomorphism of some object of \(C\). Must the Hattori-Stallings trace of \(f\) be nilpotent in \(\mathrm{THH}(C)\) ? If not, must the symmetric monoidal trace of \(f\) be nilpotent in the endomorphisms of the unit ? Update: It turns out that the answer is no to both questions, as follows from my paper on free rigid commutative algebras. But what if \(C\) is \(\mathrm{Perf}(R)\) for some commutative ring spectrum \(R\) ?
Let \(R\) be a ring spectrum. There is a morphism of spectra \(R \to \mathrm{THH}(R)\), and a corresponding morphism of groups on \(\pi_0\). Is the latter surjective on the image of \(\pi_0 \mathrm{KEnd}(\mathrm{Perf}(R)) \to \pi_0 \mathrm{THH}(R)\), 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, \(R \to \mathrm{THH}(R)\) 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:
One can construct an action of \(B\mathrm{Pic}(\mathbb{S}) \rtimes \mathbb{Z}/2\) on the \(\infty\)-category \(\mathrm{Cat}_{\mathrm{st}}\) of stable \(\infty\)-categories. Is \(B\mathrm{Pic}(\mathbb{S}) \rtimes \mathbb{Z}/2\) the whole automorphism space of \(\mathrm{Cat}_{\mathrm{st}}\) ?