What makes quantum computers fast?
No one really knows (and it's not just entanglement or superposition, as Gottesmann-Knill shows). It is something else. I think one aspect is quantum nonlocality. It's easiest to talk about nonlocality when the causal structure of the computation is fixed. For this reason, shallow circuits (i.e., circuits in NC$^0$) are relatively easy to study. It is know there is a separation between NC$^0$ and its quantum analogue QNC$^0$. I'm interested in why there is a gap, and if this gap is robust. Causal inference is helpful here.
Is quantum just Bell?
Superpositions, entanglement, no-cloning, etc. are often dogmatically declared the quintessentially quantum phenomena. No. There are many classical theories (supplemented by classical ignorance) that exhibit these features. At least what distinguishes classical from quantum are the broader quantum correlations that can be achieved in experiment. But are all the correlations "reducible" to Bell? With friends from PI, we are using tools from causal inference to answer this and similar questions.
Is gravity classical?
I've heard people say that Eppley and Hannah settled this decades ago. That's not right. Mattingly, in his gloriously titled paper "Why Eppley and Hanna isn't", decisively debunked their argument for why gravity is quantum mechanical. Currently, with friends at PI, I'm investigating if certain semiclassical theories imply polynomial time solutions to NP-complete problems. If so, then what a great reason to believe gravity is quantum!
Is Everett everything?
I think the Everett interpretation is right. But can you go "mad-dog Everettian"? That is, can everything, including you and me, emerge out of a wavefunction evolving in some Hilbert space? I'm intereseted in what mathematical structures can emerge out of mad-dog Everettianism. If not spacetime, then what is missing from the Everettian framework?
Are there any odd perfect numbers?
I don't know. I'm interested in using tools from complex analysis to address this question. In order to do this, we need a way to talk about divisors in a "continuous" way. I'm presently developing these techniques with Chai Karamchedu. It turns out this ancient problem is hard.
Is the universe isomorphic to a formal system?
Anyone who knows me knows my bible is Hofstadter's Gödel, Escher, Bach. It seems to me that theoretical physicists in search of "the ultimate theory" implicitly assume that the fundamental equations of the universe are mathematical in nature. If they are, then they comprise a formal system. That is, the structure of everything in the universe can be encoded into the structure between an alphabet of symbols. In this view, which I find quite poetic, you and I are theorems. But also in this view, the task of physics, besides finding the formal system of reality, is to find a model of the system. But that raises many questions, related to incompleteness, humans versus machines, etc. Also, what is the difference between "is" and "is isomorphic to"? This topic never fails to fascinate me.
Can artificially intelligent machines be moral?
I don't know, but I think so. I'm in the process of writing a paper that argues that this question (and many others related to it) is undecidable. Thus, the only way to judge the morality of an AI is as Turing suggested: the same way you judge if another human is moral—i.e., inductively.