My favorite fields of math are abstract algebra, algebraic topology, graph theory, and computational complexity. The latter two are my current research fields. This may seem to contradict my claim of being a pure mathematician, but I think my natural approach to research is a pure mathematicianâs approach, and I have on many occasions jokingly lamented the fact that TCS is in the CS department, instead of in the math department where it belongs. (This joke is meant as a statement about my own preferences, not a claim about how the world should be.)
Some examples of specific topics Iâve found particularly fun to explain to people: the halting problem, P vs NP and the idea of poly-time reductions, Kempeâs false proof of the four-color theorem, the basics of group theory.
Iâm guessing youâve already made up your mind on this since itâs been a few months, but since you mentioned computational complexity being your research field you might be interested to know that Scott Aaronson was persuaded by Jan Leike to spend a year at OpenAI to
⊠think about the theoretical foundations of AI safety and alignment. What, if anything, can computational complexity contribute to a principled understanding of how to get an AI to do what we want and not do what we donât want?
(Scott admitted, like you, that he basically needed to be nerd-sniped into working on problems; âthis is very important so you must work on itâ doesnât work in practice.)
Quoting Scott a bit more (and adding bullets):
So, what projects will I actually work on at OpenAI? Yeah, Iâve been spending the past week trying to figure that out. I still donât know, but a few possibilities have emerged.
First, I might work out a general theory of sample complexity and so forth for learning in dangerous environmentsâi.e., learning where making the wrong query might kill you.
Second, I might work on explainability and interpretability for machine learning: given a deep network that produced a particular output, what do we even mean by an âexplanationâ for âwhyâ it produced that output? What can we say about the computational complexity of finding that explanation?
Third, I might work on the ability of weaker agents to verify the behavior of stronger ones. Of course, if Pâ NP, then the gap between the difficulty of solving a problem and the difficulty of recognizing a solution can sometimes be enormous. And indeed, even in empirical machine learing, thereâs typically a gap between the difficulty of generating objects (say, cat pictures) and the difficulty of discriminating between them and other objects, the latter being easier. But this gap typically isnât exponential, as is conjectured for NP-complete problems: itâs much smaller than that. And counterintuitively, we can then turn around and use the generators to improve the discriminators. How can we understand this abstractly? Are there model scenarios in complexity theory where we can prove that something similar happens? How far can we amplify the generator/âdiscriminator gapâfor example, by using interactive protocols, or debates between competing AIs?
That said, these mostly lean towards theory-builders, and you mentioned upthread being more problem-solver-oriented, so they probably arenât as interesting.
What are examples of things you find cool/âaesthetic/âelegant?
My favorite fields of math are abstract algebra, algebraic topology, graph theory, and computational complexity. The latter two are my current research fields. This may seem to contradict my claim of being a pure mathematician, but I think my natural approach to research is a pure mathematicianâs approach, and I have on many occasions jokingly lamented the fact that TCS is in the CS department, instead of in the math department where it belongs. (This joke is meant as a statement about my own preferences, not a claim about how the world should be.)
Some examples of specific topics Iâve found particularly fun to explain to people: the halting problem, P vs NP and the idea of poly-time reductions, Kempeâs false proof of the four-color theorem, the basics of group theory.
Iâm guessing youâve already made up your mind on this since itâs been a few months, but since you mentioned computational complexity being your research field you might be interested to know that Scott Aaronson was persuaded by Jan Leike to spend a year at OpenAI to
(Scott admitted, like you, that he basically needed to be nerd-sniped into working on problems; âthis is very important so you must work on itâ doesnât work in practice.)
Quoting Scott a bit more (and adding bullets):
That said, these mostly lean towards theory-builders, and you mentioned upthread being more problem-solver-oriented, so they probably arenât as interesting.