what am i up to?

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# i’m giving a public lecture!

I’ll be speaking at Lectures on Tap about math research — what it looks like in practice and similarities to other creative endeavours. I’ve been reflecting on this for years now, and this will be a good excuse for me to organize and formalize my thoughts.

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# on my mind

It is crazy that we have made machines that can think. I feel caught between awe/amazement at this futuristic new technology, and concern for how it will negatively impact the world. Academia is lucky in that it is largely self-governing; I’ve been thinking a lot about what changes math-adjacent academics could take to engage with this technology productively and ethically.

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# phases of research

I’ve been thinking a lot about how research projects tend to fall into three consecutive phases.

1. Study/Exploration: You are just learning about the area, trying to develop intuition and learn techniques. This can take a long time, and sometimes you explore for a while and ultimately realize there is no path towards interesting results.
2. Discovery: Now you have a lot of intuition for the area and can prove things! This is a very exciting interval, but often short (or you get stuck here and it never ends, until you decide to give up).
3. Communication: You’ve basically determined your results but are trying to figure out how to communicate it — organizing/simplifying results, writing the paper, developing talks, etc. This phase always takes longer than I think, but I really enjoy it. It’s interesting to see how the act of communicating can help you refine the results themselves.

For my own sanity, it’s helpful to keep track of the states of my various projects:

Communication

• The BUD Walk
• Online algorithms for binary perceptron
• Public-key encryption from planted dense subgraph
• Algorithms for planted random subgraph
• Flexible Range-Searchable Encryption

Discovery

• 10x10 enumeration
• Impeaching “mayors” of universal covers (gave up for now)

Study/Exploration

• ReCom on dense random graphs
• The overlap gap property
• Average-case reductions between IS and LCIS
• Prophet inequalities with cancellation costs (on hold)
• AI and the Fourth Amendment (on hold)
• Actuarially fair insurance pricing (on hold)
• Distinguishing permutations from functions (on hold)
• Job scheduling lower bound (on hold)

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# big (TCS) questions

• What makes polynomial-time algorithms so hard to analyze? (My guess is that such algorithms are defined by a property that has nothing to do with how they find solutions.)
• Given that the low-degree conjecture is false, how can we build a good theory of average-case hardness?
• Is structure necessary for public-key cryptography?

Last updated September 2026.

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