Below you will find a few potential directions for doctoral studies. I have tried to mention some recent papers of mine so you can get a sense of my work in the corresponding area.


1. Connections between computational complexity and learning theory.

Papers "Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness" and "Constructing hard functions from learning algorithms".


2. Logical foundations of complexity theory and independence results.

Check for instance "Consistency of circuit lower bounds with bounded theories".  


3. Boolean functions and circuit complexity theory.

You may have a look at "Hardness magnification near state-of-the-art lower bounds" and at this article https://simons.berkeley.edu/news/research-vignette-lower-bounds-computational-complexity


4. The complexity of estimating computational complexity and its applications.

Papers "Randomness and intractability in Kolmogorov complexity" and "NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD circuits".


5. Randomness and pseudorandomness in computation

Check the paper "Pseudodeterministic constructions in subexponential time".


This is only a partial list, and you can contact me to discuss further possibilities.