Razborov and Rudich's natural proofs barrier roughly says that it is computationally hard to certify that a uniformly random truth table has high circuit complexity. In this work, we show that the natural proofs barrier (specifically, Rudich's conjecture that there are no NP-constructive natural properties against $P/poly$) implies the following ... more >>>

One important question in algebraic complexity is understanding the complexity of polynomial ideals (Grochow, Bulletin of EATCS 131, 2020). Andrews and Forbes (STOC 2022) studied the determinantal ideals $I^{\det}_{n,m,r}$ generated by the $r\times r$ minors of $n\times m$ matrices. Over fields of characteristic zero or of sufficiently large characteristic, they ... more >>>

Strong lower bounds of the form $2^{(1-\epsilon)n}$, where $n$ is the number of variables and $\epsilon>0$ is arbitrarily small (i.e., bounds consistent with the Strong ETH), are exceptionally rare in proof complexity. The seminal work of Beck and Impagliazzo (STOC 2013) achieved such a bound for regular resolution, and the ... more >>>