  We survey recent progress in the proof complexity of strong proof systems and
its connection to algebraic circuit complexity, showing how the synergy between
the two gives rise to new approaches to fundamental open questions, solutions
to old problems, and new directions of research. In particular, we focus on
tight connections between proof complexity lower bounds (namely, lower bounds
on the size of proofs of certain tautologies), algebraic circuit lower bounds,
and the Polynomial Identity Testing problem from derandomization theory.
