Abstract
In this talk, we will sketch the history of algebraic proof systems, such as Nullstellensatz proofs, polynomial calculus, positivestellensatz, sum of squares, and the Ideal Proof system. I will divulge some of the original motivations of these proof systems and why they failed to do what they were intended to do (so far). For failures, these proof systems have been amazingly successful, so I will also sketch what they actually have been used for, by Toni Pitassi and others. This ranges from lower bounds on algorithms such as the Groebner basis algorithm, to understanding subclasses of TFNP to lower bounds on pseudo-deterministic algorithms.