Amichai Lampert, from the University of Michigan, presents a joint IAS/Princeton number theory seminar on bounding the variables needed in Birch's theorem. Birch proved in 1957 that a collection of odd-degree forms with rational coefficients must have a nontrivial rational zero once the number of variables is large enough, though his original bounds were, in his own words, not even astronomical. Lampert reviews that argument, discusses a 1982 improvement by Schmidt that applied only to degree three, and then sketches his own result, joint work with Andrew Snowden and Tamar Ziegler, showing that for any fixed odd degree the number of variables can be taken polynomial in the number of equations. The talk runs through the technical machinery behind the proof for an audience of working number theorists.