Date of this Version
Published (2022) Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of his 75th Birthday, pp. 77-111.
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.