We will begin by recalling what it means for a sequence of integers to be log-concave. While log-concavity has been widely studied in combinatorics, such sequences are also known to arise in algebraic geometry and algebra. In 2004, George Boros and Victor Moll introduced the concept of infinite log-concavity, and conjectured that the rows of Pascal's triangle are infinitely log-concave. We will show how a stronger version of log-concavity can be used to give a computer proof of their conjecture for the first 1450 rows. We will also discuss related questions for the columns of Pascal's triangle, q-analogues and, if time permits, symmetric functions and real-rooted polynomials. Including several easily-stated conjectures along the way, we hope to convince the audience that infinite log-concavity is a fundamental concept deserving of further attention.

This is joint work with Bruce Sagan.