Lecture for a general audience: Terence Tao is UCLA's Collins Professor of Mathematics, and the first UCLA professor to win the prestigious Fields Medal. Less than a month after winning the Fields Medal, Tao was named a MacArthur Fellow. The following month, Tao was named one of "The Brilliant 10" scientists by Popular Science magazine, which called him "Math's Great Uniter" and said that "to Tao, the traditional boundaries between different mathematical fields don't seem to exist." His Colloquium is titled "Structure and Randomness in the Prime Numbers." The UCLA Science Faculty Research Colloquium Series is designed to promote interdisciplinary research. The Series is sponsored by the Divisions of Life and Physical Sciences, UCLA College. *Edit: For the question posed at [43:37], the word "Inters" should be "Integers"